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Argument principle

The argument principle, also known as Cauchy's argument principle, is a central theorem in complex analysis that establishes a relationship between the zeros and poles of a meromorphic function inside a simple closed contour and the change in the argument of the function along that contour.^[1]^[2] Specifically, for a meromorphic function f(z) analytic inside and on a simple closed positively oriented contour \gamma except for isolated zeros and poles (none on \gamma), the principle states that

\frac{1}{2\pi i} \int_\gamma \frac{f'(z)}{f(z)} \, dz = N - P,

where N is the total number of zeros inside \gamma counted with multiplicity, and P is the total number of poles inside \gamma counted with multiplicity.^[1]^[2] This integral equals the winding number of the curve f(\gamma) around the origin in the complex plane, which is the net change in the argument of f(z) as z traverses \gamma, divided by $2\pi.^[1]^[2] The proof of the argument principle follows directly from the residue theorem applied to the meromorphic function \frac{f'(z)}{f(z)}, whose residues at the zeros of f equal the multiplicities of those zeros and at the poles equal the negative of their orders, yielding the difference N - P.^[1]^[2] A geometric interpretation emphasizes the topological aspect: the number of times f(\gamma) encircles the origin reflects the imbalance between zeros and poles enclosed by \gamma.^[2] Among its most notable applications, the argument principle underpins Rouché's theorem, which facilitates counting zeros in regions by comparing functions, and provides a proof of the fundamental theorem of algebra by showing that non-constant polynomials have zeros.^[1]^[2] It also extends to the Nyquist stability criterion in control theory, where it assesses the stability of feedback systems by analyzing the encirclements of the critical point -1 in the Nyquist plot.^[1]^[2] Generalized versions apply to functions with branch points or in more abstract settings, such as Riemann surfaces, highlighting its enduring influence in modern mathematics and engineering.^[2]

Fundamentals

Statement of the Theorem

The argument principle, a fundamental result in complex analysis, asserts that if f is a meromorphic function in a domain containing a simple closed contour C and its interior, then

\frac{1}{2\pi i} \int_C \frac{f'(z)}{f(z)} \, dz = N - P,

where N is the number of zeros of f inside C counted with multiplicity, and P is the number of poles of f inside C counted with multiplicity.^[3]^[2] This result holds under the assumptions that f has finitely many zeros and poles inside C, C is positively oriented (counterclockwise), and f has no zeros or poles on C itself.^[3]^[2] An equivalent formulation expresses the principle in terms of the change in argument: as z traverses C once, the total change in \arg(f(z)) is $2\pi (N - P).^[3]^[2] For example, consider f(z) = z^2 - 1 and the unit circle |z| = 2; here N = 2 (zeros at z = \pm 1) and P = 0, so the integral equals 2 and the argument change is $4\pi.^[3]

Prerequisites and Notation

The argument principle is a fundamental result in complex analysis that relies on several key concepts from the field. A holomorphic function is defined on an open set \Omega \subset \mathbb{C} as a function f: \Omega \to \mathbb{C} that is complex differentiable at every point in \Omega, meaning the limit \lim_{h \to 0} \frac{f(z_0 + h) - f(z_0)}{h} exists for each z_0 \in \Omega.^[4] Holomorphic functions possess strong properties, such as the existence of power series expansions locally around each point. Meromorphic functions extend this notion: a function f is meromorphic on \Omega if it is holomorphic on \Omega except at a discrete set of isolated points where it has poles of finite order.^[4]^[5] Contours provide the framework for integration in the complex plane. A contour is a piecewise smooth curve \gamma: [a, b] \to \mathbb{C} where the parametrization z(t) is continuous, differentiable on subintervals with z'(t) \neq 0, and the components join end-to-end.^[6] A simple closed contour C is one where z(a) = z(b) and the curve does not intersect itself except at the endpoints, enclosing a bounded interior domain D. Such contours are assumed to be positively oriented, meaning they are traversed counterclockwise, which ensures the interior D lies to the left of the direction of travel and facilitates consistent application of integration theorems.^[6] Cauchy's integral theorem and formula form essential prerequisites. The theorem states that if f is holomorphic on a simply connected domain containing a simple closed contour C and its interior D, then \int_C f(z) \, dz = 0.^[4] The integral formula extends this: if f is holomorphic in D \cup C and z_0 \in D, then f(z_0) = \frac{1}{2\pi i} \int_C \frac{f(z)}{z - z_0} \, dz.^[7] These results underpin the analysis of function behavior inside contours. Standard notation for the argument principle involves a simple closed positively oriented contour C with interior domain D, and a function f that is holomorphic or meromorphic in D \cup C. Zeros and poles of f are counted with multiplicity: a zero at z_0 \in D has multiplicity m if f(z) = (z - z_0)^m g(z) for some holomorphic g with g(z_0) \neq 0.^[8] Similarly, a pole of order m at z_0 occurs if $1/f has a zero of multiplicity m there. The argument function \arg(f(z)) denotes the angle that the complex number f(z) makes with the positive real axis, defined up to multiples of $2\pi.^[9] The use of simple closed contours with positive orientation is crucial, as it defines a well-oriented boundary for D and aligns with the conventions of Cauchy's theorems, ensuring the principle correctly relates changes along C to interior features of f.

