Find the General Solution of the Bessel Equation

Basic definitions and propositions

Elina Shishkina , Sergei Sitnik , in Transmutations, Singular and Fractional Differential Equations with Applications to Mathematical Physics, 2020

1.1.2 Bessel functions

Bessel functions , named after the German astronomer Friedrich Bessel, are defined as solutions of the Bessel differential equation

$x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}+(x^2-{\alpha }^2)y=0,$

where α is a complex number.

The Bessel functions of the first kind, denoted by $J_{\alpha }(x)$ , are solutions of Bessel's differential equation that are finite at the origin $x=0$ . The Bessel function $J_{\alpha }(x)$ can be defined by the series

(1.13) $J_{\alpha }(x)=\sum _{m=0}^{\infty }\frac{{(-1)}^m}{m!\mathrm{\Gamma }(m+\alpha +1)}{\left(\frac{x}{2}\right)}^{2m+\alpha }.$

For noninteger α the functions $J_{\alpha }(x)$ and $J_{-\alpha }(x)$ are linearly independent. If α is integer the following relationship is valid:

$J_{-\alpha }(x)={(-1)}^{\alpha }{J}_{\alpha }(x).$

The Bessel functions of the second kind, denoted by $Y_{\alpha }(x)$ , for noninteger α are related to $J_{\alpha }(x)$ by the formula

$Y_{\alpha }(x)=\frac{J_{\alpha }(x)\cos (\alpha \pi )-J_{-\alpha }(x)}{\sin (\alpha \pi )}.$

In the case of integer order n, the function $Y_{\alpha }(x)$ is defined by taking the limit as a noninteger α tends to n:

$Y_n(x)=\underset{\alpha \to n}{\lim }{Y}_{\alpha }(x).$

Functions $Y_{\alpha }(x)$ are also called Neumann functions and are denoted by $N_{\alpha }(x)$ . The linear combination of the Bessel functions of the first and second kinds represents a complete solution of the Bessel equation:

$y(x)=C_1{J}_{\alpha }(x)+C_2{Y}_{\alpha }(x).$

Hankel functions of the first and second kind, denoted by $H_{\alpha }^{(1)}(x)$ and $H_{\alpha }^{(2)}(x)$ , respectively, are defined by the equalities

(1.14) $H_{\alpha }^{(1)}(x)=J_{\alpha }(x)+i{Y}_{\alpha }(x)$

and

(1.15) $H_{\alpha }^{(2)}(x)=J_{\alpha }(x)-i{Y}_{\alpha }(x).$

Modified Bessel functions (or occasionally the hyperbolic Bessel functions) of the first and second kind $I_{\alpha }(x)$ and $K_{\alpha }(x)$ are defined as

(1.16) $I_{\alpha }(x)=i^{-\alpha }{J}_{\alpha }(ix)=\sum _{m=0}^{\infty }\frac{1}{m!\mathrm{\Gamma }(m+\alpha +1)}{\left(\frac{x}{2}\right)}^{2m+\alpha },$

(1.17) $K_{\alpha }(x)=\frac{\pi }{2}\frac{I_{-\alpha }(x)-I_{\alpha }(x)}{\sin (\alpha \pi )},$

where α is noninteger.

In the case of integer order α, the functions $I_{\alpha }(x)$ and $K_{\alpha }(x)$ are defined by taking the limit as a noninteger α tends to $n\in \mathbb{Z}$ :

$I_n(x)=\underset{\alpha \to n}{\lim }{I}_{\alpha }(x),K_n(x)=\underset{\alpha \to n}{\lim }{K}_{\alpha }(x).$

It is obvious that $K_{\alpha }(x)=K_{-\alpha }(x)$ .

Function $I_{\nu }(r)$ is exponentially growing when $r\to \infty$ and $K_{\nu }(r)$ is exponentially decaying when $r\to \infty$ for real r and ν:

$I_{\nu }(z)\propto \frac{e^z}{\sqrt{2\pi z}}(1+O\left(\frac{1}{z}\right)),|Arg(z)|<\frac{\pi }{2},\left|z\right|\to \infty ,K_{\nu }(z)\propto \sqrt{\frac{\pi }{2}}\frac{e^{-z}}{\sqrt{z}}(1+O\left(\frac{1}{z}\right)),\left|z\right|\to \infty .$

For small arguments $0<|r|\ll \sqrt{\nu +1}$ , we have

(1.18) $I_{\nu }(r)\sim \frac{1}{\mathrm{\Gamma }(\nu +1)}{\left(\frac{r}{2}\right)}^{\nu },K_{\nu }(r)\sim \{\begin{array}{cc}-\ln \left(\frac{r}{2}\right)-\vartheta \hfill & \text{if}\nu =0,\hfill \\ \frac{\mathrm{\Gamma }(\nu )}{2^{1-\nu }}{r}^{-\nu }\hfill & \text{if}\nu >0,\hfill \end{array}$

where

$\vartheta =\underset{n\to \infty }{\lim }(-\ln n+\sum _{k=1}^n\frac{1}{k})=\underset{1}{\overset{\infty }{\int }}(-\frac{1}{x}+\frac{1}{\lfloor x\rfloor })dx$

is the Euler–Mascheroni constant [121].

Here are some of the important particular cases of Bessel functions:

$J_{\frac{1}{2}}(z)=\sqrt{\frac{2}{\pi z}}\sin (z),J_{-\frac{1}{2}}(z)=\sqrt{\frac{2}{\pi z}}\cos (z),I_{\frac{1}{2}}(z)=\sqrt{\frac{2}{\pi z}}\sinh (z),I_{-\frac{1}{2}}(z)=\sqrt{\frac{2}{\pi z}}\cosh (z),K_{\frac{1}{2}}(z)=K_{-\frac{1}{2}}(z)=\sqrt{\frac{\pi }{2z}}{e}^{-z}.$

The normalized Bessel function of the first kind $j_{\nu }$ is defined by the formula (see [242], p. 10, [317])

(1.19) $j_{\nu }(x)=\frac{2^{\nu }\mathrm{\Gamma }(\nu +1)}{x^{\nu }}{J}_{\nu }(x),$

where $J_{\nu }$ is a Bessel function of the first kind. Operator function of the type (1.19) was considered in [183,187].

