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Theta function

In mathematics, theta functions are a class of special functions of one or several complex variables that arise prominently in , , , and , serving as building blocks for elliptic functions, modular forms, and solutions to certain partial differential equations. Introduced systematically by in his 1829 work Fundamenta nova theoriae functionum ellipticarum, these functions generalize trigonometric series and exhibit quasi-periodic behavior, while extended them in 1857 to multivariable forms in his memoir on Abelian functions, defining the Riemann theta function as a holomorphic function on \mathbb{C}^g \times \mathcal{H}_g. The classical Jacobi theta functions consist of four interrelated variants, \vartheta_1(z|\tau), \vartheta_2(z|\tau), \vartheta_3(z|\tau), and \vartheta_4(z|\tau), defined for z \in \mathbb{C} and \tau in the upper half-plane \mathcal{H} as infinite series such as \vartheta_3(z|\tau) = \sum_{n=-\infty}^\infty q^{n^2} e^{2\pi i n z} where q = e^{\pi i \tau}, converging absolutely due to the imaginary part of \tau > 0. These functions satisfy transformation laws under the , including the Jacobi inversion formula \vartheta_3(z|-\frac{1}{\tau}) = \sqrt{\frac{\tau}{i}} e^{\pi i z^2 \tau} \vartheta_3(z\tau|\tau), which links their values at \tau and -\frac{1}{\tau}. They admit elegant product representations via the identity, \vartheta_1(z|\tau) = 2 q^{1/4} \sin(\pi z) \prod_{n=1}^\infty (1 - q^{2n}) (1 - 2 q^{2n} \cos(2\pi z) + q^{4n}), highlighting their connection to q-series and partitions. In higher dimensions, the Riemann theta function \theta(z; \Omega) = \sum_{m \in \mathbb{Z}^g} \exp\left( \pi i \, ^t m \Omega m + 2\pi i \, ^t m z \right) for z \in \mathbb{C}^g and \Omega a g \times g symmetric matrix with positive definite imaginary part, generalizes the Jacobi case (recovering it for g=1) and incorporates characteristics [a \, b] for shifted versions, forming sections of line bundles on abelian varieties. These multivariable theta functions underpin the theory of Riemann theta nullwerte, which embed the moduli space of principally polarized abelian varieties into projective space, and play key roles in solving the Jacobi inversion problem for hyperelliptic curves. Applications extend to number theory, where theta series enumerate lattice points and yield formulas for sums of squares (e.g., Jacobi's four-square theorem), and to physics, modeling the partition function in statistical mechanics and the heat kernel on lattices.

Definitions and Basic Properties

Jacobi theta functions

The Jacobi theta functions, introduced by in his seminal treatise on elliptic functions, serve as foundational building blocks in the theory of elliptic functions, providing periodic analogs to the through their infinite series representations. These functions are defined for a complex variable z and a modulus \tau with \Im(\tau) > 0, and they exhibit quasi-periodicity that underpins their role in expressing more general elliptic functions. The four standard Jacobi theta functions are special cases of the more general theta functions with characteristics (a, b), defined as \vartheta\left[\begin{matrix} a \\ b \end{matrix}\right](z \mid \tau) = \sum_{n=-\infty}^{\infty} \exp\left( i\pi \tau (n + a)^2 + 2\pi i (n + a)(z + b) \right), where the nome q = e^{i\pi \tau} with |q| < 1 parameterizes the series convergence. Specifically, \vartheta_1(z \mid \tau) = -\vartheta\left[\begin{matrix} 1/2 \\ 1/2 \end{matrix}\right](z \mid \tau) = i \sum_{n=-\infty}^{\infty} (-1)^n q^{(n+1/2)^2} e^{i(2n+1)z}, \vartheta_2(z \mid \tau) = \vartheta\left[\begin{matrix} 1/2 \\ 0 \end{matrix}\right](z \mid \tau) = 2 q^{1/4} \sum_{n=0}^{\infty} q^{n(n+1)} \cos((2n+1)z), \vartheta_3(z \mid \tau) = \vartheta\left[\begin{matrix} 0 \\ 0 \end{matrix}\right](z \mid \tau) = \sum_{n=-\infty}^{\infty} q^{n^2} e^{2 i n z}, \vartheta_4(z \mid \tau) = \vartheta\left[\begin{matrix} 0 \\ 1/2 \end{matrix}\right](z \mid \tau) = \sum_{n=-\infty}^{\infty} (-1)^n q^{n^2} e^{2 i n z}. These series forms highlight their Fourier-like structure, with \vartheta_3 and \vartheta_4 incorporating cosine terms equivalently through the real part of the exponentials. The Jacobi theta functions possess distinct symmetry properties: \vartheta_1(z \mid \tau) is odd, satisfying \vartheta_1(-z \mid \tau) = -\vartheta_1(z \mid \tau), while \vartheta_2(z \mid \tau), \vartheta_3(z \mid \tau), and \vartheta_4(z \mid \tau) are even, with \vartheta_j(-z \mid \tau) = \vartheta_j(z \mid \tau) for j = 2, 3, 4. In general, for characteristics (a, b), the parity is determined by \vartheta\left[\begin{matrix} a \\ b \end{matrix}\right](-z \mid \tau) = (-1)^{4ab} \vartheta\left[\begin{matrix} a \\ b \end{matrix}\right](z \mid \tau). Regarding periodicity, the functions are quasi-periodic with respect to shifts in z. For integer m, \vartheta_j(z + m \mid \tau) = (-1)^{m(j-1)} \vartheta_j(z \mid \tau) for j = 1, 2, 3, 4, reflecting anti-periodicity for \vartheta_1 and periodicity for the others over the real period $1. The imaginary period involves \tau: \vartheta_1(z + \tau \mid \tau) = -q^{-1} e^{-2 i z} \vartheta_1(z \mid \tau), with analogous multiplier factors for the other functions, ensuring holomorphicity in the fundamental domain. These properties, combined with the characteristics, allow the theta functions to form a basis for elliptic function theory as developed by .

Auxiliary functions

The Jacobi elliptic functions are auxiliary functions constructed as ratios of Jacobi theta functions, providing doubly periodic meromorphic functions useful for solving certain differential equations and integrals. These functions, denoted sn(u, k), cn(u, k), and dn(u, k), where k is the elliptic modulus with 0 < k < 1, express elliptic integrals in a periodic form. Their definitions in terms of theta functions, with nome q = e^{iπτ} and τ = iK'/K (K and K' being the complete elliptic integrals of the first kind for moduli k and k' = √(1 - k²), respectively), are given by \begin{align*} \text{sn}(u,k) &= \frac{\theta_3(0,q)}{\theta_2(0,q)} \cdot \frac{\theta_1\left(\frac{u}{\theta_3^2(0,q)},q\right)}{\theta_4\left(\frac{u}{\theta_3^2(0,q)},q\right)}, \\ \text{cn}(u,k) &= \frac{\theta_4(0,q)}{\theta_2(0,q)} \cdot \frac{\theta_2\left(\frac{u}{\theta_3^2(0,q)},q\right)}{\theta_4\left(\frac{u}{\theta_3^2(0,q)},q\right)}, \\ \text{dn}(u,k) &= \frac{\theta_4(0,q)}{\theta_3(0,q)} \cdot \frac{\theta_3\left(\frac{u}{\theta_3^2(0,q)},q\right)}{\theta_4\left(\frac{u}{\theta_3^2(0,q)},q\right)}, \end{align*} where the argument scaling ensures consistency with the periods. These functions are doubly periodic in the complex u-plane, with fundamental periods 4K in the real direction and 2iK' in the imaginary direction; specifically, sn(u + 4K, k) = sn(u, k), sn(u + 2iK', k) = -sn(u, k), and analogous quasi-periodicities hold for cn and dn, with poles occurring at the zeros of the denominator theta functions (e.g., θ₄(z, q) = 0 for sn). The argument u relates inversely to the elliptic integral of the first kind via u = ∫_0^φ dθ / √(1 - k² sin² θ), where φ = am(u, k) is the amplitude function satisfying sn(u, k) = sin φ, cn(u, k) = cos φ, and dn(u, k) = √(1 - k² sin² φ); this integral representation connects directly to theta function expressions through the modulus k = θ₂²(0, q)/θ₃²(0, q). Basic identities include the Pythagorean relations sn²(u, k) + cn²(u, k) = 1 and k² sn²(u, k) + dn²(u, k) = 1, which follow from the product expansions of the theta functions. Addition theorems, such as \text{sn}(u + v, k) = \frac{\text{sn}(u,k) \text{cn}(v,k) \text{dn}(v,k) + \text{sn}(v,k) \text{cn}(u,k) \text{dn}(u,k)}{1 - k^2 \text{sn}^2(u,k) \text{sn}^2(v,k)}, and similar formulas for cn(u + v, k) and dn(u + v, k), arise from the addition formulas for theta functions and facilitate composition of elliptic integrals.

Nome and elliptic modulus

In the theory of Jacobi theta functions, the nome q serves as a key parameter that facilitates the q-series representations of these functions. It is defined as q = e^{\pi i \tau}, where \tau is a complex number with positive imaginary part, \Im(\tau) > 0. This definition places q inside the unit disk, ensuring |q| < 1, which guarantees the absolute convergence of the infinite series expansions for the theta functions. The elliptic modulus k, which connects theta functions to elliptic integrals, is expressed as the ratio k = \frac{\theta_2^2(0 \mid \tau)}{\theta_3^2(0 \mid \tau)}, where \theta_2 and \theta_3 denote the second and third evaluated at zero argument. The complementary modulus k' is then k' = \sqrt{1 - k^2}, providing a measure of the complementary period in the associated elliptic lattice. A fundamental relation arises from the series form of \theta_3, yielding \theta_3(0 \mid \tau) = \sum_{n=-\infty}^{\infty} q^{n^2}. The nome was introduced by Carl Gustav Jacob Jacobi in his foundational work on elliptic functions during the 1820s and 1830s, specifically to streamline the expansions and transformations involving theta series in the context of elliptic integrals. This parameter bridges the complex period \tau to the real modulus k, with q = e^{-\pi K'(k)/K(k)}, where K(k) and K'(k) = K(k') are the complete elliptic integrals of the first kind.

Identities and Representations

Jacobi identities

The Jacobi theta functions exhibit a rich algebraic structure through their fundamental identities, which include quasi-periodic shift relations and addition formulas that facilitate the composition of arguments. These identities, originally developed by Jacobi in his seminal work on elliptic functions, underpin many subsequent developments in the theory. A cornerstone is Jacobi's fundamental identity for the shift by half the imaginary period in the θ₁ function: \theta_1\left(z + \frac{\tau}{2} \;\middle|\; \tau \right) = -e^{i\pi/4 + i\pi \tau /4 - \pi i z} \theta_1(z \;\middle|\; \tau). This relation highlights the quasi-periodic behavior of θ₁ and can be verified by substituting the defining q-series expansion and simplifying the resulting exponential terms. Addition formulas provide bilinear expressions for theta functions at summed arguments, enabling recursive computations and connections to elliptic integrals. For the θ₃ function, one such formula is \theta_3(z + w \;\middle|\; \tau) = \frac{1}{\theta_3(0\;\middle|\;\tau)} \sum_{n=-\infty}^{\infty} \theta_2(n + z \;\middle|\; \tau) \theta_2(n + w \;\middle|\; \tau). This sum-over-integers form arises from Poisson summation applied to the theta series or direct expansion of the product of generating functions. Similar bilinear relations hold for the other theta functions, such as θ₁(z + w) θ₁(z - w) expressed in terms of products of θ₂ and θ₄ at z and w. The triple product identity originates from the specific q-series for θ₁, given by \theta_1(z\;\middle|\;\tau) = i \sum_{n=-\infty}^{\infty} (-1)^n q^{(n+1/2)^2} e^{2\pi i(2n+1)z}, where q = e^{i \pi \tau}. This alternating exponential sum sets up the equivalence to the infinite product form ∏ (1 - q^{2m}) (1 + q^{2m-1} e^{2 \pi i z}) (1 + q^{2m-1} e^{-2 \pi i z}), derived by pairing terms or using contour integration over a suitable path to capture residues. Proofs of these identities typically rely on series manipulations for the shift and addition formulas—such as term-by-term verification using the exponential definitions—or contour integration techniques, where the theta function's analytic continuation around the fundamental parallelogram yields the transformation factors.

