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Cylindrical harmonics

Cylindrical harmonics are a class of orthogonal functions that provide complete solutions to Laplace's equation \nabla^2 \Phi = 0 in three-dimensional cylindrical coordinates (\rho, \phi, z), obtained through separation of variables.^[1] These functions take the general form \Phi_{n\kappa}(\rho, \phi, z) = J_n(\kappa \rho) e^{\pm i n \phi} e^{\pm \kappa z}, where J_n denotes the Bessel function of the first kind of order n (a non-negative integer), \kappa is a separation constant, and the exponential terms account for azimuthal and axial dependencies; alternative forms incorporate the Neumann function Y_n for singular solutions or modified Bessel functions I_n and K_n for evanescent waves.^[1]^[2] They form a basis for expanding potentials in problems exhibiting cylindrical symmetry, ensuring completeness and orthogonality over appropriate domains.^[3] In contrast to the power-law solutions that arise in two-dimensional cylindrical coordinates (independent of z), the three-dimensional cylindrical harmonics incorporate oscillatory or decaying radial behavior via Bessel functions, making them suitable for bounded or unbounded regions with finite extent along the axis.^[1] The separation process yields ordinary differential equations: the azimuthal equation \frac{d^2 \Psi}{d\phi^2} + n^2 \Psi = 0 with periodic solutions, the axial equation \frac{d^2 Z}{dz^2} - \kappa^2 Z = 0 with exponential solutions, and the radial Bessel equation \rho^2 \frac{d^2 R}{d\rho^2} + \rho \frac{dR}{d\rho} + (\kappa^2 \rho^2 - n^2) R = 0.^[3] Boundary conditions, such as those on cylindrical surfaces, often require linear combinations of these harmonics, with coefficients determined by orthogonality relations like \int_0^a \rho J_n(k_{nm} \rho) J_n(k_{n\ell} \rho) d\rho = 0 for distinct roots k_{nm}.^[3] Cylindrical harmonics find extensive applications in electromagnetostatics, acoustics, and quantum mechanics for modeling fields in waveguides, cavities, and scattering problems with axial uniformity or periodicity.^[1] For instance, they describe the potential inside a charged cylindrical capacitor or the modes of vibration in a drumhead extended axially, highlighting their role in decomposing complex boundary-value problems into solvable eigenfunction expansions.^[2]

Mathematical formulation

Cylindrical coordinate system

The cylindrical coordinate system provides a natural framework for describing points in three-dimensional space, particularly in geometries with rotational symmetry around an axis, by extending the two-dimensional polar coordinates with a vertical dimension. In this system, a point is specified by the ordered triple (\rho, \phi, z), where \rho denotes the radial distance from the z-axis (\rho \geq 0), \phi represents the azimuthal angle measured from the positive x-axis in the xy-plane (ranging from 0 to $2\pi), and z is the axial coordinate along the z-axis, identical to the Cartesian z. Geometrically, surfaces of constant \rho form infinite cylinders parallel to the z-axis, surfaces of constant \phi correspond to half-planes emanating from the z-axis, and surfaces of constant z are horizontal planes perpendicular to the z-axis.^[4]^[5] The transformation between cylindrical and Cartesian coordinates facilitates the use of this system in standard mathematical formulations. The forward transformation from cylindrical to Cartesian coordinates is given by

x = \rho \cos \phi, \quad y = \rho \sin \phi, \quad z = z,

while the inverse transformation is

\rho = \sqrt{x^2 + y^2}, \quad \phi = \atan2(y, x), \quad z = z.

These equations preserve the Euclidean distance and allow seamless conversion for problems involving cylindrical symmetry.^[4] For differential calculus in cylindrical coordinates, the scale factors account for the varying metric along each direction: h_\rho = [1](/page/1), h_\phi = \rho, and h_z = [1](/page/1). Consequently, the infinitesimal volume element is dV = \rho \, d\rho \, d\phi \, dz, which arises from the Jacobian determinant of the transformation and is essential for integrals over cylindrical volumes.^[6] The underlying polar coordinates were introduced by Leonhard Euler in the 1760s, notably in his 1766 paper on the vibrations of circular membranes.^[7] The cylindrical system extends this by adding the axial coordinate z, serving as the foundational setup for solving partial differential equations like Laplace's equation in contexts exhibiting axial symmetry.^[4]

Laplace's equation

Laplace's equation, in its general vector form, is given by \nabla^2 V = 0, where V represents a scalar potential function satisfying the condition of zero Laplacian.^[8] This partial differential equation describes harmonic functions, which arise in scenarios where the divergence of the gradient vanishes, indicating equilibrium states without sources or sinks.^[9] In cylindrical coordinates (\rho, \phi, z), Laplace's equation expands to

\frac{1}{\rho} \frac{\partial}{\partial \rho} \left( \rho \frac{\partial V}{\partial \rho} \right) + \frac{1}{\rho^2} \frac{\partial^2 V}{\partial \phi^2} + \frac{\partial^2 V}{\partial z^2} = 0.