The normalized modified Bessel function of the first kind $i_{\nu }$ is defined by the formula

(1.20) $i_{\nu }(x)=\frac{2^{\nu }\mathrm{\Gamma }(\nu +1)}{x^{\nu }}{I}_{\nu }(x),$

where $I_{\nu }$ is a modified Bessel function of the first kind.

The normalized modified Bessel function of the second kind $k_{\nu }$ is defined by the formula

(1.21) $k_{\nu }(x)=\frac{1}{2^{\nu }\mathrm{\Gamma }(1+\nu )x^{\nu }}{K}_{\nu }(x),$

where $K_{\nu }$ is a modified Bessel function of the second kind. We have

(1.22) $\frac{d{k}_{\nu }(x)}{dx}=-\frac{1}{2^{\nu }\mathrm{\Gamma }(1+\nu )x^{\nu }}{K}_{\nu +1}(x).$

Here are some of the important particular cases of normalized Bessel functions:

$j_{\frac{1}{2}}(z)=\frac{\sin (z)}{z},j_{-\frac{1}{2}}(z)=\cos (z),i_{\frac{1}{2}}(z)=\frac{\sinh (z)}{z},i_{-\frac{1}{2}}(z)=\cosh (z),k_{\frac{1}{2}}(z)=\frac{e^{-z}}{z},k_{-\frac{1}{2}}(z)=e^{-z}.$

Using formulas (9.1.27) from [2] we obtain that $j_{\nu }(t)$ is an eigenfunction of operator ${(B_{\nu })}_t=\frac{d^2}{d{t}^2}+\frac{\nu }{t}\frac{d}{dt}$ :

(1.23) ${(B_{\nu })}_t{j}_{\frac{\nu -1}{2}}(\tau t)=-{\tau }^2{j}_{\frac{\nu -1}{2}}(\tau t),$

(1.24) ${(B_{\nu })}_t{i}_{\frac{\nu -1}{2}}(\tau t)={\tau }^2{i}_{\frac{\nu -1}{2}}(\tau t),$

(1.25) ${(B_{\nu })}_t{k}_{\frac{\nu -1}{2}}(\tau t)={\tau }^2{k}_{\frac{\nu -1}{2}}(\tau t).$

Normalized Bessel functions have the following properties:

(1.26) $j_{\nu }(0)=1,j_{\nu }^{\prime }(0)=0,i_{\nu }(0)=1,i_{\nu }^{\prime }(0)=0,\underset{x\to 0}{\lim }{x}^{2\nu }{k}_{\nu }(x)=\frac{1}{2\nu },\nu >0,$

(1.27) $\underset{x\to 0}{\lim }{k}_{\nu }(x)=\frac{\mathrm{\Gamma }(-\nu )}{2^{2\nu +1}\mathrm{\Gamma }(1+\nu )},\nu <0,-\nu \notin \mathbb{N},$

(1.28) $\underset{x\to 0}{\lim }{x}^{\alpha }{k}_0(x)=0,\alpha >0,\underset{x\to 0}{\lim }\frac{1}{\ln x}{k}_0(x)=-1,$

(1.29) $\underset{x\to 0}{\lim }{x}^{2\nu +1}\frac{d{k}_{\nu }(x)}{dx}=-1,\nu >-1.$

We will use notations

(1.30) ${\mathbf{j}}_{\gamma }(x,\xi )=\prod _{i=1}^n{j}_{\frac{{\gamma }_i-1}{2}}(x_i{\xi }_i)$

and

(1.31) ${\mathbf{i}}_{\gamma }(x,\xi )=\prod _{i=1}^n{i}_{\frac{{\gamma }_i-1}{2}}(x_i{\xi }_i),$

where $\gamma =({\gamma }_1,...,{\gamma }_n)$ , ${\gamma }_1>0,...,{\gamma }_n>0$ .

Information about the Bessel functions is taken from [591].

The Struve function is a solution $y(x)$ of the nonhomogeneous Bessel differential equation:

$x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}+(x^2-{\alpha }^2)y=\frac{4{\left(\frac{x}{2}\right)}^{\alpha +1}}{\sqrt{\pi }\mathrm{\Gamma }(\alpha +\frac{1}{2})}.$

Struve functions, denoted as ${\mathbf{H}}_{\alpha }(x)$ , have the power series form

(1.32) ${\mathbf{H}}_{\alpha }(x)=\sum _{m=0}^{\infty }\frac{{(-1)}^m}{\mathrm{\Gamma }(m+\frac{3}{2})\mathrm{\Gamma }(m+\alpha +\frac{3}{2})}{\left(\frac{x}{2}\right)}^{2m+\alpha +1}.$

Another definition of the Struve function, for values of α satisfying $\mathrm{Re}\alpha >-\frac{1}{2}$ , is possible using an integral representation:

${\mathbf{H}}_{\alpha }(x)=\frac{2{\left(\frac{x}{2}\right)}^{\alpha }}{\sqrt{\pi }\mathrm{\Gamma }(\alpha +\frac{1}{2})}{\int }_0^{\frac{\pi }{2}}\sin (x\cos \tau ){\sin }^{2\alpha }(\tau )d\tau .$

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Sturm-Liouville Eigenvalue Problems and Generalized Fourier Series

George A. Articolo , in Partial Differential Equations & Boundary Value Problems with Maple (Second Edition), 2009

Normalization

> norm:=sqrt(Int(w(x)*(BesselJ(m,lambda[m,n]*x))ˆ2,x=0..b));

(2.285) $norm:=\sqrt{\underset{0}{\overset{1}{\int }}x\text{Besse}1\text{J}{(1,{\lambda }_{1,n}x)}^2\text{d}x}$

Substitution of the eigenvalue equation simplifies the norm

> norm:=radsimp(subs(BesselJ(1,lambda[1,n])=0,value(%)));