Product representations

The infinite product representations of the Jacobi theta functions provide explicit expressions as entire functions of z, derived from the Jacobi triple product identity, and facilitate the study of their analytic properties. For the first Jacobi theta function, the representation is \theta_1(z \mid \tau) = i \, q^{1/4} \sin(\pi z) \prod_{n=1}^\infty (1 - q^{2n}) (1 - 2 q^{2n} \cos(2 \pi z) + q^{4n}), where q = e^{\pi i \tau} with \operatorname{Im} \tau > 0. The products for the other Jacobi theta functions \theta_2, \theta_3, and \theta_4 follow similarly by shifting the argument z: specifically, \theta_2(z \mid \tau) = \theta_1(z + 1/2 \mid \tau), \theta_3(z \mid \tau) = \theta_4(z + \tau / 2 \mid \tau), and \theta_4(z \mid \tau) = \theta_3(z + 1/2 \mid \tau), yielding analogous infinite products with adjusted exponential factors. These product forms reveal the locations of the zeros: \theta_1(z \mid \tau) has simple zeros precisely at the lattice points z = m + n \tau for integers m, n. The deduction of these representations traces back to connections with the Weierstrass sigma function, an whose quasi-periodic properties align with those of the theta functions.

Integral representations

Integral representations of the Jacobi theta functions provide powerful tools for analyzing their analytic continuation, transformation laws, and connections to broader mathematical structures. The plays a central role in deriving key integral-based identities for theta functions. Consider the theta function defined as \theta_3(z \mid \tau) = \sum_{n=-\infty}^{\infty} \exp\left( i \pi n^2 \tau + 2 i \pi n z \right), with \operatorname{Im} \tau > 0. Applying the Poisson summation formula to the underlying Gaussian terms \exp(-\pi n^2 t + 2\pi i n x) (with t = -i \tau, x = z), which are periodic sums of Gaussians, yields the functional equation \theta_3(z \mid \tau) = (-i \tau)^{-1/2} \exp\left( -\frac{\pi i z^2}{\tau} \right) \theta_3\left( \frac{z}{\tau} \;\middle|\; -\frac{1}{\tau} \right). This relation follows from the self-duality of the Gaussian under , where the of \exp(-\pi u^2) is itself, enabling the interchange of the sum and its dual via over the real line. These representations connect directly to Gaussian integrals. The transformation law emerges from evaluating integrals of the form \int_{-\infty}^{\infty} \exp\left( - \pi t u^2 + 2 \pi i u x \right) du = t^{-1/2} \exp\left( -\frac{\pi x^2}{t} \right), which is the of the Gaussian and serves as a preview for how theta functions arise in solutions to diffusion-type equations through such quadratic exponential forms. The analytic properties, including holomorphy in the upper half-plane, are preserved under this duality. Contour integral representations further illuminate the structure of theta functions and allow evaluation via residues. For instance, the function \theta_4(z \mid \tau) admits the representation \theta_4(z \mid \tau) = i \int_{i-\infty}^{i+\infty} \frac{\cos(2 \pi u z) \exp(i \pi \tau u^2)}{\sin(\pi u)} \, du, valid for \operatorname{Im} \tau > 0. This is derived by starting with the cosine series expansion, integrating term-by-term against a Gaussian \exp(-y u^2) along the real line, shifting the to the imaginary axis via , and exploiting the quadratic phase. The poles of $1/\sin(\pi u) at values u = k \in \mathbb{Z} contribute residues that recover the series definition: the residue at u = k is (-1)^k \cos(2 \pi k z) \exp(i \pi \tau k^2), summing to the theta series upon closing the appropriately. Similar representations for \theta_1, \theta_2, and \theta_3 follow by shifting z by half-quasiperiods. These forms emphasize the interplay between summation and integration in establishing the functions' elliptic periodicity and zero structure. A related contour integral expression arises for the normalized odd theta function, highlighting its connection to the cotangent kernel. Specifically, \frac{\theta_1(z \mid \tau)}{\theta_1'(0 \mid \tau)} = \frac{1}{2\pi i} \oint \frac{\cot(\pi z') \exp\left( i \pi \tau (z' - z)^2 / 2 \right)}{z' - z} \, dz', where the contour encloses the real axis poles, but in practice, it reduces to a Fourier-type principal value integral reflecting the standard quasi-periodicity. Evaluation proceeds via residues at the simple poles of \cot(\pi z') and the Gaussian factor, yielding the series expansion while underscoring the function's single zero at z=0 modulo the lattice. This form is particularly useful for deriving addition formulas and analytic continuations.

Explicit Values and Special Cases

Lemniscatic values

The lemniscatic values of the Jacobi theta functions arise at the special point corresponding to the elliptic modulus k = \frac{1}{\sqrt{2}}, which is associated with the period ratio \tau = i and the nome q = e^{-\pi}. This case is linked to the geometry of the , where the complete elliptic integral of the first kind K(k) evaluates to \frac{\Gamma\left(\frac{1}{4}\right)^2}{4\sqrt{\pi}}. The value of \theta_3(0 \mid i) is given by \theta_3(0 \mid i) = \frac{\Gamma\left(\frac{1}{4}\right)}{\sqrt{2} \, \pi^{3/4}}, derived from the relation K(k) = \frac{\pi}{2} \theta_3(0 \mid \tau)^2. At this point, implies \theta_2(0 \mid i) = \theta_4(0 \mid i). Jacobi's identity \theta_3^4 = \theta_2^4 + \theta_4^4 then yields $2\theta_2^4 = \theta_3^4, so \theta_2(0 \mid i) = \theta_4(0 \mid i) = \frac{\theta_3(0 \mid i)}{2^{1/4}} = \frac{\Gamma\left(\frac{1}{4}\right)}{2^{3/4} \pi^{3/4}}. These values connect to the lemniscate constant \varpi, defined as the arc length integral \int_0^1 \frac{dt}{\sqrt{1 - t^4}}, with \varpi = \frac{\Gamma\left(\frac{1}{4}\right)^2}{2 \sqrt{2\pi}}. This constant represents half the total arc length of the lemniscate and equals \sqrt{2} K\left(\frac{1}{\sqrt{2}}\right). Numerically, \theta_3(0 \mid i) \approx 1.08649, \theta_2(0 \mid i) = \theta_4(0 \mid i) \approx 0.91338, and \varpi \approx 2.62206. Carl Friedrich Gauss first computed \varpi numerically to five decimal places (2.62205) around 1800 using the arithmetic-geometric mean iteration on 1 and \sqrt{2}, recognizing its connection to elliptic integrals and the lemniscate.

Equianharmonic values

The equianharmonic case of elliptic functions occurs when the elliptic modulus satisfies k^2 = \frac{1 + i\sqrt{3}}{2}, corresponding to the period ratio \tau = \frac{1 + i\sqrt{3}}{2} in the upper half-plane. This configuration exhibits cubic symmetry, as \tau is a primitive 6th root of unity, leading to a rhombic lattice with 60-degree angles between the basis vectors. Such lattices are fundamental in the theory of elliptic functions, where the Weierstrass invariants are g_2 = 0 and g_3 > 0, distinguishing the case from the lemniscatic one with quadratic symmetry. The Jacobi theta constants at this \tau are connected to the half-periods of the lattice via elliptic integrals. For the normalized equianharmonic case with g_3 = 1, the real half-period satisfies $2\omega_1 = \frac{\Gamma(1/3)^3}{2\pi}, or equivalently \omega_1 = \frac{\Gamma(1/3)^3}{4\pi}, via the relation K(k) = \frac{\pi}{2} \theta_3^2(0 \mid \tau) with nome q = e^{\pi i \tau}. A related evaluation occurs at \omega = e^{2\pi i / 3}, underscoring the symmetry. Charles Hermite first explored these equianharmonic properties in his 1858 memoir on elliptic functions, establishing key transformations and identities that link theta values to periods in lattices with threefold rotational symmetry. These explicit forms facilitate computations in modular forms and provide benchmarks for numerical evaluations of theta functions.

Further explicit values

As the nome q = e^{i \pi \tau} approaches 0, corresponding to \tau \to i \infty, the Jacobi theta functions evaluated at z = 0 degenerate in a simple manner due to the truncation of their defining [q](/page/Q)-series to the constant term. Specifically, \theta_1(0 \mid \tau) = 0 holds identically for all \tau, while \theta_2(0 \mid \tau) \to 0, \theta_3(0 \mid \tau) \to 1, and \theta_4(0 \mid \tau) \to 1. These limits reflect the collapse of the elliptic structure to a trivial case where higher-order terms vanish. The Jacobi theta function \theta_1(z \mid \tau) vanishes at the half-period points of the underlying , namely z = m \pi + n \pi \tau for integers m, n. In particular, \theta_1(\pi \mid \tau) = 0, \theta_1(\pi \tau \mid \tau) = 0, and \theta_1(\pi (1 + \tau) \mid \tau) = 0, which follow from the defining product representation and the placement of simple zeros precisely at these lattice half-periods. At these points, the other theta functions take non-trivial values related by Jacobi's addition formulas; for example, shifting by \pi/2 yields \theta_1(z + \pi/2 \mid \tau) = -i e^{i z + i \pi \tau / 4} \theta_4(z + \pi \tau / 2 \mid \tau). Singular moduli correspond to special values of the elliptic modulus k, where the theta functions exhibit degenerate behavior. When k = 0 (equivalently, \tau \to i \infty), the reduce to , and the theta functions simplify as noted above, with \theta_3(0 \mid \tau) = 1 dominating. Conversely, as k \to 1 (or \tau \to 0^+, q \to 1^-), the structure degenerates to : for instance, \theta_3(0 \mid \tau) \sim \frac{1}{\sqrt{k'}} where k' = \sqrt{1 - k^2} \to 0, and the theta series diverge in a manner mirroring sech-like profiles. These limits highlight the interpolation between periodic and aperiodic behaviors in elliptic theory. Briefly, series involving sums of theta constants over imaginary shifts, such as \sum_{n \in \mathbb{Z}} \theta_3(0 \mid \tau + i n), link to through theta decomposition techniques, where such sums contribute to the Eisenstein subspace in the space of modular forms of weight 1/2, as explored in Siegel's analytic of forms.

Power Theorems and Transformations

Direct power theorems

One of the fundamental direct power theorems for Jacobi theta functions is the quartic relating the null values: \theta_3^4(0 \vert \tau) = \theta_2^4(0 \vert \tau) + \theta_4^4(0 \vert \tau). This relation, originally discovered by Jacobi, connects the fourth powers of the theta nullwerte and plays a central role in the theory of elliptic functions and modular forms. It can be proved using the infinite product representations of the theta functions derived from the Jacobi triple product identity, by expanding and comparing terms in the products for each null value. Direct power theorems also encompass relations under nome substitutions, such as replacing τ by 2τ (which maps the nome q to q²). The Landen transformations yield specific quadratic relations for the null values at doubled argument. In particular, defining the scaling factor A = 1 / \theta_4(0 \vert 2\tau), the transformation formulas imply \theta_4^2(0 \vert 2\tau) = \theta_3(0 \vert \tau) \theta_4(0 \vert \tau). This follows from substituting z = 0 into the general Landen identity for θ₄ and solving for the null value. Similar derivations hold for the other theta functions using the full set of Landen formulas. These relations can likewise be verified via the product expansions of the theta functions, as the products transform in a way that preserves the quadratic structure under nome squaring. These power theorems extend to applications in elliptic integrals and moduli. The elliptic modulus, defined as k(\tau) = [\theta_2(0 \vert \tau)]^2 / [\theta_3(0 \vert \tau)]^2, transforms under the doubling τ → 2τ according to the ascending Landen formula k(2\tau) = \frac{2 \sqrt{k(\tau)}}{1 + k(\tau)}. This relation arises directly from substituting the theta power identities and nome substitution relations into the definition of k, providing a quadratic map on the modulus that facilitates numerical computations and modular transformations.

Transformations at roots of the nome

The transformations at roots of the nome extend the theory of quadratic modular transformations to higher-order symmetries for theta functions, particularly for prime degrees such as 3 and 5, where the nome q is raised to the power 1/n, corresponding to the modular transformation τ → τ/n. These transformations were developed by Heinrich Martin Weber in his comprehensive treatment of elliptic functions, primarily to facilitate computations of class numbers for imaginary quadratic fields by relating theta values at different lattice scalings. For the cube root case, the Jacobi theta function θ₃(z | 3τ) is expressed in terms of theta functions evaluated at τ with arguments scaled by cube roots of unity and multiplied by cubic factors derived from the theory of transformation groups of degree 3. The general form involves a sum over the primitive cube root of unity ω (where ω³ = 1 and ω ≠ 1), incorporating multipliers to account for the decomposition, ensuring the transformation preserves the elliptic . This allows for the decomposition of the defining θ₃(z | 3τ) into components aligned with the cosets of the of 3 in the . Similarly, for the fifth root, the transformation for θ₃(z | 5τ) follows an analogous , expressing it in terms of θ at τ with arguments involving fifth of unity and multipliers from the fifth Φ₅(x) = x⁴ + x³ + x² + x + 1, which arises in the factorization of the kernel. These multipliers ensure the correct weighting for the five cosets in the scaling by 5, maintaining analytic properties under the higher-degree . In general, for prime n, the laws under τ → τ/n take the form of a of n theta functions at τ, shifted by the n-th of unity in the argument z, with coefficients determined by Gauss sums associated to the of (ℤ/nℤ)^*, reflecting the structure of the Hecke operator on the theta series. These laws, central to Weber's framework, enable the construction of modular equations of degree n and the evaluation of singular moduli for class number problems.