This form is derived by applying the Laplacian operator in the orthogonal curvilinear system defined by cylindrical coordinates.^[10] Solutions to this equation in cylindrical domains model physically relevant phenomena, such as the electrostatic potential in charge-free regions where \mathbf{E} = -\nabla V and \nabla \cdot \mathbf{E} = 0, steady-state temperature distributions in homogeneous media without internal heat generation, and velocity potentials for irrotational, incompressible fluid flows where the velocity \mathbf{v} = \nabla V satisfies \nabla \cdot \mathbf{v} = 0.^[8]^[9] Boundary value problems for Laplace's equation in cylindrical geometries typically involve specifying conditions on the surfaces enclosing the domain, such as the lateral surface \rho = a or the end caps z = 0, L. Dirichlet conditions prescribe the value of V directly on the boundary, while Neumann conditions specify the normal derivative \partial V / \partial n, corresponding to fixed potential or fixed flux, respectively.^[11] For the Dirichlet problem in a bounded domain, the solution is unique, as established by the maximum principle: if two solutions satisfy the same boundary values, their difference is a harmonic function vanishing on the boundary, hence identically zero throughout the domain.^[12]

Separation of variables

To solve Laplace's equation in cylindrical coordinates using the method of separation of variables, the potential is assumed to be a product of functions each depending on one coordinate: V(\rho, \phi, z) = R(\rho) \Phi(\phi) Z(z). This ansatz exploits the coordinate system's orthogonality to decouple the partial differential equation into ordinary differential equations.^[10] Substituting the product form into Laplace's equation separates the variables, introducing separation constants to equate terms dependent on different coordinates. The azimuthal term yields the constant -n^2, the axial term yields +k^2, resulting in three independent ordinary differential equations: one for \Phi(\phi), one for Z(z), and one for R(\rho).^[13] The azimuthal equation is \frac{d^2 \Phi}{d\phi^2} + n^2 \Phi = 0, where n must be an integer to ensure the solution is single-valued, as \Phi(\phi + 2\pi) = \Phi(\phi) due to the periodic nature of the azimuthal angle.^[10] The axial equation is \frac{d^2 Z}{dz^2} - k^2 Z = 0, which admits exponential solutions for real k (evanescent or growing behavior along z) or trigonometric solutions if k is imaginary (with corresponding modified Bessel functions in the radial part). The radial equation takes the form \rho^2 \frac{d^2 R}{d\rho^2} + \rho \frac{dR}{d\rho} + (k^2 \rho^2 - n^2) R = 0, resembling the Bessel differential equation and coupling the parameters n and k.^[13] The general separated solution for each mode is thus V_n(k; \rho, \phi, z) = R_n(k, \rho) \Phi_n(\phi) Z(k, z), with the full potential obtained by summing over integer n and integrating over continuous k to match boundary conditions in unbounded or semi-infinite domains.^[10]

Component functions

Azimuthal functions

In the separation of variables approach to solving Laplace's equation in cylindrical coordinates, the azimuthal dependence is governed by the ordinary differential equation

\frac{d^2 \Phi}{d\phi^2} + n^2 \Phi = 0,

where n^2 is the separation constant chosen to produce bounded, oscillatory solutions in the angular coordinate \phi. This equation arises after assuming a product form for the potential and isolating the \phi-dependent part.^[14] The solutions must satisfy the periodicity condition \Phi(\phi + 2\pi) = \Phi(\phi) to ensure single-valuedness in the physical domain, as \phi represents the azimuthal angle around the cylinder. This constraint restricts n to non-negative integers: n = 0, 1, 2, \dots. For n = 0, the solution is a constant, \Phi_0(\phi) = A, describing axisymmetric configurations with no angular variation. For n \geq 1, the general real-valued solution is \Phi_n(\phi) = A \cos(n\phi) + B \sin(n\phi), where the coefficients A and B are determined by boundary conditions. An equivalent complex representation is \Phi_n(\phi) = C e^{i n \phi} + D e^{-i n \phi}, which facilitates analysis in contexts like Fourier expansions.^[14]^[15]^[16] These functions form an orthogonal basis over [0, 2\pi]. They are often normalized such that the constant mode is \frac{1}{\sqrt{2\pi}} for n=0, and \sqrt{\frac{2}{\pi}} \cos(n\phi) or \sqrt{\frac{2}{\pi}} \sin(n\phi) for n \geq 1, ensuring the set satisfies the orthogonality relations \int_0^{2\pi} \Phi_m(\phi) \Phi_k(\phi) \, d\phi \propto \delta_{mk}. The integer n denotes the azimuthal mode order, quantifying the number of spatial periods in the \phi-direction and reflecting the system's rotational symmetry. In quantum mechanical treatments of particles confined in cylindrical potentials, n plays the role of the azimuthal quantum number associated with angular momentum along the axis.^[14]^[17]