(2.286) $norm:=\frac{1}{2}\sqrt{2}\text{Besse}1\text{J}(0,{\lambda }_{1,n})$

Orthonormal eigenfunctions

> phi[m,n](x):=BesselJ(m,lambda[m,n]*x)/norm;

(2.287) ${\varphi }_{1,n}(x):=\frac{\text{Besse}1\text{J}(1,{\lambda }_{1,n}x)\sqrt{2}}{\text{Besse}1\text{J}(0,{\lambda }_{1,n})}$

Fourier coefficients

> F(n):=Int(f(x)*phi[m,n](x)*w(x),x=a..b);

(2.288) $F(n):=\underset{0}{\overset{1}{\int }}\frac{x^2\text{Besse}1\text{J}(1,{\lambda }_{1,n}x)\sqrt{2}}{\text{Besse}1\text{J}(0,{\lambda }_{1,\text{n}})}\text{d}x$

Substitution of the eigenvalue equation simplifies the preceding equation

> F(n):=subs(BesselJ(1,lambda[1,n])=0,value(%));

(2.289) $F(n):=-\frac{\sqrt{2}}{{\lambda }_{1,n}}$

> Series:=Sum(F(n)*phi[m,n](x),n=1..infinity);

(2.290) $Series:=\sum _{n=1}^{\infty }\left(-\frac{2\text{Besse}1\text{J}(1,{\lambda }_{1,n}x)}{{\lambda }_{1,n}\text{Besse}1\text{J}(0,{\lambda }_{1,n})}\right)$

First five terms in expansion

> Series:=sum(F(n)*phi[m,n](x),n=1..5):

> plot({Series,f(x)},x=a..b,thickness=10);

The two curves in Figure 2.19 depict the function f(x) and its series expansion approximation in terms of the Bessel functions of the first kind of order m = 1, which satisfy the given boundary conditions over the interval. Note that f(x) satisfies the given boundary conditions at the left but fails to do so at the right end point.

Figure 2.19.

EXAMPLE 2.6.3

Consider the Bessel operator with Neumann conditions. We seek the eigenvalues and corresponding orthonormal eigenfunctions for the Bessel differential equation of order m = 0 [Sturm-Liouville type for p(x) = x, q(x) = 0, w(x) = x] over the interval I = {x | 0 < x < b}. The boundary conditions are that the solution be finite at the origin and that there is a type 2 condition at the point x = b. The Sturm-Liouville problem involving the Bessel operator is nonregular over this interval. From Green's formula in Section 2.2, since p(0) = 0, we continue to obtain a condition for orthogonality over this interval if, in addition to requiring the solution to be finite at the origin, we further require that at x = b, its first derivative vanishes (see Exercise 2.34).

Bessel differential equation of order m = 0

$x^2\left(\frac{{\text{d}}^2}{\text{d}{x}^2}y(x)\right)+x\left(\frac{\text{d}}{\text{d}x}y(x)\right)+{\lambda }^2{x}^{2}y(x)=0$

Boundary conditions

$|y(0)|<\infty \text{and}{y}_x(b)=0$

SOLUTION

We consider two possibilities for values of λ. We first consider the case for λ = 0. The system basis vectors are

> restart:y1(x):=1;y2(x):=ln(x);

(2.291) $\begin{array}{l}y1(x):=1\\ y2(x):=ln(x)\end{array}$

General solution

> y(x):=C1*y1(x)+C2*y2(x);

(2.292) $y(x):=C1+C2ln(x)$

Since ln(x) fails to exist at x = 0, we set C2 = 0, giving

> y(x):=C1;

(2.293) $y(x):=C1$

Substituting into the boundary condition at the periphery yields

> eval(subs(x=a,diff(y(x),x)))=0;

(2.294) $0=0$

Thus, C1 is arbitrary and the eigenvalue and eigenfunction corresponding to λ = 0 are given as

> lambda[0]:=0;

(2.295) ${\lambda }_0:=0$

> phi[0](x):=1;

(2.296) ${\varphi }_0(x):=1$

We next consider the case for λ > 0. The system basis vectors are

> y1(x):=BesselJ(0,lambda*x);y2(x):=BessselY(0,lambda*x);

(2.297) $\begin{array}{l}y1(x):=\text{Besse}1\text{J}(0,\lambda x)\\ y2(x):=\text{Besse}1\text{Y}(0,\lambda x)\end{array}$

General solution

> y(x):=C1*y1(x)+C2*y2(x);

(2.298) $y(x):=C1\text{Besse}1\text{J}(0,\lambda ,x)+C2\text{Besse}1\text{Y}(0,\lambda x)$

Substituting the boundary condition at the origin indicates that, since the Bessel function of the second kind [Y(m, μx)] is not finite at the origin, we must set C2 = 0. Substituting the remaining condition at x = b yields

> eval(subs({x=b,C2=0},diff(y(x),x)))=0;

(2.299) $-C1\text{Besse}1\text{J}(1,\lambda b)\lambda =0$

The only nontrivial solutions to the above occur when C1 is arbitrary and λ satisfies the following eigenvalue equation

(2.300) $\text{Besse}1\text{J}(1,{\lambda }_{n}b)=0$

for n = 1, 2, 3, ….

Nonnormalized eigenfunctions are

> phi[n](x):=BesselJ(0,lambda[n]*x);

(2.301) ${\phi }_n(x):=\text{Besse}1\text{J}(0,{\lambda }_{n}x)$

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Partial Differential Equations in Physics

In Pure and Applied Mathematics, 1949

B Two-Dimensional Case f = f(r, φ)

We first develop f(r, φ) in the complex Fourier series (1.12)

(11) $f\left(r,\phi \right)=\underset{n=-\infty }{\overset{n=+\infty }{\Sigma }}{C}_n{e}^{in\phi },C_n=C_n\left(r\right)=\frac{1}{2\pi }\underset{-\pi }{\overset{+\pi }{\int }}f\left(r,\phi \right)e^{-in\phi }d\phi .$

Due to the two-dimensional equation (2)

(12) $\frac{{\partial }^{2}u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^2}\frac{{\partial }^{2}u}{\partial {\phi }^2}=\frac{1}{k}\frac{\partial u}{\partial t}$

and the generalized substitution (3)