Interpretation

Geometric Interpretation

The argument principle provides a geometric lens through which the behavior of a meromorphic function f(z) can be understood by examining how its image under a closed contour C interacts with the origin in the complex plane. Specifically, the integral \frac{1}{2\pi i} \oint_C \frac{f'(z)}{f(z)} \, dz equals the net winding number of the curve f(C) around 0, which quantifies the total number of times the image curve encircles the origin as z traverses C once in the positive direction.^[10] This winding captures the topological wrapping of the function's values, linking algebraic features like zeros and poles to a visual measure of rotation in the range plane.^[11] As z moves along the contour C, the argument of f(z), denoted \arg(f(z)), undergoes a total change equal to $2\pi times the net winding number of f(C) around 0. This change in argument reflects the cumulative angular displacement of the function's image, where each full counterclockwise rotation contributes positively and clockwise negatively. For a function with no zeros or poles on C, this net change directly corresponds to the difference between the number of zeros N and poles P inside C, counting multiplicities, as the principle asserts \frac{1}{2\pi i} \oint_C \frac{f'(z)}{f(z)} \, dz = N - P.^[10] Geometrically, zeros inside C induce positive windings, as the function maps nearby points to values that spiral toward the origin, causing f(C) to encircle 0 counterclockwise. In contrast, poles contribute negative windings, where the image spirals away from the origin, resulting in clockwise encirclements of 0. This opposition leads to the net count N - P, providing an intuitive balance between attractive (zeros) and repulsive (poles) singularities in the function's mapping.^[11] A simple visualization illustrates this for basic cases: consider f(z) = z - a with a zero at a inside C; as z traverses C, f(C) is a translated copy of C that winds once counterclockwise around 0, yielding a positive contribution of 1. For f(z) = 1/(z - b) with a pole at b inside C, the image f(C) winds once clockwise around 0, contributing -1, as the inversion reverses the orientation.^[10]

Relation to Winding Numbers

The winding number of a closed curve \gamma around a point a not on \gamma, denoted \mathrm{wind}(\gamma, a), is defined as

\mathrm{wind}(\gamma, a) = \frac{1}{2\pi i} \int_\gamma \frac{dz}{z - a}.

This integer measures the net number of times \gamma encircles a in the counterclockwise direction.^[12]^[13] The argument principle establishes a direct connection between this topological quantity and the behavior of holomorphic functions. For a holomorphic function f that is nowhere zero on a simple closed positively oriented contour C, the total change in the argument of f(z) as z traverses C, denoted \Delta_C \arg f(z), satisfies

\frac{\Delta_C \arg f(z)}{2\pi} = \mathrm{wind}(f(C), 0),

where f(C) is the image of C under f. This equality indicates that the variation in the phase of f along C corresponds precisely to the number of times the curve f(C) winds around the origin in the complex plane.^[14]^[12] For a meromorphic function f with no zeros or poles on C, the argument principle extends this relation by incorporating the zeros and poles inside C. Specifically,