Radial functions

In the separation of variables for Laplace's equation in cylindrical coordinates, the radial dependence R(\rho) satisfies the ordinary differential equation \rho^2 \frac{d^2 R}{d\rho^2} + \rho \frac{dR}{d\rho} + (k^2 \rho^2 - n^2) R = 0 when the axial separation constant yields a real wavenumber k > 0, leading to oscillatory behavior in the radial direction.^[18]^[10] This form, known as Bessel's equation of order n, arises after normalizing the radial variable, where n is the integer azimuthal separation constant.^[1] The general solution for real k is R_n(k, \rho) = A J_n(k \rho) + B Y_n(k \rho), a linear combination of the Bessel function of the first kind J_n and the second kind Y_n (also called the Neumann function).^[18] The function J_n(k \rho) is finite and analytic at \rho = 0, with series expansion J_n(x) \sim \frac{(x/2)^n}{\Gamma(n+1)} for small x, making it suitable for problems encompassing the origin.^[10] In contrast, Y_n(k \rho) diverges logarithmically at \rho = 0 for integer n, so B = 0 is typically chosen for interior domains including the axis.^[18] For large arguments, J_n(k \rho) and Y_n(k \rho) exhibit oscillatory asymptotic forms:

J_n(x) \sim \sqrt{\frac{2}{\pi x}} \cos\left(x - \frac{n\pi}{2} - \frac{\pi}{4}\right), \quad Y_n(x) \sim \sqrt{\frac{2}{\pi x}} \sin\left(x - \frac{n\pi}{2} - \frac{\pi}{4}\right),

which describe wave-like propagation away from the axis.^[10] When the axial separation constant results in an imaginary wavenumber k = i \kappa with \kappa > 0, the radial equation becomes modified to \rho^2 \frac{d^2 R}{d\rho^2} + \rho \frac{dR}{d\rho} - (\kappa^2 \rho^2 + n^2) R = 0, corresponding to evanescent (non-oscillatory) solutions.^[18]^[10] The general solution is R_n(\kappa, \rho) = C I_n(\kappa \rho) + D K_n(\kappa \rho), involving the modified Bessel functions of the first kind I_n and second kind K_n.^[1] The function I_n(\kappa \rho) grows exponentially for large \rho, with asymptotic I_n(x) \sim \frac{e^x}{\sqrt{2\pi x}}, while K_n(\kappa \rho) decays exponentially, K_n(x) \sim \frac{e^{-x}}{\sqrt{2\pi x}}; near \rho = 0, I_n behaves like \frac{(\rho/2)^n}{\Gamma(n+1)} (regular), and K_n is singular.^[10] Selection between these depends on boundary conditions, such as using I_n for bounded growth in finite domains or K_n for decay in exterior regions.^[18]^[1]

Axial functions

In the separation of variables approach to solving Laplace's equation in cylindrical coordinates, the axial dependence is governed by the ordinary differential equation (ODE) derived from the separation constant, typically denoted as k^2.^[10] For the case where the separation constant leads to evanescent behavior along the z-axis, the ODE takes the form \frac{d^2 Z}{dz^2} - k^2 Z = 0, with k real and positive.^[19] The general solution is Z(k, z) = A e^{k z} + B e^{-k z}, which can equivalently be expressed using hyperbolic functions as Z(k, z) = C \cosh(k z) + D \sinh(k z) for convenience in bounded domains.^[10] When the separation constant is negative, corresponding to oscillatory behavior, the ODE becomes \frac{d^2 Z}{dz^2} + \mu^2 Z = 0, where \mu = |k| is real and positive (with k = i \mu).^[19] The solutions are trigonometric: Z(\mu, z) = C \cos(\mu z) + D \sin(\mu z), representing standing waves along the axis.^[10] These forms are selected based on the domain and boundary conditions: exponential solutions are used for infinite or semi-infinite domains to ensure boundedness or decay at infinity, while trigonometric functions apply to finite-length cylinders where periodicity or zero boundary conditions at the ends (e.g., Z(0) = Z(L) = 0) dictate discrete values like \mu_n = n \pi / L.^[19] Normalization of the axial functions depends on whether the spectrum is discrete or continuous. For finite domains, the trigonometric solutions form an orthogonal basis, with normalization constants chosen such that \int_0^L Z_m(z) Z_n(z) \, dz = \frac{L}{2} \delta_{mn} for sine functions satisfying Dirichlet boundaries.^[19] In infinite or unbounded cases, continuous spectra are handled via Fourier transforms, where the exponential or plane-wave forms are normalized with factors like \frac{1}{\sqrt{2\pi}} to ensure \int_{-\infty}^{\infty} Z(k, z) Z^*(k', z) \, dz = \delta(k - k').^[18]