(13) $u=R_n\left(r\right)e^{in\phi }{e}^{-{\lambda }^{2}kt}$

we have the differential equation for R_n (r)

(14) $\frac{d^2{R}_n}{d{r}^2}+\frac{1}{r}\frac{d{R}_n}{dr}+\left({\lambda }^2-\frac{n^2}{r^2}\right)R_n=0.$

This is Bessel's differential equation (19.11) with ϱ = λr . Equation (1a) requires that the only permissible solutions be of the form A_nI_n (λr). On account of (1) λ must satisfy the equation I_n(λa) = 0 which, just like I ₀(λ a) = 0, has an infinity of roots:

${\lambda }_{n,1},{\lambda }_{n,2},\dots ,{\lambda }_{n,m},\dots .$

Each of these roots yields a particular solution of the form (13):

(15) $u_{n,m}=A_{n,m}{I}_n\left({\lambda }_{n,m}r\right)e^{in\phi }{e}^{-{\lambda }_{n,m}kt},$

and these solutions satisfy the differential equation (12). Through superposition we can construct from them the general solution of (14) which at the same time satisfies our boundary conditions:

(16) $u=\Sigma \Sigma u_{n,m}=\underset{n=-\infty }{\overset{+\infty }{\Sigma }}\underset{m=1}{\overset{\infty }{\Sigma }}{A}_{n,m}{I}_n\left({\lambda }_{n,m}r\right)e^{in\phi }{e}^{-{\lambda }_{n,m}^{2}kt}.$

Here the constants A_n,m must be chosen so that for t = 0 and every integer – ∞ < n < + ∞ we have the equation

(17) $C_n\left(r\right)=\underset{m=1}{\overset{\infty }{\Sigma }}{A}_{n,m}{I}_n\left({\lambda }_{n,m}r\right)$

where according to (11) the left side is a known function of r. Equation (17) necessitates the development of this function in Bessel functions I_n . This is possible due to the orthogonality of the latter, which follows from Bessel's differential equation (14) and Green's theorem as in (7) and (7a) ⁸ Using the abbreviations

$v_m=I_n\left({\lambda }_{n,m}r\right),v_l=I_n\left({\lambda }_{n,l}r\right)$

we obtain as generalization of (7)

${\left({\lambda }_{n,m}^2-{\lambda }_{n,l}^2\right)\underset{0}{\overset{a}{\int }}r{v}_m{v}_{l}dr=r\left(v_m\frac{d{v}_l}{dr}-v_l\frac{d{v}_m}{dr}\right)|}_0^a$

Here, too, the right side vanishes. We thus have for l ≠ m

(18) $\underset{0}{\overset{a}{\int }}{v}_m{v}_{l}rdr=0.$

At the same time we obtain by a passage to the limit as described in (9)

(19) $N_{n,m}=\underset{0}{\overset{a}{\int }}{v}_m^{2}rdr=\frac{a^2}{2}{\left[{I^{\prime }}_n\left({\lambda }_{n,m}a\right)\right]}^2.$

The A_nm in (17) can now be calculated in the Fourier manner from the given C_n (r) by (18) and (19) in analogy to (10). Substituting these expressions for C_n (r) in (11) we obtain

(20) $2\pi N_{n,m}{A}_{n,m}=\underset{0}{\overset{a}{\int }}\underset{-\pi }{\overset{+\pi }{\int }}f\left(r,\phi \right)I_n\left({\lambda }_{n,m}r\right)e^{-in\phi }rdrd\phi .$

which concludes the solution of (16).

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Cylinder and Sphere Problems

ARNOLD SOMMERFELD , in Partial Differential Equations in Physics, 1949

B TWO-DIMENSIONAL CASE f = f(r,φ)

We first develop f(r,φ) in the complex Fourier series (1.12)

(11) $\begin{array}{cc}f(r,\phi )={\displaystyle \sum _{n=-\infty }^{n=+\infty }{C}_n{e}^{in\phi }},& C_{n^{\dot{}}}=C_n(r)=\frac{1}{2\pi }{\displaystyle \underset{-\pi }{\overset{+\pi }{\int }}f(r,\phi )e^{-in\phi }d\phi }\end{array}.$

Due to the two-dimensional equation (2)

(12) $\frac{{\partial }^{2}u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^2}\frac{{\partial }^{2}u}{\partial {\phi }^2}=\frac{1}{k}\frac{\partial u}{\partial t}$

and the generalized substitution (3)

(13) $u=R_n(r)e^{in\phi }{e}^{-{\lambda }^{2}kt}$

we have the differential equation for R_n (r)

(14) $\frac{d^2{R}_n}{d{r}^2}+\frac{1}{r}\frac{d{R}_n}{dr}+\left({\lambda }^2-\frac{n^2}{r^2}\right)R_n=0.$

This is Bessel's differential equation (19.11) with φ = λr . Equation (1a) requires that the only permissible solutions be of the form A_nI_n (λr). On account of (1) λ must satisfy the equation I_n (λa) = 0 which, just like I ₀(λa) = 0, has an infinity of roots:

$\begin{array}{ccc}{\lambda }_{n,1},& {\lambda }_{n,2},\dots ,& {\lambda }_{n,m},\dots .\end{array}$

Each of these roots yields a particular solution of the form (13):

(15) $u_{n,m}=A_{n,m}{I}_n({\lambda }_{n,m}r)e^{in\phi }{e}^{-{\lambda }_{n,m}kt},$

and these solutions satisfy the differential equation (12). Through superposition we can construct from them the general solution of (14) which at the same time satisfies our boundary conditions:

(16) $u={\displaystyle \sum {\displaystyle \sum u_{n,m}}}={\displaystyle \sum _{n=-\infty }^{+\infty }{\displaystyle \sum _{m=1}^{\infty }{A}_{n,m}{I}_n({\lambda }_{n,m}r)e^{in\phi }{e}^{-{\lambda }_{n,m}^{2}kt}}}.$