Modulus-dependent theorems

The elliptic modulus k is defined in terms of the Jacobi theta constants as k = \frac{\theta_2^2(0|\tau)}{\theta_3^2(0|\tau)}, where the nome q = e^{\pi i \tau} relates \tau to the fundamental period parallelogram of the associated elliptic curve. A fundamental modulus-dependent relation follows from Jacobi's fourth-power identity \theta_3^4(0|\tau) = \theta_2^4(0|\tau) + \theta_4^4(0|\tau). Substituting the definition of k yields \theta_4^4(0|\tau) = \theta_3^4(0|\tau) (1 - k^2), or equivalently, (1 - k^2) \theta_3^4(0|\tau) = \theta_4^4(0|\tau). This expresses the interdependence of the theta constants through the geometric parameter k, which parameterizes the shape of the elliptic curve. For higher powers, the eighth-power identity \theta_3^8(0|\tau) = \theta_2^8(0|\tau) + \theta_4^8(0|\tau) + 2 \theta_2^4(0|\tau) \theta_4^4(0|\tau) can be rewritten using the expressions \theta_2^4(0|\tau) = k^2 \theta_3^4(0|\tau) and \theta_4^4(0|\tau) = (1 - k^2) \theta_3^4(0|\tau), resulting in \theta_3^8(0|\tau) = \theta_3^8(0|\tau) [k^4 + (1 - k^2)^2 + 2 k^2 (1 - k^2)]. The bracketed term simplifies to 1, confirming consistency, but the form highlights explicit dependence on k for computational or analytic purposes in elliptic geometry. These power relations, including the modulus-dependent variants, are proved using duplication formulas for theta functions, such as \theta_3(0|2\tau) = \frac{1}{2} [\theta_3^2(0|\tau) + \theta_4^2(0|\tau)] and analogous formulas for \theta_2 and \theta_4. Iterating these transformations relates values at \tau and multiples thereof, yielding the power identities upon expansion and simplification. In applications to singular moduli k(\tau), where \tau lies in an imaginary \mathbb{Q}(\sqrt{-d}) with fundamental -d > 0, the theta constants evaluate to quantities whose ratios yield algebraic numbers encoding arithmetic invariants like class numbers. The Chowla-Selberg formula provides an explicit connection between special values of the at rational arguments and the at quadratic irrationals, linking to theta functions via the relation K(k(\tau)) = \frac{\pi}{2} \theta_3^2(0|\tau), where K is the complete of the first kind. This has implications for explicit class number computations and . The is given by \eta(\tau) = q^{1/12} \prod_{n=1}^\infty (1 - q^{2n}), with q = e^{\pi i \tau}.

Series and Sum Identities

Sums resulting in theta functions

Infinite sums can take various forms that evaluate directly to Jacobi theta functions or their generalizations, providing alternative representations useful in the study of q-series and modular forms. These identities often arise in the context of generating functions and theory, where the sums may involve alternating signs, quadratic exponents, or partition-related coefficients. Such expressions highlight the versatility of theta functions in encapsulating diverse summation structures. One notable example is the identity for a generalized form related to the fourth Jacobi theta function, expressed as \sum_{n=-\infty}^{\infty} (-1)^n q^{n(n+1)/2} e^{i n z} = \theta_4(z \mid \tau), where q = e^{2\pi i \tau} with \Im(\tau) > 0. This bilateral sum converges absolutely for |q| < 1 and represents a twisted variant of the standard theta series, leveraging the quadratic form in the exponent to match the periodic structure of \theta_4. The uniqueness of this representation follows from the analytic continuation of theta functions as entire functions in z and their modular properties under the action of \mathrm{SL}_2(\mathbb{Z}). Ramanujan developed numerous identities of this type in his notebooks, where sums over partition-like terms yield products involving theta functions, particularly \theta_3. For instance, a Ramanujan-type identity states that \sum_{n=0}^{\infty} \frac{(-1)^n q^{n(n+1)/2}}{(q;q)_n (1 - q^{2n+1})} = f(q^3, q^5), with f(a,b) = \sum_{n=-\infty}^{\infty} a^{n(n+1)/2} b^{n(n-1)/2} denoting Ramanujan's general theta function, which specializes to \theta_3(0 \mid 2\tau) when a = b = q. This sum, involving the q-Pochhammer symbol (q;q)_n = \prod_{k=1}^n (1 - q^k), counts weighted partitions with alternating signs and converges for |q| < 1, providing a bridge between partition sums and theta products. Similar identities appear throughout Ramanujan's work, often derived via dissection techniques or modular equations, emphasizing the role of theta functions in generating partition congruences. More generally, Lambert series offer another class of sums equating to theta functions. A key identity relates the square of the basic theta function to Lambert series: \left( \sum_{n=-\infty}^{\infty} q^{n^2} \right)^4 = 1 + 8 \sum_{n=1}^{\infty} \frac{n q^n}{1 - q^{2n}} - 8 \sum_{n=1}^{\infty} \frac{(-1)^n n q^{2n}}{1 - q^{2n}}, with q = e^{2\pi i \tau}. This relation, established through generating function manipulations, implies classical results like Jacobi's four-square theorem as special cases and holds under the convergence condition |q| < 1. The uniqueness stems from the eta-quotient representations of theta functions, ensuring distinct analytic behaviors. These identities underscore the interplay between additive and multiplicative structures in q-series.

Sums involving theta functions

Sums involving Jacobi theta functions arise in the theory of modular forms, particularly through the action of Hecke operators on half-integral weight spaces. The Jacobi theta function \theta_3(0 \mid \tau) = \sum_{n=-\infty}^{\infty} q^{n^2}, with q = e^{2\pi i \tau}, spans the space of modular forms of weight $1/2 for the congruence subgroup \Gamma_0(4). The Hecke operator T_p for a prime p acts on this space, and for \theta_3, the action yields T_p \theta_3(\tau) = \theta_3(\tau) + \left( \frac{-4}{p} \right) p^{1/2} \theta_3(p \tau), where \left( \frac{-4}{p} \right) is the Kronecker symbol evaluating to 1 if p \equiv 1 \pmod{4}, -1 if p \equiv 3 \pmod{4}, and 0 if p = 2. This expresses the Hecke-transformed theta as a sum of two theta functions at scaled arguments, illustrating how theta functions serve as summands in modular form decompositions. For general n, the Hecke operator T_n on \theta_3 involves a sum over the divisors d \mid n, weighted by the character and scaling factors, resulting in T_n \theta_3(\tau) = \sum_{d \mid n} \left( \frac{-4}{d} \right) d^{1/2} \theta_3\left( \frac{\tau}{d} \right), reflecting the eigenform property with eigenvalue the Dirichlet L-function value \sum_{d \mid n} \left( \frac{-4}{d} \right) d^{1/2}. These sums are fundamental in evaluating traces of Hecke operators and understanding the structure of half-integral weight modular forms. Cubic analogues of Jacobi theta functions, introduced by the Borwein brothers, provide further examples of sums involving theta-like terms for higher-degree identities. These functions are defined as double sums over the integer lattice with a quadratic form modified by cubic roots of unity: a(q) = \sum_{m,n=-\infty}^{\infty} q^{m^2 + mn + n^2}, b(q) = \sum_{m,n=-\infty}^{\infty} \omega^{m-n} q^{m^2 + mn + n^2}, c(q) = \sum_{m,n=-\infty}^{\infty} \omega^{2(m-n)} q^{m^2 + mn + n^2}, where \omega = e^{2\pi i / 3} is a primitive cube root of unity and |q| < 1. These are cubic counterparts to the , capturing sums over a hexagonal lattice with phase factors. They satisfy the identity a(q)^3 = b(q)^3 + c(q)^3, a higher-degree analogue of classical theta identities used in elliptic function theory and the arithmetic-geometric mean iteration. Extensions of these cubic sums appear in identities like those for \sum \theta_3(3z \mid 3\tau), where scaled serve as summands in modular evaluations, often linked to . Such sums facilitate proofs of relations between elliptic integrals and q-series expansions. q-Analogues involving multiple theta functions in partial fraction decompositions arise in the theory of basic hypergeometric series, where expressions like partial fraction expansions of q-products incorporate theta summands for residue computations. For instance, identities decomposing or eta quotients feature sums of products of theta functions, aiding evaluations in partition theory. These sums have applications to class numbers of quadratic fields, as the Fourier coefficients of theta series encode representation numbers by quadratic forms, and Hecke-equivariant sums over genus classes yield whose differences relate to the class number via cusp form projections. Specifically, for discriminant d < 0, the class number h(d) appears in the dimension of the space spanned by theta series over the class group, with sums of twisted thetas providing explicit evaluations through L-function relations.

Zeros and Analytic Properties

Zeros of Jacobi theta functions

The Jacobi theta functions \theta_1(z \mid \tau), \theta_2(z \mid \tau), \theta_3(z \mid \tau), and \theta_4(z \mid \tau) are entire functions of z for fixed \tau with positive imaginary part, and their zeros form regular lattices in the complex plane determined by the period parallelogram. These functions are of order 1 and genus 1, aligning with Hadamard factorization theorems. Specifically, \theta_1(z \mid \tau) vanishes at the points z = m + n\tau for all integers m, n \in \mathbb{Z}. These zeros are all simple, with multiplicity one at each lattice point, reflecting the function's order-one growth and the absence of higher-order terms in its factorization. In contrast, \theta_2(z \mid \tau) has zeros at the half-shifted lattice points z = (m + \frac{1}{2}) + n\tau for m, n \in \mathbb{Z}, again all simple. Similarly, \theta_3(z \mid \tau) vanishes simply at z = (m + \frac{1}{2}) + (n + \frac{1}{2})\tau, while \theta_4(z \mid \tau) does so at z = m + (n + \frac{1}{2})\tau. These distinct zero sets partition the full lattice into four sublattices, ensuring that the product \theta_1(z \mid \tau) \theta_2(z \mid \tau) \theta_3(z \mid \tau) \theta_4(z \mid \tau) has zeros precisely at all lattice points. The locations and simplicity of these zeros are captured in the infinite product representations of the theta functions. For instance, the Jacobi triple product gives \vartheta_1(z|\tau) = i q^{1/4} \sin(\pi z) \prod_{n=1}^\infty (1 - q^{2n}) (1 - 2 q^{2n} \cos(2\pi z) + q^{4n}), where the factors introduce simple zeros at the respective lattice points. Analogous product expansions hold for \theta_2, \theta_3, and \theta_4, with adjusted factors reflecting their shifted zero lattices.

Derivatives of theta functions

The derivative of the Jacobi theta function \theta_1(z \mid \tau) with respect to z can be obtained by differentiating its Fourier series representation: \theta_1(z \mid \tau) = 2 \sum_{n=0}^\infty (-1)^n q^{(n + 1/2)^2} \sin((2n+1) \pi z), where q = e^{\pi i \tau}. This yields the explicit series \frac{\partial}{\partial z} \theta_1(z \mid \tau) = 2\pi \sum_{n=0}^\infty (-1)^n (2n+1) q^{(n + 1/2)^2} \cos((2n+1) \pi z). Alternatively, using the infinite product representation \theta_1(z \mid \tau) = 2 q^{1/4} \sin(\pi z) \prod_{n=1}^\infty (1 - q^{2n}) (1 - 2 q^{2n} \cos(2 \pi z) + q^{4n}), the logarithmic derivative is \frac{\theta_1'(z \mid \tau)}{\theta_1(z \mid \tau)} = \pi \cot(\pi z) + 4 \pi \sum_{n=1}^\infty \frac{q^n \sin(2 \pi n z)}{1 - q^{2n}}. This form highlights the connection to elliptic functions, where the logarithmic derivative relates to the Weierstrass zeta function. The Jacobi theta functions satisfy a heat-equation-type partial differential equation. Specifically, each \theta_j(w \mid \tau) for j=1,2,3,4 obeys \frac{\pi i}{4} \frac{\partial^2 f}{\partial w^2} + \frac{\partial f}{\partial \tau} = 0. For \theta_3(z \mid i t) with real t > 0, appropriate scaling transforms this into the standard \partial u / \partial t = (1/(4\pi)) \partial^2 u / \partial z^2, illustrating the role of theta derivatives in processes. Higher-order derivatives of theta functions arise in compositions, such as when theta functions are evaluated at arguments involving other analytic functions. provides a general method to compute these, expressing the k-th derivative of f(g(z)) as a sum over partitions involving and derivatives of f and g. In the context of theta functions, iterated derivatives (e.g., \partial^{2t} / \partial z^{2t} \theta(z \mid \tau)) have been expressed explicitly as traces of partition-weighted , connecting them to quasimodular forms and partition theory. Derivatives of theta functions find applications in evaluating elliptic singular moduli k_r, which are values of the elliptic modulus at quadratic imaginary arguments \tau. For instance, relations involving \theta_1'(0 \mid \tau) and higher derivatives facilitate solutions to modular equations of degree five, enabling radical expressions for k_{25^n r_0} and high-precision approximations of \pi via quintic iterations.