\mathrm{wind}(f(C), 0) = N - P,

where N is the number of zeros of f inside C counted with multiplicity, and P is the number of poles inside C counted with multiplicity. Thus, the winding number of the image curve around 0 equals the net excess of zeros over poles, providing a topological interpretation of the function's analytic structure.^[15]^[13] Under the assumption that f has no zeros or poles on C, the image f(C) forms a closed curve in the complex plane that does not pass through the origin. This ensures that the winding number \mathrm{wind}(f(C), 0) is well-defined and finite, as the curve avoids the singularity at 0, allowing for a continuous argument along f(C).^[12]^[14]

Proof

Proof via Residue Theorem

To prove the argument principle using the residue theorem, consider a function f(z) that is meromorphic in a domain containing a simple closed positively oriented contour C and its interior, with f(z) analytic and nonzero on C. The singularities of f inside C are isolated poles, and f has finitely many zeros and poles inside C. Define the logarithmic derivative g(z) = \frac{f'(z)}{f(z)}, which is meromorphic in the same domain.^[2]^[16]^[17] The function g(z) has simple poles precisely at the zeros and poles of f(z), and is holomorphic elsewhere inside and on C. To see this, suppose z_0 is an isolated zero of f of order m > 0, so near z_0, f(z) = (z - z_0)^m h(z) where h(z_0) \neq 0 and h is holomorphic at z_0. Then,

g(z) = \frac{f'(z)}{f(z)} = \frac{m(z - z_0)^{m-1} h(z) + (z - z_0)^m h'(z)}{(z - z_0)^m h(z)} = \frac{m}{z - z_0} + \frac{h'(z)}{h(z)},

where \frac{h'(z)}{h(z)} is holomorphic at z_0. Thus, g(z) has a simple pole at z_0 with residue m.^[2]^[16] Similarly, if z_0 is a pole of f of order n > 0, then near z_0, f(z) = (z - z_0)^{-n} k(z) where k(z_0) \neq 0 and k is holomorphic at z_0. Differentiating gives

f'(z) = -n (z - z_0)^{-n-1} k(z) + (z - z_0)^{-n} k'(z),

g(z) = \frac{f'(z)}{f(z)} = \frac{-n (z - z_0)^{-n-1} k(z) + (z - z_0)^{-n} k'(z)}{(z - z_0)^{-n} k(z)} = -\frac{n}{z - z_0} + \frac{k'(z)}{k(z)},

where \frac{k'(z)}{k(z)} is holomorphic at z_0. Hence, g(z) has a simple pole at z_0 with residue -n.^[2]^[16] By the residue theorem applied to the meromorphic function g(z) over the contour C,

\int_C g(z) \, dz = \int_C \frac{f'(z)}{f(z)} \, dz = 2\pi i \sum \operatorname{Res}(g, z_k),

where the sum is over all singularities z_k of g inside C. The residues sum to \sum m_j - \sum n_k, where the m_j are the orders of the zeros of f inside C (total number N, counted with multiplicity) and the n_k are the orders of the poles (total number P, counted with multiplicity). Thus,

\int_C \frac{f'(z)}{f(z)} \, dz = 2\pi i (N - P),

and dividing by $2\pi i yields

\frac{1}{2\pi i} \int_C \frac{f'(z)}{f(z)} \, dz = N - P.

This establishes the argument principle, as g(z) is holomorphic except at the zeros and poles of f, and the contour avoids these points.^[2]^[16]^[17]

Proof Using Argument Changes

The proof using argument changes directly examines the total variation in the argument of the meromorphic function f(z) as z traverses the closed contour C, relating it to the zeros and poles inside C. Assume C is a positively oriented simple closed contour, f is meromorphic in a domain containing C and its interior, with no zeros or poles on C, and the singularities inside C are isolated. Parameterize C by a smooth map \gamma: [0, 1] \to \mathbb{C} such that \gamma(0) = \gamma(1) and \gamma'(t) \neq 0 for all t \in [0,1]. The image curve is f \circ \gamma, and the total change in argument along this path is \Delta \arg (f \circ \gamma) = \int_0^1 \frac{d}{dt} \arg f(\gamma(t)) \, dt.^[18] To express this change rigorously, note that for a smooth path w(t) in \mathbb{C} \setminus \{0\}, the differential of the argument is d \arg w = \Im (dw / w). Substituting w(t) = f(\gamma(t)), we have dw = f'(\gamma(t)) \gamma'(t) \, dt, so

d \arg f(\gamma(t)) = \Im \left( \frac{f'(\gamma(t)) \gamma'(t) \, dt}{f(\gamma(t))} \right).