Series expansion

General solution form

The individual cylindrical harmonics, which form the basis solutions to Laplace's equation in cylindrical coordinates, are products of the separated component functions: V_n(k; \rho, \phi, z) = R_n(k, \rho) \Phi_n(\phi) Z(k, z), where R_n(k, \rho) is typically the Bessel function of the first kind J_n(k \rho) for regularity at the origin, \Phi_n(\phi) involves azimuthal trigonometric terms such as \cos(n \phi) or \sin(n \phi) with integer order n \geq 0, and Z(k, z) consists of axial exponentials like e^{\pm k z} or hyperbolic functions such as \sinh(k z) and \cosh(k z) depending on boundary conditions.^[10]^[3] The general solution for the potential V(\rho, \phi, z) is obtained by superposing these harmonics via a double expansion over the azimuthal index n and the separation constant k:

V(\rho, \phi, z) = \sum_{n=0}^{\infty} \int_{0}^{\infty} dk \, \left[ A_n(k) J_n(k \rho) \cos(n \phi) + B_n(k) J_n(k \rho) \sin(n \phi) \right] e^{\pm k z},

where the coefficients A_n(k) and B_n(k) are determined by boundary data, and variants may replace the exponentials with sine or cosine functions for oscillatory axial behavior or include Neumann functions Y_n(k \rho) in annular regions.^[10] For infinite domains in z, the continuous spectrum requires an integral over k; in finite z-domains (e.g., height L) with boundary conditions such as V=0 at z=0 and \rho=a, the axial functions take the form \sinh(k z) and k is quantized by the radial boundary condition to discrete values k_{nm} given by the roots of J_n(k_{nm} a)=0, leading to a double sum over n and m.^[10] In radially bounded domains (e.g., \rho < a), the Fourier-Bessel series aspect emerges by imposing Dirichlet or Neumann conditions at \rho = a, which discretize k to the roots k_{n m} of J_n(k a) = 0 (or its derivative for Neumann), yielding a double sum:

V(\rho, \phi, z) = \sum_{n=0}^{\infty} \sum_{m=1}^{\infty} \left[ A_{n m} J_n(k_{n m} \rho) \cos(n \phi) + B_{n m} J_n(k_{n m} \rho) \sin(n \phi) \right] Z(k_{n m}, z).

This form leverages the orthogonality of Bessel functions over [0, a] to expand arbitrary radial profiles.^[10]^[3] The series converges uniformly in bounded domains excluding singularities, such as sources or boundaries where the potential is discontinuous, due to the completeness and orthogonality of the basis functions analogous to Fourier expansions.^[10]

Coefficient determination

The coefficients in the cylindrical harmonic series expansion are determined primarily through the orthogonality properties of the basis functions, which allow the projection of the given boundary data onto the complete set of eigenfunctions satisfying the boundary conditions. This process involves computing inner products over the domain or boundary, ensuring the expansion satisfies the governing equation and boundaries. For boundary value problems, the coefficient A_{nm} for the mode with azimuthal order n and radial wavenumber k_m is obtained via surface integrals matching the specified potential on the boundaries; for the inhomogeneous Poisson equation \nabla^2 V = -f, an analogous volume integral formula applies:

A_{nm} = \frac{\int_V \Phi_n^*(\phi) R_{nm}^*(\rho) Z_m^*(z) f(\mathbf{r}) \, dV}{\int_V |\Phi_n(\phi)|^2 |R_{nm}(\rho)|^2 |Z_m(z)|^2 \, dV},

where f(\mathbf{r}) is the source function, \Phi_n, R_{nm}, and Z_m are the normalized azimuthal, radial, and axial component functions, respectively, and the integrals are performed over the volume V of the domain.^[3] In boundary value problems with Dirichlet conditions on a finite cylinder of radius \rho = a, the radial wavenumbers k_m are selected as the roots of the Bessel function equation J_n(k_m a) = 0, ensuring the radial functions R_{nm}(\rho) = J_n(k_m \rho) vanish at the boundary. The coefficients are then derived by expanding the boundary data in a Fourier series in the azimuthal angle \phi and axial coordinate z, combined with the orthogonality of the Bessel functions in \rho. For instance, in an axisymmetric case (n=0) with specified potential V_0(\rho) at z = L, the coefficient simplifies to

A_{0m} = \frac{2}{a^2 [J_1(k_m a)]^2 \sinh(k_m L)} \int_0^a \rho \, d\rho \, J_0(k_m \rho) V_0(\rho),