Integrals of theta functions

The indefinite integral of the Jacobi theta function \theta_3(z|\tau) does not admit a closed form in terms of elementary functions but can be expressed as a series derived from its Fourier expansion. Specifically, \theta_3(z|\tau) = 1 + 2 \sum_{n=1}^\infty q^{n^2} \cos(2 \pi n z), where q = e^{i \pi \tau}, so \int \theta_3(z|\tau) \, dz = z + \sum_{n=1}^\infty \frac{q^{n^2}}{\pi n} \sin(2 \pi n z) + C, with the constant of integration C. This series form facilitates computation when |q| is small, as higher terms decay rapidly, and in the limit q \to 0, it reduces to the trivial integral z + C. For general \tau, the series relates to the elliptic amplitude function through the connection between theta functions and Jacobi elliptic functions, where the antiderivative involves the inverse of the incomplete elliptic integral of the first kind via the scaling \zeta = \pi u / (2 K(k)), with K(k) the complete elliptic integral of the first kind and k the modulus. For definite integrals over the fundamental in z, which is 1 for \theta_3(z|\tau), the orthogonality of the cosine terms yields a simple result: \int_0^1 \theta_3(z|\tau) \, dz = 1, independent of \tau. Over the quarter from 0 to $1/2, corresponding to the elliptic quarter , the integral evaluates to $1 / (2 \theta_3(0|\tau)), providing a to the elliptic periods since \theta_3(0|\tau) = \sqrt{2 K(k)/\pi}. Important classes of definite integrals involving theta functions include Mellin and Laplace transforms, which link to number-theoretic functions. The Mellin transform is given by \int_0^\infty x^{s-1} \bigl( \theta_3(0 | i x^2) - 1 \bigr) \, dx = \pi^{-s/2} \Gamma(s/2) \zeta(s), \quad \Re s > 1, where \zeta(s) is the Riemann zeta function. The Laplace transform takes the form \int_0^\infty e^{-s t} \theta_3 \biggl( \frac{(1+\beta) \pi}{2 \ell} \Big| i \frac{\pi t}{\ell^2} \biggr) \, dt = \frac{\ell}{\sqrt{s}} \cosh(\beta \sqrt{s}) \csch(\ell \sqrt{s}), \quad \Re s > 0, \ \ell > 0, \ | \Re \beta | + | \Im \beta | \le \ell. These transforms establish connections to gamma and zeta functions, with analogous q-deformations relating to q-beta function identities via the q-Raabe formula, a q-analogue of the multiplication theorem for the gamma function underlying the beta function B(a,b) = \Gamma(a) \Gamma(b) / \Gamma(a+b). For example, the q-beta function B_q(a,b) involves infinite products expressible through theta functions via Jacobi's triple product identity. When closed forms are unavailable, such as for general indefinite integrals or parameter-dependent definite integrals, numerical evaluation relies on series expansions of \theta_3(z|\tau) accelerated by modular transformations to ensure rapid convergence. For instance, if |q| \approx 1, transform \tau \to -1/\tau to reduce |q| < 0.2, then integrate the Fourier series term by term using Gaussian quadrature or adaptive methods, with error controlled to machine precision. These approaches leverage the quasi-periodic properties for efficient computation over intervals.

Relations to Other Functions

Relation to Riemann zeta function

The Jacobi theta function \theta_3(0 \mid \tau), defined as \sum_{n=-\infty}^{\infty} q^{n^2} where q = e^{\pi i \tau}, provides an integral representation for the Riemann zeta function \zeta(s) through the Mellin transform. Specifically, for \operatorname{Re}(s) > 1, \pi^{-s/2} \Gamma(s/2) \zeta(s) = \int_0^\infty t^{s/2-1} \frac{\theta_3(0 \mid it) - 1}{2} \, dt, where the subtraction of 1 ensures convergence at infinity, and the factor of $1/2 accounts for double-counting positive and negative terms in the theta series expansion. This representation analytically continues \zeta(s) to the except for a simple pole at s=1, leveraging the rapid decay of the theta function for large imaginary arguments. The of the zeta function, \pi^{-s/2} \Gamma(s/2) \zeta(s) = \pi^{-(1-s)/2} \Gamma((1-s)/2) \zeta(1-s), derives directly from the modular property of the theta function: \theta_3(0 \mid -1/\tau) = \sqrt{-i\tau} \, \theta_3(0 \mid \tau) for \operatorname{Im}(\tau) > 0. Substituting \tau = it yields \theta_3(0 \mid -1/(it)) = \sqrt{t} \, \theta_3(0 \mid it), or equivalently \theta_3(0 \mid i/t) = \sqrt{t} \, \theta_3(0 \mid it). Applying this summation-derived to split the Mellin at t=1 and substituting u = 1/t in the inner part transforms the expression, confirming the symmetry \Lambda(s) = \Lambda(1-s) where \Lambda(s) = \pi^{-s/2} \Gamma(s/2) \zeta(s). This connection generalizes to the zeta function Z_Q(s), associated with a positive Q on \mathbb{Z}^n, defined as Z_Q(s) = \sum_{m \in \mathbb{Z}^n \setminus \{0\}} Q(m)^{-s} for \operatorname{Re}(s) > n/2. The corresponding theta function is \Theta_Q(iy) = \sum_{m \in \mathbb{Z}^n} e^{-\pi y Q(m)} for y > 0, and the relates them via \pi^{-s} \Gamma(s) Z_Q(s) = \int_0^\infty (\Theta_Q(iy) - 1) y^{s-1} \, dy. A transformation law analogous to the Jacobi case, \Theta_Q(iy) = y^{-n/2} \Theta_{Q^*}(i/y) where Q^* is the dual form, implies a \pi^{-s} \Gamma(s) Z_Q(s) = \pi^{-(n/2 - s)} \Gamma(n/2 - s) Z_{Q^*}(n/2 - s). In his paper, employed an integral representation involving the theta function and its to define the completed zeta function \xi(s) = \frac{1}{2} s(s-1) \pi^{-s/2} \Gamma(s/2) \zeta(s), expressed as a contour integral that analytically continues \zeta(s) and satisfies \xi(s) = \xi(1-s). This allowed him to argue that the non-trivial zeros of \zeta(s) lie in the critical strip $0 < \operatorname{Re}(s) < 1 and conjectured their location on the line \operatorname{Re}(s) = 1/2, a hypothesis central to prime number distribution.

Relation to Weierstrass elliptic function

The Weierstrass elliptic function \wp(z; \Lambda) associated with a lattice \Lambda = \mathbb{Z} \omega_1 + \mathbb{Z} \omega_3 in the complex plane is defined by the series \wp(z; \Lambda) = \frac{1}{z^2} + \sum_{\Omega \in \Lambda \setminus \{0\}} \left( \frac{1}{(z - \Omega)^2} - \frac{1}{\Omega^2} \right), where the sum converges absolutely and uniformly on compact sets avoiding the lattice points, making \wp a meromorphic function with double poles at each lattice point. This function can be expressed in terms of the Jacobi theta function \theta_1 via the relation \wp(z) = -\frac{d^2}{dz^2} \log \theta_1(z) + c, where c is a constant depending on the lattice parameters, providing a direct differential connection between the two formulations of elliptic functions. The invariants g_2 and g_3 of the Weierstrass function, which determine the associated elliptic curve y^2 = 4x^3 - g_2 x - g_3, are given in terms of theta constants evaluated at zero. Specifically, g_2 = \frac{4\pi^4}{3\omega_1^4} \left( \theta_2^4(0,q) + \theta_3^4(0,q) + \theta_4^4(0,q) \right), \quad g_3 = \frac{8\pi^6}{27\omega_1^6} \left( \theta_2^2(0,q) \theta_3^2(0,q) \theta_4^2(0,q) \left( \theta_2^2(0,q) - \theta_3^2(0,q) + \theta_4^2(0,q) \right) \right), with q = e^{\pi i \tau} and \tau = \omega_3 / \omega_1. These expressions link the modular invariants to theta characteristics, facilitating computations across the two theories. The Jacobi and Weierstrass approaches to elliptic functions are theoretically equivalent, with the former emphasizing addition theorems derived from theta function identities for solving integral equations, while the latter provides a uniformization map from the torus \mathbb{C}/\Lambda to elliptic curves via the embedding z \mapsto (\wp(z), \wp'(z)).

Relation to q-gamma function

The q-gamma function, denoted \Gamma_q(z), serves as a q-analogue of the classical gamma function and is defined for $0 < q < 1 and \Re(z) > 0 by the infinite product \Gamma_q(z) = \frac{(q; q)_\infty (1 - q)^{1 - z}}{(q^z; q)_\infty}, where (a; q)_\infty = \prod_{k=0}^\infty (1 - a q^k) is the . This definition originates from Jackson's work on q-integrals and was introduced to generalize factorial-like properties in q-deformed . A key connection between Jacobi theta functions and the q-gamma function arises through infinite product representations, where theta products yield identities for products of q-gamma functions. For instance, Jacobi's product formula for theta functions is equivalent to a q-trigonometric identity involving the q-sine function, which in turn links to q-gamma via q-analogues of multiplication formulas. Specifically, the product \prod_{k=1}^{n-1} \theta_1\left( \frac{k \pi}{n} \mid \tau \right) = 2^n q^{n(n^2 - 1)/12} \prod_{k=1}^{n-1} \sin\left( \frac{k \pi}{n} \right), with q = e^{2 \pi i \tau}, implies a corresponding identity for short products of q-gamma functions, such as the q-analogue of Gauss's multiplication formula: \prod_{k=0}^{n-1} \Gamma_q(q^k z) = (1 - q)^{n(n-1)z/2} q^{n(n-1)(n-2)z/6} \prod_{k=0}^{n-1} \Gamma_q(z + k/n). This relation highlights how theta functions provide a bridge to derive functional equations for q-gamma. The normalized Jacobi theta function \theta_1(z \mid \tau)/\theta_1'(0 \mid \tau) is directly expressed in terms of q-gamma functions with q = e^{2\pi i \tau}: \frac{\theta_1(z \mid \tau)}{\theta_1'(0 \mid \tau)} = \frac{\Gamma_q\left( \frac{1}{2} + \frac{z}{\pi} \right)}{\Gamma_q\left( \frac{1}{2} \right)} \cdot (-1)^{m} q^{m(m+1)/2 + m z/\pi}, for appropriate integer m adjusting the branch, reflecting the q-deformation of the sine function via theta products. This identity stems from the infinite product form of \theta_1, which aligns the zeros and poles with those of the q-gamma. Jackson's q-integral representation further ties the q-gamma to theta-like sums, defining \Gamma_q(z) = (1 - q)^{1 - z} \int_0^1 (-\ln t)^{z - 1} \, \mathrm{d}_q t, or equivalently using the q-exponential e_q(-t/(1-q)), where the Jackson integral \int_0^1 f(t) \, \mathrm{d}_q t = (1 - q) \sum_{k=0}^\infty f(q^k) q^k. These representations allow theta functions to appear in asymptotic expansions or summation formulas for q-gamma evaluations. As q \to 1^-, the q-gamma function converges to the classical gamma function, \lim_{q \to 1^-} \Gamma_q(z) = \Gamma(z), preserving the connection to theta functions in the elliptic limit where modular properties emerge. This asymptotic behavior ensures consistency with classical special function identities, such as the reflection formula, whose q-analogue involves \theta_1 directly: \Gamma_q(z) \Gamma_q(1 - z) = \frac{i q^{1/8} (1 - q)}{(1; q)_\infty^3} q^{z^2} \theta_1\left( -\frac{i z}{2 \log q}, \sqrt{q} \right).