Integrating gives the total change

\Delta \arg f = \Im \int_0^1 \frac{f'(\gamma(t))}{f(\gamma(t))} \gamma'(t) \, dt = \Im \int_C \frac{f'(z)}{f(z)} \, dz,

where the last equality follows from the change of variables dz = \gamma'(t) \, dt along the path C. This path integral representation links the argument variation to the logarithmic derivative f'/f.^[18] A fuller derivation connects this to the complex logarithm. Consider \log f(z) = \ln |f(z)| + i \arg f(z), so d \log f = df / f = d \ln |f| + i \, d \arg f. Along the closed path C, the real part satisfies \Delta \ln |f| = 0 because |f| returns to its initial value. Thus,

\int_C \frac{df}{f} = i \Delta \arg f,

or equivalently,

\Delta \arg f = \frac{1}{i} \int_C \frac{f'(z)}{f(z)} \, dz = -i \int_C \frac{f'(z)}{f(z)} \, dz.

This shows that the argument change is purely imaginary times the contour integral of the logarithmic derivative. To evaluate it, consider the local behavior at the singularities, which determines the net contribution.^[18] Near a zero of order m at z_0 inside C, write f(z) = (z - z_0)^m g(z) where g is analytic and g(z_0) \neq 0. On a small circle |z - z_0| = \epsilon traversed positively, \arg (z - z_0) increases by $2\pi, while \arg g(z) changes by approximately 0 for small \epsilon. Thus, the argument of f(z) changes by $2\pi m. Similarly, near a pole of order m at z_0, f(z) = (z - z_0)^{-m} h(z) with h(z_0) \neq 0 and h analytic, so the argument changes by -2\pi m on the small circle. To compute the global change, deform C (via homotopy, assuming no other singularities crossed) into small circles around each zero and pole, connected by line segments that cancel in pairs (forth and back changes in argument sum to zero). The net \Delta \arg f along the original C is therefore the sum of local contributions: $2\pi N - 2\pi P = 2\pi (N - P), where N and P are the numbers of zeros and poles inside C, counted with multiplicity. This establishes the relation without relying on global residue summation.^[18]

Applications

Zero and Pole Counting

The argument principle enables the counting of zeros and poles of a meromorphic function f inside a simple closed contour C by computing the contour integral

\frac{1}{2\pi i} \oint_C \frac{f'(z)}{f(z)} \, dz = N - P,

where N is the number of zeros and P the number of poles inside C, both counted with multiplicity.^[2] This integral can be evaluated analytically via the residue theorem, since \frac{f'(z)}{f(z)} is meromorphic with simple poles at the zeros and poles of f, and the residue at a zero of multiplicity m is +m while at a pole of order m it is -m.^[19] Alternatively, for numerical computation, the integral equals the winding number of the image curve f(C) around the origin, given by