leveraging the Bessel orthogonality relation \int_0^a \rho \, d\rho \, J_0(k_m \rho) J_0(k_{m'} \rho) = \frac{a^2}{2} [J_1(k_m a)]^2 \delta_{mm'}.^[3] An alternative method employs the Green's function approach for inhomogeneous equations, where the coefficients arise directly from expanding the source terms in the cylindrical harmonic basis. The Green's function G(\mathbf{r}, \mathbf{r}') satisfying \nabla^2 G = -\delta(\mathbf{r} - \mathbf{r}') with appropriate boundaries is expressed as a series \sum_{nm} A_{nm} \Phi_n(\phi) R_{nm}(\rho) Z_m(z) \Phi_n(\phi') R_{nm}(\rho') Z_m(z'), and the coefficients A_{nm} are determined by matching the source singularity, often yielding closed-form expressions involving the eigenmode norms. This is particularly useful for problems with distributed sources inside enclosures. Numerically, the infinite series is truncated at a finite order N for practical computation, with convergence assessed by monitoring residuals or error norms; typically, N \sim 10-50 terms suffice for moderate precision in electrostatic applications. For large n, fast algorithms exploiting the separability—such as discrete Fourier transforms for the \phi-dependence and efficient Bessel function evaluations—reduce computational cost from O(N^3) to near-linear scaling. As a non-detailed example in the axisymmetric (n=0) case, the coefficient A_0(k) is proportional to \int V(z) \cos(k z) \, dz normalized by \int J_0^2(k \rho) \rho \, d\rho.^[20]

Properties

Orthogonality relations

The azimuthal functions in cylindrical harmonics, arising from the separation of variables in Laplace's equation, are the standard Fourier basis functions: \cos(n\phi) and \sin(n\phi) for integer n \geq 0. These functions satisfy orthogonality relations over the interval [0, 2\pi] with respect to the uniform measure d\phi. Specifically, for n, m \geq 1,

\int_0^{2\pi} \cos(n\phi) \cos(m\phi) \, d\phi = \pi \delta_{nm},

and similarly for the sine functions,

\int_0^{2\pi} \sin(n\phi) \sin(m\phi) \, d\phi = \pi \delta_{nm},

while cross terms vanish: \int_0^{2\pi} \cos(n\phi) \sin(m\phi) \, d\phi = 0. For the constant term (n = m = 0), the integral is $2\pi.^[21] The radial functions, typically Bessel functions of the first kind J_n(k \rho) for fixed azimuthal order n, exhibit orthogonality over a finite disk of radius a when k_m and k_l are chosen such that J_n(k_m a) = 0 and J_n(k_l a) = 0 (i.e., roots of the Bessel function to satisfy boundary conditions like a conducting cylinder). The relation is

\int_0^a J_n(k_m \rho) J_n(k_l \rho) \, \rho \, d\rho = \frac{a^2}{2} [J_{n+1}(k_m a)]^2 \delta_{ml},

with the weight \rho \, d\rho arising from the cylindrical volume element.^[22] The axial functions, for finite height h with appropriate boundary conditions (e.g., Dirichlet), take forms like \sinh(\kappa_m (z)) or \sin(k_l z) depending on the separation case, satisfying

\int_0^h Z_m(z) Z_l(z) \, dz \propto \delta_{ml},

via Sturm-Liouville theory.^[3] In three dimensions, the full cylindrical harmonics \Phi_{n\kappa}(\rho, \phi, z) = J_n(\kappa \rho) e^{\pm i n \phi} e^{\pm \kappa z} (or real combinations) are orthogonal over cylindrical domains with finite extent in z or suitable boundary conditions. The integral over the volume element \rho \, d\rho \, d\phi \, dz separates into the product of the individual component orthogonality relations, yielding proportionality to

\delta_{nn'} \delta_{mm} \delta_{ll'}\ ) for discrete modes in finite radius \(a

and height h, where m, l index radial and axial eigenvalues, respectively. For unbounded domains, continuous spectra lead to Dirac deltas in the corresponding variables.^[3] These orthogonality relations follow from Sturm-Liouville theory applied to the separated ordinary differential equations. The radial Bessel equation can be cast in self-adjoint Sturm-Liouville form:

\frac{d}{d\rho} \left( \rho \frac{d R}{d\rho} \right) + \left( k^2 \rho - \frac{n^2}{\rho} \right) R = 0,

with boundary conditions ensuring the operator is self-adjoint, leading to orthogonality of eigenfunctions R_m and R_l for distinct eigenvalues k_m^2 and k_l^2 with respect to the weight \rho. The azimuthal part follows from the periodicity and the structure of the angular operator.^[22] Normalization constants are chosen to make the harmonics have unit norm over the domain. For the azimuthal part, the normalized functions are \frac{1}{\sqrt{2\pi}} e^{i n \phi} (or \frac{1}{\sqrt{\pi}} \cos(n\phi), \frac{1}{\sqrt{\pi}} \sin(n\phi) for n \geq 1, and \frac{1}{\sqrt{2\pi}} for n=0). For the radial part in a cylinder of radius a, the normalized eigenfunction is \sqrt{\frac{2}{a^2 [J_{n+1}(k_m a)]^2}} J_n(k_m \rho). The full 3D normalized harmonic then incorporates these factors, often yielding an L^2-norm of 1 over the volume.^[23]