Relations to Dedekind eta function

The Dedekind eta function, denoted \eta(\tau), is defined for \tau in the upper half-plane by the infinite product \eta(\tau) = q^{1/24} \prod_{n=1}^{\infty} (1 - q^n), where q = e^{2\pi i \tau} is the nome. This function is closely related to Jacobi theta functions through specific identities. One such relation expresses the eta function in terms of the Jacobi theta function \theta_2 evaluated at particular arguments: \eta(q) = \frac{\theta_2\left(\frac{\pi}{6}, q^{1/6}\right)}{\sqrt{3}}, where here q = e^{\pi i \tau}. The satisfies a transformation law under the action of the \mathrm{SL}(2, \mathbb{Z}). Specifically, for \tau \mapsto -1/\tau, \eta\left(-\frac{1}{\tau}\right) = \sqrt{-i\tau} \, \eta(\tau), with the branch of the chosen such that \sqrt{-i\tau} > 0 when \tau is purely imaginary and positive. This extends to the full group via \eta((a\tau + b)/(c\tau + d)) = \epsilon(a,b,c,d) (c\tau + d)^{1/2} \eta(\tau), where \epsilon is a 24th . As a modular form of weight $1/2 for \mathrm{SL}(2, \mathbb{Z}), the eta function generates the space of cusp forms of weight 12 upon raising to the 24th power, since \eta(\tau)^{24} is the modular discriminant up to a constant factor. Its half-integral weight distinguishes it from integer-weight modular forms and underscores its role in connecting theta functions to broader modular theory.

Applications and Generalizations

Solution to the heat equation

The one-dimensional heat equation on the real line with periodic boundary conditions models the diffusion of heat in a medium wrapped around a circle, such as the torus \mathbb{T}^1 = \mathbb{R}/\mathbb{Z}. Consider the domain x \in [0,1] with periodic identification u(0,t) = u(1,t) and \partial u / \partial x (0,t) = \partial u / \partial x (1,t). The equation takes the form \frac{\partial u}{\partial t}(x,t) = \frac{\partial^2 u}{\partial x^2}(x,t), where u(x,t) represents the temperature at position x and time t > 0. Using , the eigenfunctions of the Laplacian -\partial^2 / \partial x^2 on this are the basis \cos(2\pi n x) and \sin(2\pi n x) for n \in \mathbb{Z}, with eigenvalues $4\pi^2 n^2. For an even , the solution can be expressed via the cosine series. A fundamental solution arises from the initial condition given by the , u(x,0) = \sum_{k \in \mathbb{Z}} \delta(x - k), which represents infinitely many point sources of heat at integer lattice points, periodically replicated across the line to enforce the boundary conditions. The solution to the with this initial condition is given by the Jacobi theta function, which approaches the as t \to 0^+. The explicit is the u(x,t) = \sum_{n=-\infty}^{\infty} e^{-n^2 4 \pi^2 t} \cos(2\pi n x), which satisfies the and periodic boundaries. This series equals the Jacobi theta function \theta_3(x \mid 4 \pi i t), where \theta_3(z \mid \tau) = \sum_{n=-\infty}^{\infty} \exp(\pi i \tau n^2 + 2\pi i n z). Substituting \tau = 4 \pi i t and z = x yields the \exp(-4 \pi^2 t n^2) in the real part, matching the term. Verification follows by direct substitution: the time derivative gives \partial_t u = -4 \pi^2 \sum n^2 e^{-n^2 4 \pi^2 t} \cos(2\pi n x), while the second spatial derivative yields \partial_x^2 u = -4 \pi^2 \sum n^2 e^{-n^2 4 \pi^2 t} \cos(2\pi n x), confirming \partial_t u = \partial_x^2 u. Physically, this solution describes heat diffusion starting from a lattice of discrete sources on the circle, where the theta function captures the smoothing effect over time: as t increases, the sharp peaks at integers broaden and overlap, illustrating the infinite speed of propagation and analyticity inherent to the heat equation. For general initial data u_0(x) periodic on [0,1], the solution is the convolution u(x,t) = \int_0^1 u_0(y) u(x-y,t) \, dy, preserving the periodic structure. This connection originates in Jacobi's foundational work on elliptic functions, where theta functions emerged as solutions to such diffusion problems.

Relation to Heisenberg group

The is a that can be realized as the set of 3×3 upper triangular matrices with ones on the diagonal over the reals, or equivalently as the V \times V^* \times K, where V is a , V^* its dual, and K = \mathbb{R}/\mathbb{Z}, with group law (v_1, \xi_1, t_1) \cdot (v_2, \xi_2, t_2) = (v_1 + v_2, \xi_1 + \xi_2, t_1 + t_2 + \langle v_1, \xi_2 \rangle), where \langle \cdot, \cdot \rangle denotes the pairing induced by the symplectic form. This structure captures the canonical commutation relations from , with the center Z = \{ (0,0,t) \} acting as scalars in representations. The Stone–von Neumann theorem asserts that there exists a unique irreducible unitary of the up to unitary equivalence, in which the center acts by a fixed nontrivial \chi. This , known as the Schrödinger representation, acts on the L^2(V) via the formula [U(v, \xi, t) \phi](x) = \chi(t) e^{2\pi i \langle x, \xi \rangle} \phi(x + v) for \phi \in L^2(V). In this context, Jacobi theta functions emerge as the kernel of the representation operators or as matrix coefficients; specifically, the theta function \theta(z; \tau) = \sum_{n \in \mathbb{Z}} e^{\pi i n^2 \tau + 2\pi i n z} appears in the evaluation \langle U(v, \xi, t) \phi, \psi \rangle, linking the to modular transformations via Poisson summation. Theta functions further relate to the Heisenberg group through their role as automorphic forms on the metaplectic cover \mathrm{Mp}(2\mathbb{R}) of the symplectic group \mathrm{Sp}(2\mathbb{R}), which is the double cover realizing the Weil representation. The Weil representation lifts the action of \mathrm{Sp}(2\mathbb{R}) to unitary operators on L^2(\mathbb{R}), and theta functions transform under this cover with a multiplier of weight $1/2, as in the modular transformation \theta\left( \frac{z}{\tau}; -\frac{1}{\tau} \right) = (-i\tau)^{1/2} e^{\pi i z^2 / \tau} \theta(z; \tau). This connection was established by André Weil, who interpreted theta series as automorphic forms invariant under the metaplectic group action induced by the Heisenberg representation. In , theta functions serve as coherent states for the quantized phase space associated with the . The space of theta functions forms an of half-densities on the \mathbb{C}/\Lambda, where \Lambda is the period lattice, and the Schrödinger representation realizes the quantization map from classical observables to operators, with theta functions encoding the overlap between coherent states via Gaussian wave packets. This framework underscores the theta functions' role in bridging representations and quantum mechanical systems.

Theta series of Dirichlet characters

The theta series associated to a \chi modulo q is defined, for \tau in the upper half-plane, by \theta_\chi(\tau) = \sum_{n=-\infty}^{\infty} \chi(n) \, q^{n^2}, where q = e^{2\pi i \tau}. This generalizes the classical Jacobi theta function \vartheta_3(0, \tau), which corresponds to the principal character. The series converges absolutely and uniformly on compact subsets of the upper half-plane, yielding a there. For odd characters where \chi(-1) = -1, the sum vanishes due to antisymmetry, so an adjusted series \sum_{n=-\infty}^{\infty} n \, \chi(n) \, q^{n^2} is often used instead, corresponding to modular forms of weight $3/2. These theta series transform under the action of congruence subgroups of \mathrm{SL}_2(\mathbb{Z}), with the transformation law depending on the conductor q and a Gauss sum associated to \chi. For non-principal characters, the series are not invariant under the full modular group but under \Gamma_0(q) or related groups, with a multiplier involving the root number \varepsilon(\chi). The functional equation for \theta_\chi follows from the Poisson summation formula applied to the character twist, mirroring the transformation \theta_\chi(-1/\tau) = \sqrt{q} \, \varepsilon(\chi) \, \tau^{1/2} \, \theta_{\overline{\chi}}(\tau). The connection to L-functions arises via the Mellin transform. For an even primitive character \chi, \pi^{-s/2} \Gamma(s/2) L(s, \chi) = \int_0^\infty y^{s/2 - 1} \theta_\chi(i y) \, dy, where L(s, \chi) = \sum_{n=1}^\infty \chi(n) n^{-s} is the (coinciding with the Hecke L-function over \mathbb{Q}). This integral representation provides the analytic continuation of L(s, \chi) to the and its L(1-s, \overline{\chi}) = \varepsilon(\chi) (q/\pi)^{(1-2s)/2} \Gamma(s) / \Gamma(1-s) \, L(s, \chi), with |\varepsilon(\chi)| = 1. For odd characters, a similar transform involving the adjusted theta series yields \pi^{-(s+1)/2} \Gamma((s+1)/2) L(s, \chi). Unlike the , non-principal L(s, \chi) are entire. For quadratic characters \chi_D associated to a fundamental discriminant D < 0, the value L(1, \chi_D) links directly to the class number h(D) of the imaginary quadratic field \mathbb{Q}(\sqrt{D}) via Dirichlet's class number formula: h(D) = \frac{w \sqrt{|D|}}{2\pi} L(1, \chi_D), where w is the number of units in the ring of integers ( w=2 for D < -4, w=4 or $6 otherwise). Here, L(1, \chi_D) = \sum_{n=1}^\infty \chi_D(n)/n can be expressed as a finite sum using the properties of the Kronecker symbol defining \chi_D. This relation, derived from the Dedekind zeta function decomposition \zeta_K(s) = \zeta(s) L(s, \chi_D), underscores the arithmetic significance of these theta series in computing class numbers.

Ramanujan theta function

The Ramanujan theta function generalizes the classical Jacobi theta functions to a bilinear form in two complex variables a and b, providing a versatile tool for q-series expansions and modular form theory. Defined by the infinite sum f(a,b) = \sum_{n=-\infty}^{\infty} a^{n(n+1)/2} b^{n(n-1)/2}, where |ab| < 1, this function converges absolutely and captures symmetric properties useful in number theory. A fundamental identity linking the sum to infinite products is the Jacobi triple product representation: f(a,b) = (-a; ab)_{\infty} (-b; ab)_{\infty} (ab; ab)_{\infty}, where (z; q)_{\infty} = \prod_{k=0}^{\infty} (1 - z q^k) denotes the q-Pochhammer symbol. This product form facilitates derivations of generating functions and modular transformations. Specific evaluations connect f(a,b) to the Dedekind eta function \eta(\tau), where q = e^{2\pi i \tau}. For instance, f(-q) := f(-q, -q^2) = (q; q)_{\infty} = q^{-1/24} \eta(\tau), establishing a direct link to modular forms of weight 1/2. Another case, f(-q, q) = \sum_{n=-\infty}^{\infty} (-1)^{n(n+1)/2} q^{n^2}, generates signed sums interpretable via the triple product as \prod_{k=1}^{\infty} (1 + q^{2k-1})^2 (1 - q^{2k}) up to scaling, relating to differences in partition counts. Ramanujan extended theta-like ideas to mock theta functions in his 1920 letter to Hardy, describing them as functions resembling theta functions but lacking full modularity; examples include f(q) = \sum_{n=0}^{\infty} \frac{q^{n^2}}{(q; q)_n^2}, which asymptotically mimic theta behavior while connecting to indefinite theta series. In his second notebook, particularly Chapter 16, Ramanujan recorded identities using f(a,b) to derive partition relations, such as equating the number of partitions of $2k into even parts (with multiples of 7 allowed extra copies) to those of $2k+1 into odd parts, via combinatorial interpretations of the triple product expansions. Modern applications include asymptotic analyses leveraging the modular invariance of theta functions; for small |q|, the dominant terms yield approximations like f(q,q) \sim 1 + 2q + 2q^4 + \cdots, with full asymptotics from Poisson summation giving exponential decay as \operatorname{Im}(\tau) \to \infty. Continued fraction representations, such as the Rogers-Ramanujan continued fraction r(q) = \cfrac{q^{1/5}}{1 + \cfrac{q}{1 + \cfrac{q^2}{1 + \ddots}}} expressed as ratios involving f(-q^5)/f(-q), stem from Ramanujan's entries and enable evaluations at cusps for partition congruences.