\frac{1}{2\pi} \Delta_C \arg f(z) = N - P,

which can be approximated by discretizing the contour and tracking the continuous change in the argument of f(z) along it, ensuring branches are handled to avoid jumps exceeding \pi.^[14] For polynomials, which have no poles (P=0), the argument principle on a sufficiently large circle |z|=R (with R chosen to enclose all zeros) yields N = degree of the polynomial. On such a circle, f(z) \approx a_n z^n where a_n is the leading coefficient and n the degree, so \arg f(z) \approx \arg a_n + n \arg z, and traversing the circle once produces an argument change of $2\pi n.^[13] A representative example is f(z) = z^2 - 1 on the contour C: |z-1| = 1. This circle encloses the zero at z=1 (simple zero) but not at z=-1, with no poles. Parameterizing C and computing \Delta_C \arg f(z) gives $2\pi, so N - P = 1, confirming one zero inside.^[14] For the sine function on |z| = \pi/2, the contour encloses the simple zero at z=0 with no others or poles inside; the argument change along the circle is $2\pi, yielding N - P = 1.^[19] The counts N and P include multiplicities, so a zero of order m contributes m to N. For instance, in f(z) = (z-1)(z-i)^3(z-2)^5 on |z| = 1.5 centered at the origin, the contour encloses the zero at z=1 (multiplicity 1) and at z=i (multiplicity 3), giving \oint_C \frac{f'(z)}{f(z)} \, dz = 8\pi i, so N - P = 4. A key limitation is that f must have no zeros or poles on C itself, as this would make \frac{f'(z)}{f(z)} undefined there; if such points exist, the contour can be deformed slightly to avoid them while preserving the enclosed region.^[2] The method assumes f is meromorphic inside and on C with finitely many singularities.^[14]

Connections to Other Theorems

The argument principle serves as a foundational tool for proving Rouché's theorem, which states that if f and g are holomorphic inside and on a closed contour C, with no zeros of f on C, and |g(z)| < |f(z)| for all z on C, then f and f + g have the same number of zeros inside C, counting multiplicities.^[19] The proof applies the argument principle to the function h(z) = f(z) + g(z) divided by f(z), showing that the change in argument of h(z)/f(z) along C is zero because |g(z)/f(z)| < 1 implies the image lies inside the unit disk without encircling the origin, thus equating the zero counts of f + g and f.^[20] The argument principle provides a proof of the fundamental theorem of algebra, which states that every non-constant polynomial with complex coefficients has at least one complex root. For a polynomial p(z) of degree n \geq 1, consider a large circle |z| = R enclosing all roots. As R \to \infty, p(z) \sim a_n z^n, so the image p(C) winds around the origin n times, giving N = n zeros inside C by the argument principle (no poles). Thus, p(z) has exactly n roots counting multiplicity.^[2] Hurwitz's theorem, which asserts that if a sequence of holomorphic functions \{f_n\} converges uniformly on compact subsets to a holomorphic function f on a domain, then either f is identically zero or the zeros of f are limits of zeros of the f_n, relies on the argument principle to preserve zero counts under such convergence.^[19] Specifically, for a compactly contained contour, the argument principle applied to f_n shows that the number of zeros inside remains bounded, and uniform convergence ensures the argument change for f matches the limit, preventing sudden disappearance of zeros unless f \equiv 0. In control theory, the argument principle underpins the Nyquist stability criterion, which determines the stability of a closed-loop feedback system. For a transfer function G(s), the Nyquist plot of G(j\omega) as \omega varies from -\infty to \infty (completed with a semicircle in the right half-plane) is analyzed. The number of encirclements of the critical point -1 by this plot equals the difference between the number of unstable poles and zeros of $1 + G(s), via the argument principle applied to $1 + G(s). The system is stable if there are no right half-plane poles of the closed-loop transfer function, i.e., the plot encircles -1 exactly as many times as the open-loop unstable poles (clockwise for standard convention).^[2]^[21] Jensen's formula, which for a holomorphic function f on the disk |z| < R with f(0) \neq 0 and zeros z_k (counting multiplicities) states

\log |f(0)| + \frac{1}{2\pi} \int_0^{2\pi} \log |f(Re^{i\theta})| \, d\theta = \sum_k \log \frac{R}{|z_k|},

derives directly from the argument principle applied to circles of radius r < R.^[4] The principle equates the argument change of f on |z| = r to $2\pi times the number of zeros inside, and integrating the logarithmic derivative f'/f yields the mean value of \log |f| on the circle, leading to the formula upon taking limits as r \to R.^[22]