Here the constants A_n,m must be chosen so that for t = 0 and every integer − ∞ < n < + ∞ we have the equation

(17) $C_n(r)={\displaystyle \sum _{m=1}^{\infty }{A}_{n,m}{I}_n({\lambda }_{n,m}r)}$

where according to (11) the left side is a known function of r. Equation (17) necessitates the development of this function in Bessel functions I_n . This is possible due to the orthogonality of the latter, which follows from Bessel's differential equation (14) and Green's theorem as in (7) and (7a) ⁸ . Using the abbreviations

$\begin{array}{cc}{v}_m=I_n({\lambda }_{n,m}r),& v_l=I_n({\lambda }_{n,l}r)\end{array}$

we obtain as generalization of (7)

$\left({\lambda }_{n,m}^2-{\lambda }_{n,l}^2\right){\displaystyle \underset{0}{\overset{a}{\int }}r{v}_m{v}_{l}dr}={r\left(v_m\frac{d{v}_l}{dr}-v_l\frac{d{v}_m}{dr}\right)|}_0^a$

Here, too, the right side vanishes. We thus have for l ≠ m

(18) ${\displaystyle \underset{0}{\overset{a}{\int }}{v}_m{v}_{l}rdr}=0.$

At the same time we obtain by a passage to the limit as described in (9)

(19) $N_{n,m}={\displaystyle \underset{0}{\overset{a}{\int }}{v}_m^{2}rdr}=\frac{a^2}{2}{[{I^{\prime }}_n\left({\lambda }_{n,m}a\right)]}^2.$

The A_nm in (17) can now be calculated in the Fourier manner from the given C_n(r) by (18) and (19) in analogy to (10). Substituting these expressions for C_n (r) in (11) we obtain

(20) $2\pi N_{n,m}{A}_{n,m}={\displaystyle \underset{0}{\overset{a}{\int }}{\displaystyle \underset{-\pi }{\overset{+\pi }{\int }}f(r,\phi )I_n({\lambda }_{n,m}r)e^{-in\phi }rdrd\phi }}.$

which concludes the solution of (16).

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Eigenfunctions and Eigen Values

ARNOLD SOMMERFELD , in Partial Differential Equations in Physics, 1949

B THE EIGENFUNCTIONS OF UNLIMITED MANY-DIMENSIONAL SPACE

From the potential equation we pass to the wave equation. For a function which depends only on r the wave equation is, according to (3),

(6) $\frac{d^{2}u}{d{r}^2}+\frac{p+1}{r}\frac{du}{dr}+k^{2}u=0.$

If we set u = r^{−p /2} w , then we obtain the Bessel differential equation with index p/2 for w. Hence (6) is integrated by

(6a) $u=r^{-p/2}{I}_{p/2}(kr),$

and also by

(6b) $u=r^{-p/2}{H}_{p/2}^1(kr),$

(6c) $u=r^{-p/2}{H}_{p/2}^2(kr).$

The function in (6b) behaves asymptotically like

$\begin{array}{cc}C{e}^{ikr}/r\frac{p+1}{2},& C=\sqrt{\frac{2}{k\pi }}\end{array}e{-}^{\frac{p+1}{2}\frac{i\pi }{2}}$

and satisfies the radiation condition (28.7)

$\underset{e\to \infty }{\text{Lim}{r}^{\frac{p+1}{2}}}\left(\frac{\partial u}{\partial r}-iku\right)=0;$

In the same manner (6c) satisfies the absorption condition. Hence (6b,c) represent the radiated and absorbed spherical waves in (p + 2)-dimensional space. This remains valid for a general position of the source point with the solutions

(7) $\begin{array}{cc}U=R^{-p/2}{H}_{p/2}(kR),& R^2=r^2-2r{r}_0\cos +r_0^2.\end{array}$

The function in (6a) may be called "eigenfunction of spherical symmetry." We now want to find the general eigenfunctions of zonal symmetry. They are of the form

(8) $u_n(r,\vartheta )={\upsilon }_n(r)P_n(\cos \vartheta |p).$

From the equation (5c) of P_n we find the differential equation of v_n

$\left(\frac{d^2}{d{r}^2}+\frac{p+1}{r}\frac{d}{dr}+k^2-\frac{n(n+p)}{r^2}\right){\upsilon }_n=0.$

If we treat this equation as we did (6) by setting v_n = r^{−p /2} w then for w we obtain the Bessel differential equation with index n + p/2, and hence as the solution which is finite for r = 0

$w=I_{n+p/2}(kr).$

Hence the eigenfunction becomes

(8a) $u_n=r^{-p/2}{I}_{n+p/2}(kr)P_n(\cos \vartheta |p).$

According to §26 any two of these eigenfunctions are mutually orthogonal, both in the continuous spectrum 0 < k < ∞, and in the discrete spectrum n = 0,1,2, ….

For two eigenfunction u_n,u_m with equal k but different indices we obtain from (2b,c) and (8):

(9) ${\displaystyle \int u_n{u}_{m}d\tau ={\displaystyle \underset{0}{\overset{\infty }{\int }}{I}_{n+p/2}(kr)I_{m+p/2}(kr)rdr{\displaystyle \underset{0}{\overset{\pi }{\int }}{P}_n(\cos \vartheta |p)P_m(\cos \vartheta |p){\sin }^p\vartheta d\vartheta {\Omega }_{\phi }.}}}$

where ω _φ is as in (2f). Due to the fact that neither ω _φ nor the integral with respect to r vanish and due to the orthogonality of u_n and u_m we obtain:

(10) $\begin{array}{cc}{\displaystyle \underset{0}{\overset{\pi }{\int }}{P}_n(\cos \vartheta |p)P_m(\cos \vartheta |p){\sin }^p\vartheta d\vartheta =0,}& m\ne n.\end{array}$

Note the characteristic factor sin^vϱ in (10), which in the three-dimensional case (p = 1) becomes the customary factor sin ϱ for the Legendre polynomials. While in the customary analytic derivation of (10) this factor might appear artificial, it follows in our many-dimensional approach directly from the meaning of dτ.