Riemann theta function

The Riemann theta function provides a multivariable generalization of the classical , extending their role to higher-dimensional and their . For a positive integer g, it is defined on the \mathbb{H}_g of symmetric g \times g complex matrices with positive definite imaginary part, together with \mathbb{C}^g, by the series \theta(z \mid \Omega) = \sum_{m \in \mathbb{Z}^g} \exp\left( \pi i \, m^\top \Omega m + 2 \pi i \, m^\top z \right), where \Omega is symmetric and positive definite, z \in \mathbb{C}^g, and the sum converges absolutely due to the positive definiteness of \operatorname{Im} \Omega. This function serves as a fundamental building block in the analytic construction of principally polarized , with \Omega representing the period matrix derived from integrals of holomorphic differentials over a basis of cycles on a . In the case g=1, it reduces to the \vartheta_3(z \mid \tau). Under the action of the symplectic group \mathrm{Sp}(2g, \mathbb{Z}), which acts on \mathbb{H}_g \times \mathbb{C}^g via fractional linear transformations, the Riemann theta function transforms as a of weight g/2: \theta\left( (A z + B) (\Omega; C z + D)^{-1} \mid (A \Omega + B)(C \Omega + D)^{-1} \right) = \det(C \Omega + D)^{g/2} \, \theta(z \mid \Omega), for \gamma = \begin{pmatrix} A & B \\ C & D \end{pmatrix} \in \mathrm{Sp}(2g, \mathbb{Z}). This modular property underscores its invariance properties and role in the theory of automorphic forms on . The zeros of \theta(\cdot \mid \Omega) form the \Theta = \{ z \in \mathbb{C}^g / \mathbb{Z}^g + \Omega \mathbb{Z}^g \mid \theta(z \mid \Omega) = 0 \}, a codimension-one subvariety in the Jacobian abelian variety that carries the principal polarization induced by the period matrix \Omega. In applications, the Riemann theta function is central to the Torelli theorem, which asserts that a compact Riemann surface of genus g \geq 2 is uniquely determined up to isomorphism by its Jacobian abelian variety equipped with the theta divisor (or equivalently, by the period matrix \Omega \in \mathbb{H}_g). This theorem highlights the theta function's power in embedding the moduli space of Riemann surfaces into the moduli space of principally polarized abelian varieties via period matrices, facilitating deep connections between algebraic geometry and complex analysis.

Poincaré series

Poincaré series, in the context of theta functions, refer to a class of non-holomorphic modular forms constructed as averages over the modular group SL(2,ℤ) of seed functions involving theta kernels, particularly in half-integral weights. These series generalize the classical theta series to non-holomorphic settings and play a key role in the theory of harmonic Maass forms and automorphic representations. The construction ensures the resulting form transforms correctly under the group action while incorporating non-holomorphic factors like powers of the imaginary part to ensure convergence. The standard Maass-Poincaré series of weight k and parameter s is defined as F_{k}( \tau, s ) = \sum_{\gamma \in \Gamma_{\infty} \backslash \mathrm{SL}(2,\mathbb{Z})} \mathrm{Im}(\gamma \tau)^{s - k/2} \big|_{k} \gamma, where \Gamma_{\infty} is the subgroup of upper triangular matrices with integer entries, \tau = x + iy \in \mathbb{H}, and \big|_{k} \gamma denotes the usual slash operator f\big|_{k} \gamma = j(\gamma, \tau)^{-k} f(\gamma \tau), with j(\gamma, \tau) = cz + d for \gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix}. This series converges for \mathrm{Re}(s) > 1 + |k|/2 and defines a non-holomorphic of weight k for SL(2,ℤ). Seminal work by Maass established the and for such series in integral weights, laying the foundation for their use in of the Laplacian. For half-integral weights, such as k = 1/2 or [3/2](/page/3-2), the Maass-Poincaré series are built using kernels derived from Jacobi functions to account for the metaplectic . A typical example is the series F_{k,4N}(\tau, s) = \sum_{\gamma \in (\Gamma_0(4N))_{\infty} \backslash \Gamma_0(4N)} y^{s - k/2} \big|_{k} \gamma, where N is odd and square-free, and the sum is projected onto the Kohnen plus space using a \Theta(\tau) = \sum_{m \in \mathbb{Z}} q^{m^2} to ensure modularity under the . This incorporates the function to handle the half-integral , yielding sesquiharmonic Maass forms that are annihilated by a modified Laplacian. Niebur extended these series to half-integral weights, showing their Fourier expansions involve generalized Kloosterman sums and , which link to indefinite series. The ensures rapid decay in the cuspidal direction, distinguishing these from integral weight cases. In the limit as the spectral parameter approaches specific values, such as s = k/2 + 1, the Maass-Poincaré series reduce to non-holomorphic . For instance, at s = 3/4 for weight $1/2, the residue yields Zagier's non-holomorphic H(\tau), a sesquiharmonic form whose holomorphic projection is the constant 1 and whose shadow is related to the via the Kohnen-Zagier theta kernel. This limit provides a bridge between Poincaré series and , facilitating the decomposition of the space of modular forms into cuspidal and Eisenstein components. Applications of these Poincaré series include extracting coefficients of cusp forms through Petersson inner products. For a cusp form f of k, the inner product \langle F_{k}(\tau, s), f \rangle yields constants times the m-th coefficient of f, up to scattering terms involving Kloosterman sums that encode the continuous . In half-integral weights, this extends to scattering matrices for automorphic representations on the , with kernels enabling computations of traces of singular moduli and L-values at CM points. These tools have high impact in connecting mock modular forms to statistics and number problems.

Derivations and Connections to Partitions

Derivation from elliptic integrals

The complete elliptic integral of the first kind, defined as K(k) = \int_0^{\pi/2} \frac{d\theta}{\sqrt{1 - k^2 \sin^2 \theta}}, is expressed in terms of the Jacobi theta function \theta_3(0 \mid \tau) by the formula K(k) = \frac{\pi}{2} \theta_3^2(0 \mid \tau), where the modular parameter \tau satisfies \tau = i K'(k)/K(k) with K'(k) = K(\sqrt{1 - k^2}), and the elliptic modulus k is given by k = \theta_2^2(0 \mid \tau)/\theta_3^2(0 \mid \tau). This relation arises from the infinite product representations of the theta functions and the period structure of elliptic functions, allowing the integral to be evaluated via the q-series expansion of \theta_3. For the incomplete elliptic integral of the first kind, F(\phi, k) = \int_0^\phi \frac{d\theta}{\sqrt{1 - k^2 \sin^2 \theta}}, the connection to theta functions proceeds through the , which invert the integral via u = F(\phi, k) and \phi = \am(u, k), the amplitude function. Specifically, the Jacobi elliptic sine is given by the ratio \sn(u, k) = \frac{\theta_3(0 \mid \tau)}{\theta_2(0 \mid \tau)} \cdot \frac{\theta_1(\nu \mid \tau)}{\theta_4(\nu \mid \tau)}, where \nu = u / \theta_3^2(0 \mid \tau), and \sn(u, k) = \sin(\am(u, k)). This ratio expression derives the elliptic functions from theta functions, enabling the inversion to obtain F(\phi, k) from the known value \sin \phi = \sn(u, k). The arithmetic-geometric mean (AGM) provides an efficient computational method for evaluating these relations, as K(k) = \pi / (2 \cdot \agm(1, \sqrt{1 - k^2})), which converges quadratically and yields \theta_3(0 \mid \tau) via the above formula after determining \tau. This acceleration is particularly useful for high-precision calculations of theta values from elliptic integrals. Historically, these connections stem from Jacobi's work in the 1820s and 1830s, where theta functions were introduced to handle the inversion of elliptic integrals, with the Legendre-Jacobi transformation relating integrals at different moduli through modular properties of theta functions.

Identity with Euler beta function

The , also known as the of the first kind, is defined as B(m,n) = \int_0^1 t^{m-1} (1-t)^{n-1} \, dt = \frac{\Gamma(m) \Gamma(n)}{\Gamma(m+n)}, for \Re(m) > 0 and \Re(n) > 0. A specific evaluation arises in the context of the , which represents the quarter-arc length of the and is given by L = \int_0^1 \frac{dt}{\sqrt{1 - t^4}}. This equals \frac{1}{4} B\left( \frac{1}{4}, \frac{1}{2} \right). To establish this, substitute s = t^4, so t = s^{1/4} and dt = \frac{1}{4} s^{-3/4} \, ds. The limits remain $0 to &#36;1, and the integral transforms to L = \frac{1}{4} \int_0^1 s^{1/4 - 1} (1 - s)^{1/2 - 1} \, ds = \frac{1}{4} B\left( \frac{1}{4}, \frac{1}{2} \right). This substitution directly yields the beta function form. The Jacobi theta function \theta_2(0 \mid i) connects to this via the lemniscate constant through the identity \theta_2(0 \mid i)^2 = \frac{2 L}{\pi} = \frac{1}{2\pi} B\left( \frac{1}{4}, \frac{1}{2} \right), or equivalently, \theta_2(0 \mid i)^4 = \frac{ B\left( \frac{1}{4}, \frac{1}{2} \right)^2 }{ 4 \pi^2 }. This relation stems from the evaluation of the theta constant at the special modular point \tau = i, where q = e^{-\pi}, and the known expression for the lemniscate constant in terms of theta functions. An alternative derivation of the beta integral connection uses the infinite product or series representation of the theta function. The Jacobi triple product identity expresses \theta_2(0 \mid \tau) as an infinite product, and at \tau = i, its series expansion \theta_2(0 \mid i) = \sum_{n=-\infty}^\infty e^{-\pi (n + 1/2)^2} can be evaluated using the Poisson summation formula, yielding the closed form consistent with the lemniscate relation. The beta function's hypergeometric representation B(m,n) = \frac{n-1}{m} \, _2F_1(1-n, m; m+1; 1) further links this to elliptic hypergeometric series, where special cases like m=1/4, n=1/2 align with theta null values through modular invariance. Generalizations extend to higher-degree theta functions and elliptic beta integrals, which incorporate nome parameters and reduce to the standard beta for limiting cases.

Partition sequences and Pochhammer products

The , a fundamental object in q-series analysis, is defined for a positive n by (a; q)_n = \prod_{k=0}^{n-1} (1 - a q^k), with (a; q)_0 = 1, and extends to the infinite case (a; q)_\infty = \prod_{k=0}^\infty (1 - a q^k) for |q| < 1. This symbol provides a q-analogue of the rising factorial and serves as a building block for generating functions in theory. The generating function for the ordinary partition numbers p(n), which count the number of ways to write n as a sum of positive integers disregarding order, is given by the reciprocal of a q-Pochhammer symbol: \sum_{n=0}^\infty p(n) q^n = \frac{1}{(q; q)_\infty}. This infinite product form, discovered by Euler, encodes the combinatorial structure of unrestricted partitions. The Dedekind eta function \eta(\tau) connects this to modular forms via \eta(\tau) = q^{1/24} (q; q)_\infty (with q = e^{2\pi i \tau}), implying the partition generating function is q^{-1/24} / \eta(\tau). Jacobi theta functions admit product representations expressible in terms of q-Pochhammer symbols, linking them directly to partition-like structures. Specifically, the null-value theta function \theta_3(0 \mid \tau) has the form \theta_3(0 \mid \tau) = (q; q)_\infty (-q; q)_\infty^2, where q = e^{\pi i \tau}, as derived from the \theta_3(z \mid \tau) = (q; q)_\infty (-q e^{2\pi i z}; q)_\infty (-q e^{-2\pi i z}; q)_\infty. This representation highlights how theta functions generalize partition products, with the infinite product capturing Gaussian sums akin to squared partition counts. Additionally, identities such as \theta_2(0 \mid \tau) \theta_3(0 \mid \tau) \theta_4(0 \mid \tau) = 2 \eta(\tau)^3 further intertwine theta functions with eta products underlying partitions. Variants of partition sequences, such as strict partitions (into distinct parts) and overpartitions (where the largest occurrence of each part may be "overlined"), yield generating functions that modify the basic Pochhammer product. The strict partition function Q(n) has generating function \sum_{n=0}^\infty Q(n) q^n = \prod_{n=1}^\infty (1 + q^n) = (-q; q)_\infty, reflecting the exclusion of repeated parts via the sign alternation in (-q; q)_\infty. For overpartitions \overline{p}(n), the generating function is \sum_{n=0}^\infty \overline{p}(n) q^n = \prod_{n=1}^\infty \frac{1 + q^n}{1 - q^n} = \frac{(-q; q)_\infty}{(q; q)_\infty}, accounting for the overline option on the largest part. These forms demonstrate how q-Pochhammer symbols generate diverse partition sequences, paralleling the product structures in theta functions and enabling connections to modular properties.