Generalizations and Extensions

Multivariable and Other Variants

In several complex variables, the argument principle generalizes to holomorphic functions f: U \subset \mathbb{C}^n \to \mathbb{C} defined on a domain U. Unlike the one-variable case, the boundary of U is a (2n-1)-dimensional manifold, so the classical line integral is replaced by integration over currents or using sheaf cohomology to capture the topological degree or winding number. For practical computations, such as counting zeros in a polydisc, one can fix all but one variable and apply the one-variable argument principle iteratively; for example, the number of solutions to f(z_1, \dots, z_n) = 0 in the unit polydisc can be obtained by integrating the number of zeros in z_n over the previous variables. A concrete example is the equation f(z,w) = zw - 1 = 0 on the torus \mathbb{T}^2 = S^1 \times S^1, where the argument principle, applied slicewise, counts the single solution per fundamental domain, reflecting the degree of the map (z,w) \mapsto zw.^[23] In algebraic geometry, the argument principle underlies the computation of the degree of a holomorphic map between compact Riemann surfaces, equating it to the number of preimages of a point (counted with multiplicity), obtained via the integral \frac{1}{2\pi i} \int_{\partial U} \frac{df}{f - a} for generic a. This extends to intersection theory on Riemann surfaces, where the intersection number of divisors is given by the degree of the line bundle, and the argument principle counts intersections by winding numbers around curves. For instance, for a map f: X \to Y of degree d, the pullback of a canonical divisor intersects the fundamental class with multiplicity d.^[24] A further generalization applies to functions with branch points on Riemann surfaces. For a holomorphic function on a Riemann surface, the argument principle can be adapted using the Riemann-Hurwitz formula or by considering the covering map, where the change in argument accounts for branching. This connects the number of zeros and poles to the genus and degree of the surface. In quaternionic analysis, for slice-regular functions f: B(0,R) \subset \mathbb{H} \to \mathbb{H}, the argument principle states that \frac{1}{4\pi I} \int_{\partial B_I(0,r)} \frac{f_s'(z)}{f_s(z)} dz equals the difference between the multiplicities of zeros and orders of poles inside the ball, where f_s is the symmetrized function and I is an imaginary unit. Post-2000 developments, such as those for Fueter-regular functions, remain active but less standardized.^[25]

Historical Development

Origins and Discovery

The argument principle has its roots in the foundational work of Augustin-Louis Cauchy on complex integration during the 1820s and 1830s, which established key theorems for contour integrals of holomorphic functions. In his 1825 memoir Mémoire sur les intégrales définies prises entre des limites imaginaires, Cauchy proved the independence of line integrals for analytic functions along paths in simply connected domains, providing the groundwork for later results involving residues and function behavior inside contours. This development built on his earlier 1823 explorations of definite integrals and laid the analytic framework essential for counting singularities in the complex plane.^[26] The explicit discovery of the argument principle is attributed to Cauchy, who first stated and proved a version of it in a memoir presented on November 27, 1831, to the Royal Academy of Sciences in Turin during his political exile in Italy. In this work, published in the academy's transactions, Cauchy related the total change in the argument of a meromorphic function along a closed contour to the difference between the number of zeros and poles enclosed by the contour, marking the theorem's initial formulation as a direct consequence of his residue calculus. Known in early literature as "Cauchy's argument principle," it emerged amid his investigations into the properties of analytic functions and their singularities. The principle gained further formalization through Bernhard Riemann's 1851 doctoral dissertation, Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse, where he incorporated argument changes to analyze conformal mappings and multi-valued functions. Riemann applied these ideas to count zeros, poles, and branch points on Riemann surfaces, particularly in the study of elliptic functions, where understanding the distribution of singularities was crucial for global function theory. This contextual embedding highlighted the principle's topological significance in complex analysis.^[27]

Key Developments and Contributors

Following its initial formulation, the argument principle underwent significant developments in the late 19th and early 20th centuries, with key figures expanding its scope in complex analysis and related fields. Bernhard Riemann played a central role in its formalization during his lectures on the theory of functions in 1861, aspects of which were published posthumously in 1899 as Elliptische Functionen, where he integrated it with residue calculus to evaluate integrals and study analytic functions.^[28] In the 1880s, Henri Poincaré contributed to the study of automorphic functions and Fuchsian groups, forging connections between complex analysis and topology through the examination of fundamental domains and transformation properties. Building on developments in the field, Jacques Hadamard in the 1890s worked on analyses of entire functions, contributing to refinements in growth estimates and derivations supporting Émile Picard's theorems on value distribution. The 20th century saw further extensions, notably by Lars Ahlfors in the 1940s, who utilized the argument principle in the geometric theory of Riemann surfaces to count branch points and analyze conformal mappings.^[29] Rolf Nevanlinna advanced its role in value distribution theory during the 1920s, incorporating it to quantify the frequency of value attainments by meromorphic functions and establish theorems on exceptional values.^[30] In the computational realm, Peter Henrici in the 1960s developed numerical methods leveraging the argument principle for reliable root-finding of analytic functions, emphasizing contour integration techniques in applied complex analysis.^[31] Overall, the principle proved pivotal in establishing the fundamental theorem of algebra, where applications to large circular contours demonstrate that non-constant polynomials possess zeros within sufficiently expansive regions.^[2]