We also note the corresponding normalizing integral for m = n

(11) $N={\displaystyle \underset{0}{\overset{\pi }{\int }}{[P_n(\cos \vartheta |p)]}^2{\sin }^p\vartheta d\vartheta =}\frac{\Gamma (n+p)}{2^{p-1}(n+p/2)n!}\frac{\pi }{\Gamma (p/2)\Gamma (p/2)}$

which is a generalization of the normalizing integral for ordinary zonal spherical harmonics: $N=1/\left(1+\frac{1}{2}\right)$ for p = 1. The proof of (11) starts from the defining equation (5) of the Gegenbauer polynomials.

With the help of (2e) we can replace (11) by:

(11a) $N=\frac{p}{2}\frac{\Gamma (n+p)}{(n+p/2)n!\Gamma (p)}{\Omega }_{\vartheta }.$

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Partial Differential Equations in Physics

In Pure and Applied Mathematics, 1949

B. The Eigenfunctions Of Unlimited Many-Dimensional Space

From the potential equation we pass to the wave equation. For a function which depends only on r the wave equation is, according to (3),

(6) $\frac{d^{2}u}{d{r}^2}+\frac{p+1}{r}\frac{du}{dr}+k^{2}u=0.$

If we set $u=r^{-p/2}w,$ then we obtain the Bessel differential equation with index p/2 for w. Hence (6) is integrated by

(6a) $u=r^{-p/2}{I}_{p/2}\left(kr\right),$

and also by

(6b) $u=r^{-p/2}{H}_{p/2}^1\left(kr\right),$

(6c) $u=r^{-p/2}{H}_{p/2}^2\left(kr\right).$

The function in (6b) behaves asymptotically like

$C{e}^{ikr}/r^{\frac{p+1}{2}},C=\sqrt{\frac{2}{k\pi }}{e}^{-\frac{p+1}{2}\frac{i\pi }{2}}$

and satisfies the radiation condition (28.7)

$\underset{r\to \infty }{lim}{r}^{\frac{p+1}{2}}\left(\frac{\partial u}{\partial r}-iku\right)=0;$

In the same manner (6c) satisfies the absorption condition. Hence (6b, c) represent the radiated and absorbed spherical waves in (p + 2)-dimensional space. This remains valid for a general position of the source point with the solutions

(7) $U=R^{-p/2}{H}_{p/2}\left(kR\right),R^2=r^2-2r{r}_{0}cos\theta +r_0^2.$

The function in (6a) may be called "eigenfunction of spherical symmetry." We now want to find the general eigenfunctions of zonal symmetry. They are of the form

(8) $u_n\left(r,\vartheta \right)=v_n\left(r\right)P_n\left(cos\theta |p\right).$

From the equation (5c) of P_n we find the differential equation of v_n

$\left(\frac{d^2}{d{r}^2}+\frac{p+1}{r}\frac{d}{dr}+k^3-\frac{n\left(n+p\right)}{r^2}\right)v_n=0.$

If we treat this equation as we did (6) by setting $v_n=r^{-p/2}w$ then for w we obtain the Bessel differential equation with index n + p/2, and hence as the solution which is finite for r = 0

$\omega =I_{n+p/2}\left(kr\right).$

Hence the eigenfunction becomes

(8a) $u_n=r^{-p/2}{I}_{n+p/2}\left(kr\right)P_n\left(cos\vartheta |p\right).$

According to §26 any two of these eigenfunctions are mutually orthogonal, both in the continuous spectrum 0 < k < ∞, and in the discrete spectrum n = 0,1,2, … .

For two eigenfunction u_n,u_m with equal k but different indices we obtain from (2b, c) and (8):

(9) $\int u_n{u}_{m}d\tau =\underset{0}{\overset{\infty }{\int }}{I}_{n+p/2}\left(kr\right)I_{m+p/2}\left(kr\right)rdr\underset{0}{\overset{\pi }{\int }}{P}_n\left(cos\vartheta |p\right)P_m\left(cos\vartheta |p\right){sin}^p\vartheta d\vartheta {\Omega }_{\phi }.$

where Ω_φ is as in (2f). Due to the fact that neither Ω_φ nor the integral with respect to r vanish and due to the orthogonality of u_n and u_m we obtain:

(10) $\underset{0}{\overset{\pi }{\int }}{P}_n\left(cos\vartheta |p\right)P_m\left(cos\vartheta |p\right){sin}^p\vartheta d\vartheta =0,m\ne n.$

Note the characteristic factor sin^ν ϑ in (10), which in the three-dimensional case (p = 1) becomes the customary factor sin ϑ for the Legendre polynomials. While in the customary analytic derivation of (10) this factor might appear artificial, it follows in our many-dimensional approach directly from the meaning of dτ.

We also note the corresponding normalizing integral for m = n

(11) $N=\underset{0}{\overset{\pi }{\int }}{\left[P_n\left(cos\vartheta |p\right)\right]}^2{sin}^{\gamma }\vartheta d\vartheta =\frac{\Gamma \left(n+p\right)}{2^{p-1}\left(n+p/2\right)n!}\frac{\pi }{\Gamma \left(p/2\right)\Gamma \left(p/2\right)}$

which is a generalization of the normalizing integral for ordinary zonal spherical harmonics: $N=1/\left(n+\frac{1}{2}\right)$ for p = 1. The proof of (11) starts from the defining equation (5) of the Gegenbauer polynomials.

With the help of (2e) we can replace (11) by:

(11a) $N=\frac{p}{2}\frac{\Gamma \left(n+p\right)}{\left(n+p/2\right)n!\Gamma \left(p\right)}{\Omega }_{\vartheta }.$

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Semiconductor Nanowires II: Properties and Applications

Sudha Mokkapati , ... Chennupati Jagadish , in Semiconductors and Semimetals, 2016

2.1 Solutions to Maxwell's Equations: Guided and Leaky Modes

Maxwell's equations (Griffiths, 1981) in a nonmagnetic (relative magnetic permeability   =   1), nonconducting material are written as

(1) $\nabla \cdot \mathbf{D}=0\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}$

(2) $\nabla \cdot \mathbf{B}=0\nabla \times \mathbf{H}=\frac{\partial \mathbf{D}}{\partial t}$

where E is the electric field, D is the displacement vector, B is the magnetic field, and H is the auxiliary field. They are related to each other by

(3) $\mathbf{D}=ϵ\left(r\right)\mathbf{E}\mathbf{B}=\mu \mathbf{H}$

ϵ(r) is the permittivity of the material and, in principle, could be position dependent, while μ is the magnetic permeability of the material and related to the refractive index, n of the material.