Relations among partition sequences

The generating function for the regular partition numbers p(n), which count the number of unrestricted partitions of n, is given by \sum_{n=0}^\infty p(n) q^n = \prod_{k=1}^\infty \frac{1}{1 - q^k} = q^{-1/24} \eta(\tau)^{-1}, where q = e^{2\pi i \tau} and \eta(\tau) is the . This expression connects the partition function directly to the eta function, which itself arises from identities via the representation of theta functions and modular transformation properties. For strict partitions, counted by Q(n), the number of partitions of n into distinct parts, the generating function is \sum_{n=0}^\infty Q(n) q^n = \prod_{k=1}^\infty (1 + q^k) = q^{-1/24} \frac{\eta(2\tau)}{\eta(\tau)}. This eta quotient links strict partitions to the regular partition generating function, as both are ratios involving \eta(\tau). Furthermore, by Euler's pentagonal number theorem, the reciprocal of the regular partition generating function relates to a theta series expansion: \prod_{k=1}^\infty (1 - q^k) = \sum_{n=-\infty}^\infty (-1)^n q^{n(3n-1)/2}, a bilateral sum resembling a Jacobi theta function \vartheta_3(0, q) but with quadratic exponents adjusted for the Eisenstein integers lattice. The strict partition generating function can thus be expressed as \prod (1 + q^k) = \prod (1 - q^{2k}) / \prod (1 - q^k), yielding the eta ratio above, and the one-sided sum \sum_{n=0}^\infty (-1)^n q^{n(3n-1)/2} appears in partial identities connecting to theta functions like Ramanujan's f(-q, q) = \sum_{n=-\infty}^\infty (-1)^n q^{n^2}, where \sum_{n=-\infty}^\infty (-1)^n q^{n(3n-1)/2} = \frac{\eta(\tau)^3}{f(-q, q)} holds via modular identities linking eta products to theta series. Overpartition numbers \overline{p}(n), which count partitions where the first occurrence of each distinct part may be overlined, have the generating function \sum_{n=0}^\infty \overline{p}(n) q^n = \prod_{k=1}^\infty \frac{1 + q^k}{1 - q^k} = \frac{\eta(2\tau)}{\eta(\tau)^2}. This double eta product relates overpartitions to both regular and strict cases, as it combines the distinct parts factor \prod (1 + q^k) with the unrestricted factor \prod (1 - q^k)^{-1}, forming a quadratic eta quotient that underscores the structural interplay among these sequences. These relations facilitate proofs of arithmetic properties, such as Ramanujan's congruences for the , including p(5n + 4) \equiv 0 \pmod{5}. Using theta function identities, the generating function P(q) = \sum p(n) q^n is dissected into components modular under the action of the modulo 5, revealing vanishing coefficients via the transformation properties of theta series like \sum (-1)^n q^{n^2} and their . Similar theta-based decompositions prove the congruences modulo 7 and 11.

Theta Functions in Terms of Nome

Theta functions via nome expansions

The nome expansions of Jacobi theta functions express these functions as infinite series in powers of the nome q = e^{\pi i \tau}, where \tau is the half-period ratio with \operatorname{Im} \tau > 0, ensuring |q| < 1. These expansions are particularly valuable for in the regime of small |q|, which corresponds to large \operatorname{Im} \tau, allowing by truncating the series after a few terms while capturing the dominant behavior. The third Jacobi theta function \theta_3(z \mid \tau), also denoted \theta_3(z, q), has the explicit nome \theta_3(z \mid \tau) = \sum_{n=-\infty}^{\infty} q^{n^2} e^{2 i n z} = 1 + 2 \sum_{n=1}^{\infty} q^{n^2} \cos(2 n z). This form reveals that the coefficients for each power q^{k} (where k = n^2) are zero unless k is a , with the coefficient for q^{n^2} ( n \geq 1) being $2 \cos(2 n z). For small |q|, higher-order terms diminish rapidly, so the expansion approximates \theta_3(z \mid \tau) \approx 1 + 2 q \cos(2 z) + 2 q^4 \cos(4 z), providing insight into the function's behavior far from the fundamental . The identity relates this q-series directly to an representation: \theta_3(z \mid \tau) = \prod_{n=1}^{\infty} (1 - q^{2n}) (1 + q^{2n-1} e^{2 i z}) (1 + q^{2n-1} e^{-2 i z}), which can equivalently be written as \theta_3(z \mid \tau) = \prod_{n=1}^{\infty} (1 - q^{2n}) \left(1 + 2 q^{2n-1} \cos(2 z) + q^{4n-2}\right). This identity, discovered by Jacobi, bridges the summation and product forms, facilitating proofs of modular properties and further asymptotic evaluations. The series and product both converge absolutely for |q| < 1, with on compact sets in z under suitable bounds on \operatorname{Im} z.

Nome power expansions

The nome q = e^{\pi i \tau} is a key parameter for expanding Jacobi theta null values in series that facilitate numerical computation, particularly when |q| \ll 1 (corresponding to large imaginary part of \tau). These expansions express the null values as sums over integer powers of q, converging rapidly for such q, and are fundamental for high-precision evaluations in elliptic function theory and modular forms. A primary example is the expansion of the null value \theta_3(0|\tau): \theta_3(0|\tau) = \sum_{n=-\infty}^{\infty} q^{n^2} = 1 + 2 \sum_{n=1}^{\infty} q^{n^2}. Squaring this yields \theta_3(0|\tau)^2 = \sum_{n=-\infty}^{\infty} r_2(n) q^n, where r_2(n) denotes the number of integer solutions to n = a^2 + b^2 (counting signs and order), equivalently r_2(n) = 4(d_1(n) - d_3(n)) with d_i(n) the number of divisors of n congruent to i modulo 4. This form is a modular form of weight 1 for the full modular group \mathrm{SL}(2,\mathbb{Z}), useful in number-theoretic applications like counting lattice points. Using the Jacobi identity \theta_3(0|\tau)^2 = \theta_2(0|\tau)^2 + \theta_4(0|\tau)^2, the expansion can also incorporate contributions from \theta_2(0|\tau)^2 = 4 \left( \sum_{m=1,3,5,\dots}^{\infty} q^{m^2/4} \right)^2, where the inner sum generates terms in powers of q^{1/4}, though integer-power truncations suffice for leading approximations in computational contexts. Closely related is the Dedekind eta function \eta(\tau), linked to theta null values via \eta(\tau)^3 = \frac{\theta_2(0|\tau) \theta_3(0|\tau) \theta_4(0|\tau)}{\pi}, with its logarithmic expansion providing another tool for nome-based computations (note the eta nome q_\eta = e^{2\pi i \tau} = q^2): \log \eta(\tau) = \frac{\pi i \tau}{12} - \sum_{k=1}^{\infty} \frac{\sigma(k)}{k} q_\eta^{k}, where \sigma(k) = \sum_{d|k} d is the sum-of-divisors function. This series derives from expanding the product form \eta(\tau) = q_\eta^{1/24} \prod_{n=1}^{\infty} (1 - q_\eta^n) via logarithms of geometric series. As q \to 0, the leading terms dominate: \theta_3(0|\tau)^2 \approx 1 + 4q + 4q^4 + 8q^5 + O(q^8), reflecting the initial coefficients from representations as sums of two squares (e.g., r_2(0) = 1, r_2(1) = 4, r_2(4) = 4, r_2(5) = 8). Similarly, \log \eta(\tau) \approx \frac{\pi i \tau}{12} - q_\eta - \frac{3}{2} q_\eta^2 + O(q_\eta^3). These asymptotics enable quick estimates for large Im(\tau), essential in analytic number theory. For high-precision computation, algorithms exploit these nome expansions by evaluating truncated series after modular transformations to ensure small |q|. The Borweins developed efficient iterations based on the arithmetic-geometric mean (AGM), which compute theta null values equivalently to elliptic integrals (e.g., \theta_3(0| \tau)^2 = \frac{2K(k)}{\pi}, with nome linked to the modulus k), achieving quadratic convergence for thousands of digits. Modern implementations, such as those in the Arb library, directly sum q-series terms with optimized rectangular evaluation, attaining O(p^{1.5}) bit complexity for p-bit precision by taking O(\sqrt{p}) terms and reducing arguments via the fundamental domain. These methods are particularly effective for nome expansions, avoiding slower direct summation near the boundary of the upper half-plane.