References

[1]
https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_(Orloff)/12:_Argument_Principle/12.01:_Principle_of_the_Argument
[2]
[PDF] 18.04 S18 Topic 11: Argument Principle - MIT OpenCourseWare
The argument principle (or principle of the argument) is a consequence of the residue theorem. It connects the winding number of a curve with the number of ...
[3]
STUDIES IN THE HISTORY OF COMPLEX FUNCTION THEORY II
One sees, for instance, that the Cauchy Integral Theorem was already present in Cauchy's first work from the year 1814 (but only for the case of rectangles with ...
[4]
[PDF] Introduction to Complex Analysis
The Argument Principle and Applications. Theorem 4.18 (Argument principle for holomorphic functions). Let Ω ⊂ C be open and f : Ω → C be holomorphic, with f 6≡ ...
[5]
Meromorphic Function -- from Wolfram MathWorld
A meromorphic function is a single-valued function that is analytic in all but possibly a discrete subset of its domain, and at those singularities it must go ...
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[PDF] Section 4.39. Contours
Jan 15, 2020 · Such a curve is positively oriented when it is traced out in a counterclockwise direction as t ranges from a to b.
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[PDF] 18.04 S18 Topic 4: Cauchy's integral formula - MIT OpenCourseWare
We start with a statement of the theorem for functions. After some examples, we'll give a gener- alization to all derivatives of a function. After some more ...
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None
### Definition and Explanation of Multiplicity of a Zero for Holomorphic Functions
[9]
Complex Argument -- from Wolfram MathWorld
The complex argument of a number z is implemented in the Wolfram Language as Arg[z]. The complex argument can be computed as arg(x+iy)=tan^(-1)(y/x).
[10]
[PDF] Complex Analysis with Applications Princeton University MAT330 ...
Jan 27, 2023 · Figure 2: The geometric interpretation of complex conjugation. ... Before tending to the proof of the argument principle, it is instructive to ...
[11]
[PDF] Domain Coloring and the Argument Principle - Scholar Commons
Jan 26, 2017 · This domain coloring of the squaring function gives a first hint of why this is such a powerful technique to visualize the argument principle: ...
[12]
[PDF] Math 3228 - Week 7 • Winding numbers • The argument principle
Since 0 - " = - " , this is again consistent with the argument principle. • Informally, what the argument principle is telling us is that every zero of a ...
[13]
[PDF] The argument principle • Applications to
The argument principle establishes a striking relationship between number of zeros − number of poles of f in a domain D.
[14]
[PDF] V.3. The Argument Principle
May 1, 2018 · In the. Argument Principle we relate the value of an integral to winding numbers of zeros or poles. In Rouche's Theorem, a quantity related ...
[15]
[PDF] The Residue Theorem
Since g′/g is analytic at z0, it follows that f′/f will have a simple pole at z0 with residue −k. Applying the residue theorem, the result follows.
[16]
[PDF] The Residue Theorem and its consequences
This writeup presents the Argument Principle, the winding number, Rouché's Theorem, the Lo- cal Mapping Theorem, the Open Mapping Theorem, the Hurwitz Theorem, ...
[17]
[PDF] Ahlfors, Complex Analysis
In Chapter 4 there is a new and simpler proof of the general form of. Cauchy's theorem. It is due to A. F. Beardon, who has kindly permitted me to reproduce ...<|control11|><|separator|>
[18]
[PDF] Lecture 6 - Argument principle, Rouché's theorem and consequences
Recall the Residue theorem, which allows to compute curve integrals of meromorphic functions in terms of their residues at the poles enclosed by that curve.
[19]
[PDF] 12. Rouches Theorem Looking at how we used the argument ...
Looking at how we used the argument principle in lecture 11, it is hopefully clear that it is often very useful to identify the dominant term and exploit ...
[20]
[PDF] Contents 1. Review: Complex numbers and functions of a complex ...
Hurwitz's theorem. This theorem shows that a uniform limit of analytic functions with no zeros in a domain is either identically zero, or else an analytic ...
[21]