For modes propagating along the axis of the waveguide, an implicit dependence of the form ${\mathrm{e}}^{-\mathrm{i}\left(\omega t-k_{z}z\right)}$ is implied where the waveguide axis is along the z-axis, ω is the angular frequency, and k _z is the propagation constant for the mode. For solutions with this dependence on z and t, Maxwell's equations reduce to (Snyder and Love, 1983)

(4) ${\{\nabla }_t^2+n^2{k}^2\}\mathbf{E}+\nabla \left\{\mathbf{E}\bullet \nabla \left[ln\left(n^2\right)\right]\right\}=k_z^2\mathbf{E}$

III–V semiconductor nanowires can be approximated as cylinders with constant permittivity surrounded by air and in this case, Maxwell's equations reduce to Helmholtz's equation or the scalar wave equation:

(5) ${\{\nabla }_t^2+n^2{k}^2\}\mathbf{E}=k_z^2\mathbf{E}$

where k is the wave number of the mode. A similar equation can be written for H. For electromagnetic waves with functional dependence of the form ${\mathrm{e}}^{-\mathrm{i}\left(\omega t-k_{z}z\right)}$ , the transverse field components can be written entirely in terms of the z-component. So, we only require to solve Eq. (5) for the z-components of E and H.

In cylindrical coordinates, assuming azimuthal dependence of the form e^imϕ where m is a positive integer, Eq. (5) can be written as

(6) $\frac{{\partial }^2{E}_z}{\partial {\rho }^2}+\frac{1}{\rho }\frac{\partial E_z}{\partial \rho }+\left({\gamma }^2-\frac{m^2}{{\rho }^2}\right)E_z=0$

where ${\gamma }^2=n^2{k}^2-k_z^2$ . A similar equation can be written for H _z . Equation (6) takes the form of the well known Bessel's differential equation with a change of variable $s=\gamma \rho$ . With an added constraint that the solutions decay to 0 at infinity, the solutions take the form (Okamoto, 2006; Snyder and Love, 1983):

(7) ${\mathrm{E}}_z=\left\{\begin{array}{c}{A}_e{J}_m\left(\gamma \rho \right){\mathrm{e}}^{\mathrm{i}m\varphi }\mathrm{for}\rho <a\\ B_e{K}_m\left(\beta \rho \right){\mathrm{e}}^{\mathrm{i}m\varphi }\mathrm{for}\rho >a\end{array}\right.$

(8) ${\mathrm{H}}_z=\left\{\begin{array}{c}{A}_h{J}_m\left(\gamma \rho \right){\mathrm{e}}^{\mathrm{i}m\varphi }\mathrm{for}\rho <a\\ B_h{K}_m\left(\beta \rho \right){\mathrm{e}}^{\mathrm{i}m\varphi }\mathrm{for}\rho >a\end{array}\right.$

where a is the radius of the nanowire, J _m is the Bessel's function of first kind of order m, and K _m is the modified Bessel function of the second kind of order m. A _e,h and B _e,h are constants that are to be determined using boundary conditions (Griffiths, 1981) at the nanowire–air interface:

(9) $D_1^{\perp }-D_2^{\perp }=0;{\mathbf{E}}_1^{\parallel }-{\mathbf{E}}_2^{\parallel }=0$

(10) $B_1^{\perp }-B_2^{\perp }=0;{\mathbf{H}}_1^{\parallel }-{\mathbf{H}}_2^{\parallel }=0$

The boundary conditions also give us the characteristic equation (Okamoto, 2006; Snyder and Love, 1983):

(11) $\left(\frac{1}{p}\frac{J_m^{\prime }\left(p\right)}{J_m\left(p\right)}+\frac{1}{q}\frac{K_m^{\prime }\left(q\right)}{K_m\left(q\right)}\right)\left(\frac{n^2}{p}\frac{J_m^{\prime }\left(p\right)}{J_m\left(p\right)}+\frac{1}{q}\frac{K_m^{\prime }\left(q\right)}{K_m\left(q\right)}\right)=m^2{n}_{\mathrm{eff}}^2\frac{V^4}{p^4{q}^4}$

where $a\gamma =\mathit{ka}\sqrt{n^2-n_{\mathrm{eff}}^2}$ ; $q=a\beta =\mathit{ka}\sqrt{n_{\mathrm{eff}}^2-1\text{;}}$ and V is the waveguide parameter given by $V=\mathit{ka}\sqrt{n^2-1}$ . n _eff is the effective index of the Eigen mode given by $n_{\mathrm{eff}}=\frac{k_z}{k}$ . The characteristic equation needs to be solved for n _eff or k _z , and this gives us the dispersion and field components for all the Eigen modes supported in the nanowire. Solutions to the characteristic equation that satisfy $1<\mathrm{R}\mathrm{e}\left(n_{\mathrm{eff}}\right)<\mathrm{R}\mathrm{e}\left(n\right)$ are the guided Eigen modes and solutions with $\mathrm{R}\mathrm{e}\left(n_{\mathrm{eff}}\right)<1$ are the leaky eigen modes. Im(n _eff) determines the propagation loss for the Eigen modes.