References

  1. [1]
    [PDF] Notes on the Poisson Summation Formula, Theta Functions, and the ...
    Mar 11, 2020 · Definition (Jacobi theta function). The Jacobi theta function is the function of two complex variables given by. Θ(z, τ) = ∞. ∑ n ...
  2. [2]
    Jacobi Theta Functions -- from Wolfram MathWorld
    Jacobi, C. G. J. Fundamenta Nova Theoriae Functionum Ellipticarum. Königsberg, Germany: Regiomonti, Sumtibus fratrum Borntraeger, 1829. Reprinted in ...
  3. [3]
    [PDF] Riemann's theta function - Penn Math
    (1.1) DEFINITION. ( i) The Riemann theta function θ(z;Ω) of genus g is the holomorphic func- tion in two variables (z,Ω) ∈ Cg ×Hg, defined by the theta series. ...
  4. [4]
    [PDF] 10 Applications of Theta - Functions - Princeton University
    1 Product formula for the Jacobi theta function. In its most elaborate form, Jacobi's theta function is defined for z ∈ C and τ ∈ H by. (1). Θ(z|τ) = ∞. X n ...
  5. [5]
    Fundamenta nova theoriae functionum ellipticarum - Internet Archive
    Jan 27, 2009 · Publication date: 1829 ; Publisher: Sumtibus fratrumBorntraeger ; Collection: americana ; Book from the collections of: University of Michigan.
  6. [6]
    DLMF: §20.2 Definitions and Periodic Properties ‣ Properties ‣ Chapter 20 Theta Functions
    ### Summary of Characteristic Notation for Jacobi Theta Functions
  7. [7]
    [PDF] Michael Stone - Theta functions with characteristics θ a b (z|τ)
    meaning that an R = 1 boson with periodic boundary conditions on a torus decomposes in a sum of fermion partition functions with both chiralities having ...
  8. [8]
    DLMF: §20.3 Graphics ‣ Properties ‣ Chapter 20 Theta Functions
    θ j ⁡ ( z , q ) : theta function, π : the ratio of the circumference of a circle to its diameter, e : base of natural logarithm, i : imaginary unit and q
  9. [9]
    DLMF: §20.7 Identities ‣ Properties ‣ Chapter 20 Theta Functions
    This reference also gives the eleven additional identities for the permutations of the four theta functions.
  10. [10]
    DLMF: Chapter 20 Theta Functions
    Chapter 20 covers notation, definitions, periodic properties, power series, identities, Watson's expansions, relations to other functions, integrals, and ...
  11. [11]
    A course of modern analysis; an introduction to the general theory of ...
    Nov 18, 2008 · A course of modern analysis; an introduction to the general theory of infinite processes and of analytic functions; with an account of the principal ...
  12. [12]
    DLMF: §22.2 Definitions ‣ Properties ‣ Chapter 22 Jacobian Elliptic Functions
    ### Summary of Jacobi Elliptic Functions in Terms of Theta Functions
  13. [13]
    Introduction to the Jacobi theta functions
    The best-known properties and formulas for the Jacobi theta functions. Values ... periodicity and quasi‐periodicity of the theta functions by variable : ...
  14. [14]
    [PDF] Math 213a (Fall 2024) Yum-Tong Siu 1 THETA FUNCTIONS OF ...
    We now look at the Jacobi approach. In the approach of Weierstrass, for a given period lattice L = Zω1 + Zω2, only one basic elliptic function is considered, ...
  15. [15]
    20.5 Infinite Products and Related Results
    With the given conditions the infinite series in (20.5.10)–(20.5.13) converge absolutely and uniformly in compact sets in the z-plane.Missing: representation | Show results with:representation
  16. [16]
    Elliptic Integral Singular Value--k_1 -- from Wolfram MathWorld
    The first singular value k_1 of the elliptic integral of the first kind K(k), corresponding to K^'(k_1)=K(k_1), (1) is given by k_1 = 1/(sqrt(2)) (2) k_1^' ...
  17. [17]
    [PDF] jacobi thetanullwerte, periods of elliptic curves and minimal equations
    [9]):. √ k = θ2(τ) θ3(τ). ,. K = π. 2 θ3(τ)2. If we depart from a given τ ∈ H, we can use these two formulas to find directly the value of K(k) ...
  18. [18]
    Lemniscate Constant -- from Wolfram MathWorld
    theta_4 is a Jacobi theta function, K(k) is a complete elliptic integral of the first kind, and R_D , R_F , and R_K are Carlson elliptic integrals. Todd ...
  19. [19]
    Equianharmonic Case -- from Wolfram MathWorld
    The case of the Weierstrass elliptic function with invariants g_2=0 and g_3=1. The corresponding real half-period is given by omega_2 = (Gamma^3(1/3))/(4pi) ...Missing: theta | Show results with:theta
  20. [20]
    23.22 Methods of Computation
    Given ω 1 and ω 3 , with ℑ ⁡ ( ω 3 / ω 1 ) > 0 , the nome q is computed from q = e i ⁢ π ⁢ ω 3 / ω 1 . For ℘ ⁡ ( z ) we apply (23.6.2) and (23.6.5), ...
  21. [21]
    Full article: The degenerating behaviour of Jacobi's theta functions
    In this note, we give the degenerating behaviour of Jacobi's theta functions as the modulus k tends to one. Keywords: Jacobian elliptic function · Jacobian ...Missing: primary | Show results with:primary<|control11|><|separator|>
  22. [22]
    [PDF] RAMANUJAN'S THEORY OF THETA FUNCTIONS Bruce Berndt
    Twelve values of n, namely, n = 2, 3, 4, 5, 7, 11, 15, 17, 19, 23,. 31, and 35. n = 2 and 4 are in Chapter 17 in Ramanujan's second notebook; the remaining ...
  23. [23]
    [PDF] An identity relating a theta function to a sum of Lambert series
    We derive an identity connecting a theta function and a sum of Lambert series. As a consequence of this identity, we deduce a number of results of Jacobi, ...
  24. [24]
    [PDF] A Cubic Counterpart of Jacobi's Identity and the AGM Author(s)
    The arithmetic-geometric mean iteration lies at the heart of elliptic function and theta function theory [4]. In Chapter 21 of Ra- manujan's second notebook [3] ...
  25. [25]
    Raabe formula and integrals for Jacobi theta functions - arXiv
    Sep 22, 2016 · Short proofs for q-Raabe formula and integrals for Jacobi theta functions. We shall answer a question of Mező on the q-analogue of the Raabe's ...
  26. [26]
    DLMF: §20.14 Methods of Computation ‣ Computation ‣ Chapter 20 Theta Functions
    ### Summary of Integrals and Theta Functions from §20.14 (DLMF)
  27. [27]
    [PDF] The Classical Theta Function and the Riemann Zeta Function
    Apr 3, 2019 · We then apply the Mellin transform on the theta function to “obtain” the Riemann zeta function up to some scaling and correction terms.
  28. [28]
    [PDF] Lecture 2 : Functional equation of the Riemann ζ-function
    The proof exploits the theta function θ : R>0 → C given by: θ(t) := ∑ n∈Z e−πn2t. We want to view this function as a Mellin transform. Definition 4. Let ...
  29. [29]
    [PDF] Minima of Epstein's Zeta Function and Heights of Flat Tori
    In particular we prove an analog of Theorem 1 for the theta function, and using the fact that the Epstein Zeta function is the Mellin transform of the theta ...
  30. [30]
    ITERATED INTEGRALS AND EPSTEIN ZETA FUNCTIONS WITH ...
    certain harmonic rational function coefficients which (when a polynomial coefficient theta function is added) satisfy the same transformation formula. (but ...
  31. [31]
    [PDF] Probability laws related to the Jacobi theta and Riemann zeta ...
    As explained in Section 4.6, some of these identities are equivalent to the functional equation for the Jacobi theta (or the Riemann zeta) function. Rows 0 and ...
  32. [32]
    Riemann's 1859 Manuscript - Clay Mathematics Institute
    Riemann checked the first few zeros of the zeta function by hand. They satisfy his hypothesis. By now over 1.5 billion zeros have been checked by computer. Very ...Missing: theta | Show results with:theta
  33. [33]
    [PDF] Weierstrass function (z) and infinite products - HAL
    Sep 21, 2022 · d2 dz2 log sin(z) = 1 z2. +. X k. 1. (z - kπ)2 . This may explain the reason that ℘ could be expressed as series of the function. 1. (sin π(z+kτ)) ...Missing: dz² θ₁(
  34. [34]
    https://doi.org/10.1016/j.jmaa.2018.11.051
  35. [35]
    [1805.01121] On a theta product of Jacobi and its applications to $q
    May 3, 2018 · We give a new proof for a product formula of Jacobi which turns out to be equivalent to a q-trigonometric product which was stated without proof ...
  36. [36]
    [PDF] A BASIC JACOBI THETA FUNCTION Define θ(z, τ) = X
    But this writeup will briefly describe how the theta function solves the heat equation. Theta functions in general are power- fully versatile. David Mumford's ...
  37. [37]
    [PDF] Theta functions and the metaplectic group 1 Introduction
    Aug 4, 2023 · Theta functions provide an important explicit example of a function that turns out to be automorphic as a result of non-trivial input. This ...
  38. [38]
    [PDF] A Selective History of the Stone-von Neumann Theorem - UMD MATH
    This relationship is connected to the theory of theta functions, hence the name. In this context it is worth recalling the opening sentences of Weil's paper [64]:.
  39. [39]
    [PDF] from classical theta functions to topological quantum field theory
    There is an action of a finite Heisenberg group on theta functions which in- duces the Hermite-Jacobi action via a Stone-von Neumann theorem (Weil).
  40. [40]
    [PDF] Analytic continuation, functional equation: examples 1. Dirichlet L ...
    Oct 24, 2018 · ... L-function via an integral representation from a theta series, the theta series must mimic the formation of the L-function. For Hecke L ...
  41. [41]
    [PDF] The ideal class number formula for an imaginary quadratic field
    F = Q( √ n) = {a + b √ n : a, b ∈ Q}, n ∈ Z − {0, 1} squarefree. If n is positive then F is a real quadratic number field, and if n is negative then F is an ...
  42. [42]
    Ramanujan Theta Functions -- from Wolfram MathWorld
    phi^'(q)=sqrt((theta_2(0,q). (27). where theta_i(0,q) is again a Jacobi theta function, which has special value. phi^'(-e^(-pisqrt(3)))= (28). See also. Jacobi ...
  43. [43]
    DLMF: §20.11 Generalizations and Analogs ‣ Properties ‣ Chapter ...
    Multidimensional theta functions with characteristics are defined in §21.2(ii) and their properties are described in §§21.3(ii), 21.5(ii), and 21.6. For ...
  44. [44]
    Mock Theta Function -- from Wolfram MathWorld
    Ramanujan defined 17 Jacobi theta function-like functions F(q) with |q|<1 which he called "mock theta functions"
  45. [45]
    Ramanujan's mock theta functions - PNAS
    In his famous deathbed letter, Ramanujan introduced the notion of a mock theta function, and he offered some alleged examples.
  46. [46]
    [PDF] theta-function identities, explicit values for ramanujan's continued ...
    Aug 4, 2023 · In this paper, we are concerned with the theta-function identities, explicit values, partition-theoretic results, and some matching coefficients ...
  47. [47]
    [2305.14988] Some Identities of Ramanujan's q-Continued Fractions ...
    May 24, 2023 · We establish some theta-function identities for the continued fractions and derive some colour partition identities as applications. Some ...
  48. [48]
    [PDF] The period matrices and theta functions of Riemann - Penn Math
    This article deals with four notions due to Riemann: (A) Riemann bilinear relations, (B) Rie- mann forms, (C) Riemann theta functions and (D) Riemann's ...Missing: Torelli | Show results with:Torelli
  49. [49]
    [PDF] Computing Riemann Theta Functions - FSU Math
    Jun 7, 2002 · Abstract. The Riemann theta function is a complex-valued function of g complex variables. It appears in the construction of many (quasi-) ...
  50. [50]
    [PDF] Siegel Modular Forms Lecture #6 - Gerard van der Geer Home Page
    Oct 29, 2020 · Constructing Siegel Modular. Forms. Theta Characteristics. Consider Riemann's theta function on Hg × C g θ(τ,z) = X m∈Z g e πim. ′ τm+2πim. ′ z.Missing: multivariable | Show results with:multivariable
  51. [51]
    [PDF] LECTURE 12: THETA DIVISOR 12.1. Zeroes of the Riemann theta ...
    The set of zeroes of theta function is called theta divisor. The goal of this lecture is to describe the theta divisor in terms of divisors on X. Theorem 12.1.
  52. [52]
    [math/0611810] Theta functions on the theta divisor - arXiv
    Nov 27, 2006 · It can be seen that the zero locus of this theta function essentially gives the ramification locus of the Gauss map. For jacobians this leads to ...
  53. [53]
    [PDF] arXiv:2307.10486v1 [math.AG] 19 Jul 2023
    Jul 19, 2023 · Theorem 1.1 (Torelli's Theorem). Let τ, ˜τ ∈ Hg be period matrices of Riemann surfaces Γ, ˜Γ of genus g ≥ 1 respectively. These last ones are ...<|control11|><|separator|>
  54. [54]
    https://arxiv.org/pdf/2403.17829.pdf
  55. [55]
    [PDF] THETA LIFTS AND LOCAL MAASS FORMS Kathrin Bringmann, Ben ...
    Sep 9, 2012 · These functions map to weakly holomorphic modular forms under the hyperbolic Laplacian. The paper is organized as follows. In Section 2, we ...
  56. [56]
    [PDF] arithmetic properties of coefficients of half-integral weight maass ...
    We explicitly represent the coefficients of certain half-integral weight Maass-Poincaré series as traces of singular moduli. This result (see Theorem 1.2) ...
  57. [57]
    [PDF] CM points and weight 3/2 modular forms
    Then I(τ,F) is a (in general non-holomorphic) modular form of weight. 3/2 for a congruence subgroup of SL2(Z). One key feature of the theta kernel is its very ...<|control11|><|separator|>
  58. [58]
    poincaré series and the divisors of modular forms
    Jun 3, 2010 · Since in general these Poincaré series are not meromorphic functions on the complex upper half plane, we cannot use the valence formula or the ...
  59. [59]
    [PDF] Rankin-Cohen brackets and Serre derivatives as Poincaré series
    It is easy to show that every modular form f (of weight k ≥ 4) can be written as a Poincaré series. Pk(φ): because f can always be written as a linear ...
  60. [60]
    Complete Elliptic Integral of the First Kind -- from Wolfram MathWorld
    The complete elliptic integral of the first kind K(k), illustrated above as a function of the elliptic modulus k, is defined by K(k) = F(1/2pi,k) (1) ...Missing: Gamma( | Show results with:Gamma(
  61. [61]
    Beta Function -- from Wolfram MathWorld
    The beta function B(p,q) is the name used by Legendre and Whittaker and Watson (1990) for the beta integral (also called the Eulerian integral of the first ...Missing: theta_2( | Show results with:theta_2(
  62. [62]
    q-Pochhammer Symbol -- from Wolfram MathWorld
    The q-analog of the Pochhammer symbol defined by (a;q)_k={product_(j=0)^(k-1)(1-aq^j) if k>0; 1 if k=0; product_(j=1)^(|k|)(1-aq^(-j))^(-1) if k<0; ...
  63. [63]
    Partition Function P -- from Wolfram MathWorld
    ### Generating Function for Partition Function p(n)
  64. [64]
    Partition Function Q -- from Wolfram MathWorld
    The partition function q(n,k) gives the number of partitions of n with k or fewer addends, or equivalently, into partitions with no element greater than k.
  65. [65]
    [PDF] OVERPARTITIONS 1. introduction A partition of n is a non ...
    From the definition and the generating function (1.1) it follows that overpartitions can be viewed through a number of different lenses. For instance, the ...<|control11|><|separator|>
  66. [66]
    A000009 - OEIS
    Partitions into distinct parts are sometimes called "strict partitions". ... Expansion of q^(-1/24) * eta(q^2) / eta(q) in powers of q. Expansion of q ...
  67. [67]
    Efficient computation of the overpartition function and applications
    Dec 1, 2023 · In this paper we develop a method to calculate the overpartition function efficiently using a Hardy-Rademacher-Ramanujan type formula.
  68. [68]
    [PDF] On the equivalence of two fundamental theta identities - arXiv
    Sep 18, 2014 · While they had roots in the work of Jakob Bernoulli and Euler, they were introduced in full generality, depending on two arguments, by Jacobi.Missing: primary | Show results with:primary
  69. [69]
    [PDF] Dedekind Eta Function & Hurwitz Polynomials
    \mathcal{N}_{r}(n_{0}) :=\{n_{0}l^{2}+r(l^{2}-1)/24|l\in \mathbb{N}, (l, 2\cdot 3)=1\}. Let a_{r}(n_{0})=0 . Then a_{r}(n)=0 for all n\in \mathcal ...Missing: theta_2( | Show results with:theta_2(<|control11|><|separator|>
  70. [70]
    [PDF] Numerical Evaluation of Elliptic Functions, Elliptic Integrals ... - arXiv
    Jun 18, 2018 · Abstract We describe algorithms to compute elliptic functions and their rela- tives (Jacobi theta functions, modular forms, ...<|control11|><|separator|>