[PDF] basic results arising from Cauchy's theorem 1. Maximum modulus ...
Nov 3, 2014 · Maximum modulus principle. 2. Open mapping theorem ... To this end, consider an argument-principle integral which counts the number of zeros.
[22]
[PDF] The Riemann Mapping Theorem - UChicago Math
The argument principle is especially useful because it is such a simple formula for counting zeros. This principle will be used several times in later proofs, ...
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[PDF] Advanced Complex Analysis - Harvard Mathematics Department
One way to make the proof more transparent and constructive is to employ the argument principle. ... Theorem 3.13 (Jensen's formula) Let f(z) be a holomorphic ...
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[PDF] A function is called meromorphic on a region if it is analytic in this ...
2 to meromorphic functions. Theorem 5.7.3. (Meromorphic Counting Theorem) ... Either of the preceding theorems is also known as the argument principle be-.
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[PDF] Notes for Analysis and Geometry of Several Complex Variables ...
Jun 30, 2020 · First prove the one variable case by the argument principle. ... Every polynomial p in three complex variables according to the maximum principle.<|control11|><|separator|>
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[PDF] Several Complex Variables - UCR Math Department
By the argument principle,. #of zeros of hz0 inside |zn| = is. 1. 2πi. Z. |ζ|=. ∂f. ∂zn. (z0,ζ) f(z0,ζ) dζ. Holomorphic function of z0 near 0 is integer valued ...
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[PDF] Parametric argument principle and its applications to CR functions ...
regarded as a general form of the argument principle. ... Argument Principle, the holomorphic mappings Φt(ζ) ... phic functions of several complex variables, Duke ...
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[PDF] A STEIN CRITERION VIA DIVISORS FOR DOMAINS OVER STEIN ...
In this paper we prove the following result: THEOREM 1. Let Y be a Stein manifold of dimension n. Then a domain X over Y is Stein provided the following two ...
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[PDF] Riemann Surfaces - Berkeley Math
In algebraic geometry, this is done by a procedure called normalization. ... be bounded in terms of the coefficients (e.g. by the argument principle).
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[PDF] CS 812 Lecture 39 Monday 4/27/20 Factoring Polynomials With ...
Apr 27, 2020 · Background. Polynomial factoring is a key task in algebra. As taught in high school, it is not particularly algorithmic.
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The Argument Principle for Quaternionic Slice Regular Functions
Notice that, by the final statement of Theorem 3.2, real singularities are completely analogous to singularities of holomorphic functions of one complex ...
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https://arxiv.org/pdf/2311.16649.pdf
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[PDF] FOUNDATIONS OF A GENERAL THEORY OF FUNCTIONS OF A ...
This is the LATEX-ed version of an English translation of Bernhard Riemann's. 1851 thesis, which marked the beginning of the geometrical theory of complex.
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Picard's Theorems - ScienceDirect.com
In 1879 Picard proved that an entire function takes on every value with at most one exception, (Picard's “Little Theorem”).
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[PDF] RIEMANN SURFACES
A Riemann surface is, in the first place, a surface, and its properties depend to a very great extent on the topological character of the surface.
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The inverse problem of the Nevanlinna theory - Project Euclid
Value.distribution of F* (Lemma 14) . ... That w is a homeomorphism depends on the argument principle (that the argument principle applies to quasi ...
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[PDF] cs9 THE QD-ALGOR ITHM AS A METHOD FOR' FINDING THE ...
Aitken [1] and Henrici [6] have used these for the same purpose of rootfinding as treated here. However, theorem 2.4 by means of which the existence of a ...