Completeness

Cylindrical harmonics, obtained through separation of variables in cylindrical coordinates for Laplace's equation, form a complete orthonormal basis for square-integrable functions in L² spaces over finite cylindrical domains that satisfy the relevant boundary conditions, such as Dirichlet or Neumann conditions on the lateral surface.^[19] This completeness ensures that any sufficiently smooth potential or field in such a domain can be uniquely expanded in terms of these harmonics, leveraging the orthogonality established in prior analyses.^[19] Parseval's theorem applies to these expansions, stating that for a function V expressed as V(\mathbf{r}) = \sum_n c_n \psi_n(\mathbf{r}), where \psi_n are the cylindrical harmonics and c_n the coefficients, the equality \int |V|^2 \, dV = \sum_n |c_n|^2 holds over the cylindrical volume, preserving the L² norm.^[24] The radial component's completeness, rooted in Fourier-Bessel series, follows from Sturm-Liouville theory applied to Bessel's equation with appropriate boundary conditions at finite radius.^[25] Convergence of the series occurs pointwise for continuous functions within the domain and in the L² norm for more general square-integrable functions; however, near discontinuities, the partial sums exhibit overshoot known as the Gibbs phenomenon, with the overshoot amplitude approaching approximately 8.95% of the jump height, analogous to standard Fourier series.^[26] Proofs of completeness for the Bessel functions often rely on generating function expansions or criteria akin to Weyl's for spectral completeness in self-adjoint operators.^[27] Limitations arise when addressing singularities, where expansions may require principal value interpretations to handle non-integrable behaviors, and for exterior domains (r > R), the basis shifts to include Hankel functions for outgoing waves rather than the interior Bessel functions discussed here.^[19]

Example: Point source in enclosure

Solution inside conducting cylinder

The solution for the electrostatic potential inside a grounded infinite conducting cylinder due to a point charge considers an infinite cylindrical tube of radius a, with the surface held at zero potential (V = 0 at \rho = a). A point charge q is located at position (\rho_0, \phi_0, z_0) where \rho_0 < a, in cylindrical coordinates (\rho, \phi, z). This setup satisfies Poisson's equation \nabla^2 V = -\frac{q}{\epsilon_0} \delta(\mathbf{r} - \mathbf{r}_0) inside the cylinder, with the Dirichlet boundary condition enforced on the conducting surface.^[28] The free-space Green's function for a point charge, \frac{1}{4\pi |\mathbf{r} - \mathbf{r}_0|}, can be expanded in cylindrical harmonics using modified Bessel functions to facilitate imposition of the boundary condition. The expansion takes the form

\frac{1}{|\mathbf{r} - \mathbf{r}_0|} = \frac{1}{\pi} \sum_{n=0}^\infty \epsilon_n \cos[n(\phi - \phi_0)] \int_0^\infty dk \, \cos[k(z - z_0)] \, I_n(k \rho_<) K_n(k \rho_>),

where \rho_< = \min(\rho, \rho_0), \rho_> = \max(\rho, \rho_0), \epsilon_0 = 1, \epsilon_n = 2 for n \geq 1, and I_n, K_n are the modified Bessel functions of the first and second kind, respectively. This representation arises from the Fourier series in the azimuthal direction and Fourier cosine transform in the axial direction, reflecting the symmetry of the problem.^[29] To account for the grounded boundary, the full potential inside the cylinder is obtained via the Green's function method, modifying the free-space expansion to ensure V(\rho = a, \phi, z) = 0. The explicit solution is

V(\rho, \phi, z) = \frac{q}{2\pi^2 \epsilon_0} \sum_{n=0}^\infty \epsilon_n \cos[n(\phi - \phi_0)] \int_0^\infty dk \, \cos[k(z - z_0)] \, I_n(k \rho_<) \left[ K_n(k \rho_>) - \frac{I_n(k \rho_>) K_n(k a)}{I_n(k a)} \right],

where the correction term -\frac{I_n(k \rho_>) K_n(k a)}{I_n(k a)} enforces the zero potential at the boundary. Equivalently, in complex exponential form for the azimuthal sum, it is

V(\rho, \phi, z) = \frac{q}{2 \pi^2 \epsilon_0} \sum_{m=-\infty}^\infty e^{i m (\phi - \phi_0)} \int_0^\infty dk \, \cos[k(z - z_0)] \, I_m(k \rho_<) \left[ K_m(k \rho_>) - \frac{I_m(k \rho_>) K_m(k a)}{I_m(k a)} \right].

This ensures the radial function for \rho > \rho_0 is proportional to I_m(k a) K_m(k \rho) - K_m(k a) I_m(k \rho), which vanishes at \rho = a.^[28] The derivation proceeds by solving Poisson's equation using separation of variables in cylindrical coordinates, expanding the source term (Dirac delta) in a complete set of eigenfunctions for the azimuthal and axial directions. The azimuthal part yields the Fourier series with \cos[n(\phi - \phi_0)], while the infinite extent in z requires a continuous Fourier cosine transform over k > 0, leading to the \cos[k(z - z_0)] factor. For each mode (n, k), the radial equation is the modified Bessel equation, with solutions I_n(k \rho) (regular at \rho = 0) for the inner region \rho < \rho_0 and a linear combination of I_n and K_n for \rho_0 < \rho < a to satisfy the boundary condition. Continuity of the potential and the appropriate jump in the radial derivative at \rho = \rho_0 (determined by the delta function source) fix the coefficients, resulting in the given expansion. Mode matching across the source location ensures the correct singularity. An image charge method is not directly applicable here due to the complexity of the cylindrical geometry for a point source, requiring the infinite mode sum instead.^[28] Physically, each term in the expansion represents a mode that decays exponentially as e^{-k |z - z_0|} (via the Fourier representation of the cosine) away from the source plane at z = z_0, confining the influence along the axis. Higher azimuthal orders n capture finer angular variations in the potential around the off-axis source position, with contributions diminishing for large n due to the oscillatory nature of the Bessel functions. Near the source, the potential approaches the free-space Coulomb form, while far from it or near the boundary, the conducting surface screens the field, inducing opposite surface charges that maintain equipotential zero.^[28]

Free-space comparison

In the free-space scenario, the domain is unbounded without a boundary at \rho = a, and the potential due to a point charge q located at \mathbf{r}_0 = (\rho_0, \phi_0, z_0) is given by V(\mathbf{r}) = \frac{q}{4\pi \epsilon_0 |\mathbf{r} - \mathbf{r}_0|}.^[30] For a source at the origin (z_0 = 0, \rho_0 = 0), the problem is axisymmetric (n=0), and the potential expands as

V(\rho, \phi, z) = \frac{q}{4\pi \epsilon_0} \int_0^\infty dk \, e^{-k |z|} J_0(k \rho).

This representation follows from the integral properties of Bessel functions, where the integral evaluates to \frac{1}{\sqrt{\rho^2 + z^2}}, matching the exact form \frac{q}{4\pi \epsilon_0 \sqrt{\rho^2 + z^2}}.^[30] For a general off-axis source position, the expansion incorporates the full azimuthal dependence:

V(\rho, \phi, z) = \frac{q}{4\pi \epsilon_0} \sum_{n=0}^\infty \epsilon_n \cos[n(\phi - \phi_0)] \int_0^\infty dk \, J_n(k \rho_0) J_n(k \rho) e^{-k |z - z_0|},

with \epsilon_0 = 1 and \epsilon_n = 2 for n > 0. This form arises from the Fourier series in the azimuthal angle combined with the completeness of the Bessel functions in the radial direction for the unbounded domain.^[30] For large |z - z_0|, the potential exhibits asymptotic decay as $1/|z - z_0|, dominated by the low-k contribution in the integral, where J_n(k \rho_0) \approx 1 near k=0. The integral can be evaluated using Weber's discontinuous integral theorem, which provides the limiting behavior for such Bessel integrals. Unlike the free-space case, the solution inside the infinite conducting cylinder modifies the radial functions with a boundary-enforcing term but retains the continuous spectrum over k \in [0, \infty). Both reflect the infinite extent in z, allowing a Fourier transform rather than discrete modes (which would apply to finite-length cylinders). This highlights the role of the cylindrical boundary in screening the field without discretizing the axial wavenumber.^[30]

Applications

Electrostatics

Cylindrical harmonics are employed in electrostatics to solve Laplace's equation ∇²Φ = 0 for potentials in geometries exhibiting cylindrical symmetry, such as infinite lines of charge, conducting cylinders, and coaxial configurations. These solutions arise from separation of variables in cylindrical coordinates (ρ, φ, z), yielding radial functions involving Bessel functions J_n and modified Bessel functions K_n (or I_n), angular terms cos(nφ) or sin(nφ), and axial dependence e^{±κz} or linear in z for specific cases.^[31]^[32] A fundamental application is the electrostatic potential due to an infinite line charge with uniform linear density λ along the z-axis. By symmetry, the potential Φ depends only on the radial distance ρ, reducing Laplace's equation to the n=0, k=0 case of the cylindrical harmonic expansion. The solution is

\Phi(\rho) = -\frac{\lambda}{2\pi \epsilon_0} \ln \rho + C,

where C is a constant determined by a reference potential, often set to zero at some ρ_0. This logarithmic form reflects the two-dimensional nature of the field, derived from integrating the electric field E = (λ/(2π ε_0 ρ)) \hat{ρ} obtained via Gauss's law.^[33] For a conducting cylinder of radius a in an external field or with specified boundary conditions, the potential is expanded using cylindrical harmonics separately inside (ρ < a) and outside (ρ > a) the cylinder. Inside, the solution uses modified Bessel functions of the first kind I_n(κ ρ) to remain finite at ρ=0, while outside it employs modified Bessel functions of the second kind K_n(κ ρ) to ensure decay at infinity. The general forms are

\Phi_\text{in}(\rho, \phi, z) = \sum_{n=0}^\infty \int_0^\infty d\kappa \, [A_n(\kappa) I_n(\kappa \rho) + B_n(\kappa) K_n(\kappa \rho)] [C_n(\kappa) \cos(n\phi) + D_n(\kappa) \sin(n\phi)] e^{\pm \kappa z},

but K_n is typically discarded inside for regularity, and for exterior,

\Phi_\text{out}(\rho, \phi, z) = \sum_{n=0}^\infty \int_0^\infty d\kappa \, [E_n(\kappa) I_n(\kappa \rho) + F_n(\kappa) K_n(\kappa \rho)] [G_n(\kappa) \cos(n\phi) + H_n(\kappa) \sin(n\phi)] e^{\pm \kappa z},

with I_n discarded outside for boundedness. Boundary conditions at ρ = a—such as continuity of Φ and normal derivative (or specified potential)—determine the coefficients A_n, F_n, etc., often via orthogonality of the harmonics. For example, in a uniform axial field, only the n=0, linear z terms contribute, yielding a uniform field inside an uncharged conducting cylinder.^[31]^[32] The capacitance per unit length of coaxial cylinders, with inner radius a and outer radius b (b > a), both conducting and sharing the same axis, is a direct application. The potential between them, assuming the inner at potential V_0 and outer grounded, is the n=0 azimuthal-independent solution:

\Phi(\rho) = A \ln \rho + B,

matched to boundaries to give A = -V_0 / \ln(b/a) and B = V_0 \ln b / \ln(b/a). The charge per unit length on the inner cylinder is λ = 2π ε_0 V_0 / \ln(b/a), yielding capacitance C = 2π ε_0 / \ln(b/a). This result, essential for high-voltage cables, emerges from the logarithmic harmonic without higher-order terms.^[34] In cylindrical coordinates, a multipole expansion analogous to the spherical case represents the potential due to localized charge distributions near the z-axis. For sources offset slightly from the axis, the far-field potential expands as

\Phi(\rho, \phi) \approx \frac{\lambda}{2\pi \epsilon_0} \ln \frac{1}{\rho} + \sum_{n=1}^\infty \frac{1}{\rho^n} (p_n \cos n\phi + q_n \sin n\phi),

where the monopole is the total line charge λ, and higher multipoles p_n, q_n characterize azimuthal asymmetries, decaying as powers of 1/ρ for large ρ. This expansion facilitates analysis of off-axis charges or multipolar sources in cylindrical systems, such as in accelerator electrodes.^[30] The use of cylindrical harmonics in electrostatics traces to early 19th-century developments in potential theory.^[35]

Heat conduction

In steady-state heat conduction problems within cylindrical domains, the temperature field T satisfies Laplace's equation \nabla^2 T = 0 in source-free regions, subject to thermal boundary conditions such as prescribed temperatures or heat fluxes at surfaces. This equation arises from the balance of heat fluxes in the absence of time dependence or internal generation, enabling separation of variables in cylindrical coordinates (\rho, \phi, z) to yield solutions in terms of cylindrical harmonics.^[36] For an insulated cylinder containing an internal heat source, the governing equation becomes Poisson's equation \nabla^2 T = -[q](/page/Q)/[k](/page/K), where [q](/page/Q) is the volumetric heat generation rate and [k](/page/K) is the thermal conductivity. The temperature distribution is expressed as a series expansion using cylindrical harmonics, incorporating terms like \cos(n\phi) J_n(k \rho) to capture azimuthal and radial variations, with coefficients determined by orthogonality to match the source distribution. These Bessel functions J_n ensure bounded solutions at the axis, while insulation at the outer surface imposes zero radial heat flux, \partial T / \partial \rho = 0 at \rho = a. In radial heat flow through pipes under steady-state conditions, the axisymmetric solution (n=0, no axial variation) yields a logarithmic temperature profile T(\rho) = A \ln \rho + B, where A and B are constants set by inner and outer boundary temperatures or fluxes. This form reflects the diverging area with radius in two-dimensional steady flow, leading to a thermal resistance proportional to \ln(r_o / r_i) for a pipe of inner radius r_i and outer radius r_o. For annular regions in multi-layer pipes, the solutions in each layer for pure conduction without axial gradients involve logarithmic functions for n=0, matched at interfaces for continuity of temperature and heat flux. For extended surfaces like fins with surface convection, modified Bessel functions of the first kind I_n and second kind K_n are used. For cases with axial variation leading to exponential dependence, modified Bessel functions apply in the radial direction.^[37]^[38] A practical application arises in modeling temperature distributions within nuclear fuel rods, where internal heat generation from fission drives steady-state conduction in a finite-length cylindrical geometry. Here, discrete modes from cylindrical harmonics, including Bessel functions for radial profiles and Fourier terms for azimuthal and axial dependencies, are used to compute the temperature field, accounting for cladding and coolant boundaries to predict thermal margins.^[39]

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