For $m=0$ , we get azimuthally symmetric solutions for E _z and H _z (Eqs. 7 and 8). Two classes of solutions are possible depending on whether Eq. (11) is solved for $\left(\frac{1}{p}\frac{J_m^{\prime }\left(p\right)}{J_m\left(p\right)}+\frac{1}{q}\frac{K_m^{\prime }\left(q\right)}{K_m\left(q\right)}\right)=0$ or $\left(\frac{n^2}{p}\frac{J_m^{\prime }\left(p\right)}{J_m\left(p\right)}+\frac{1}{q}\frac{K_m^{\prime }\left(q\right)}{K_m\left(q\right)}\right)=0$ , corresponding to TE and TM modes, respectively. For TE modes, $E_z=0$ and for TM modes, $H_z=0$ . Solving Eq. (11) for $m\ne 0$ gives solutions with all six nonzero field (E and H) components. In this case, the E _z and H _z components do not have azimuthal symmetry and the Eigen modes can be classified as either HE or EH modes depending on whether E _z varies as cos mϕ or sin mϕ. Different orders of the TE, TM, HE, or EH modes are distinguished from each other by two subscripts. The first subscript denotes the azimuthal order and the second subscript denotes the radial order. For TE and TM modes, because of the azimuthal symmetry the first subscript is always 0.

Figure 1 shows the dispersion (n _eff vs. diameter at a fixed wavelength) for lowest order TE, TM, HE, and EH modes in a nanowire with n  =   3.4, in air. The corresponding E _z , H _z , and |E|² profiles are shown on the right. HE₁₁ mode is the fundamental mode and does not have a cutoff. The lowest order TE and TM modes have a cutoff at 200   nm diameter and other modes are supported in larger diameter nanowires. With increasing nanowire diameter, n _eff of a given mode increases and approaches n—the refractive index of the nanowire. For $\mathrm{R}\mathrm{e}\left(n_{\mathrm{eff}}\right)~1$ , the Eigen mode has large evanescent fields and as n _eff increases, the field confinement inside the nanowire increases.

Figure 1. Dispersion relation at 880   nm for the lowest order HE, TE, TM, and EH modes supported in a nanowire of index 3.4 as a function of diameter (right). Normalized profiles for the axial (z) components of electric and magnetic field and the electric field intensity for these modes in a 600   nm diameter nanowire are also shown (left).

The dispersion relationship shown in Fig. 1 can also be expressed in terms of k _z and ω as shown in Fig. 2A . The slope of this dispersion relationship is related to the velocity at which the power associated with a guided mode propagates along the nanowire axis (Griffiths, 1981; Maslov and Ning, 2007). This is known as the group velocity and is defined by $v_{\mathrm{g}}=\frac{\mathrm{d}\omega }{\mathrm{d}{k}_z}$ . Figure 2B shows variation of v _g with nanowire diameter for the lowest order HE, TE, TM, and EH modes. By proper choice of nanowire diameter, the group velocity of the guided modes can be significantly reduced to enhance interaction of electromagnetic fields with nanowire material which has significant implications for optoelectronic devices as will be discussed later on in this chapter.

Figure 2. (A) Dispersion relation (ω vs. k_z ) and (B) group velocity, v _g for the lowest order HE, TE, TM, and EH modes supported in a nanowire of index 3.4.

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Partial Differential Equations

George B. Arfken , ... Frank E. Harris , in Mathematical Methods for Physicists (Seventh Edition), 2013

Circular Cylindrical Coordinates

Curvilinear coordinate systems introduce additional nuances into the process for separating variables. Again we consider the Helmholtz equation, now in circular cylindrical coordinates. With our unknown function ψ dependent on ρ, φ, and z, that equation becomes, using Eq. (3.149) for ∇ ²:

(9.53) $\nabla {}^2\psi (\rho ,\phi ,z)+k^2\psi (\rho ,\phi ,z)=0,$

or

(9.54) $\frac{1}{\rho }\frac{\partial }{\partial \rho }\left(\rho \frac{\partial \psi }{\partial \rho }\right)+\frac{1}{{\rho }^2}\frac{{\partial }^2\psi }{\partial {\phi }^2}+\frac{{\partial }^2\psi }{\partial z^2}+k^2\psi =0.$

As before, we assume a factored form ³ for ψ,

(9.55) $\psi (\rho ,\phi ,z)=P(\rho )\mathrm{\Phi }(\phi )Z(z).$

Substituting into Eq. (9.46), we have

(9.56) $\frac{\mathrm{\Phi }Z}{\rho }\frac{d}{d\rho }\left(\rho \frac{dP}{d\rho }\right)+\frac{PZ}{{\rho }^2}\frac{d^2\mathrm{\Phi }}{d{\phi }^2}+P\mathrm{\Phi }\frac{d^{2}Z}{d{z}^2}+k^{2}P\mathrm{\Phi }Z=0.$

All the partial derivatives have become ordinary derivatives. Dividing by PΦZ and moving the z derivative to the right-hand side yields

(9.57) $\frac{1}{\rho P}\frac{d}{d\rho }\left(\rho \frac{dP}{d\rho }\right)+\frac{1}{{\rho }^2\mathrm{\Phi }}\frac{d^2\mathrm{\Phi }}{d{\phi }^2}+k^2=-\frac{1}{Z}\frac{d^{2}Z}{d{z}^2}.$

Again, a function of z on the right appears to depend on a function of ρ and φ on the left. We resolve this by setting each side of Eq. (9.57) equal to the same constant. Let us choose ⁴ −l ². Then

(9.58) $\frac{d^{2}Z}{d{z}^2}=l^{2}Z$

and

(9.59) $\frac{1}{\rho P}\frac{d}{d\rho }\left(\rho \frac{dP}{d\rho }\right)+\frac{1}{{\rho }^2\mathrm{\Phi }}\frac{d^2\mathrm{\Phi }}{d{\phi }^2}+k^2=-l^2.$

Setting

(9.60) $k^2+l^2=n^2,$

multiplying by ρ ², and rearranging terms, we obtain

(9.61) $\frac{\rho }{P}\frac{d}{d\rho }\left(\rho \frac{dP}{d\rho }\right)+n^2{\rho }^2=-\frac{1}{\mathrm{\Phi }}\frac{d^2\mathrm{\Phi }}{d{\phi }^2}.$

We set the right-hand side equal to m ², so

(9.62) $\frac{d^2\mathrm{\Phi }}{d{\phi }^2}=-m^2\mathrm{\Phi },$

and the left-hand side of Eq. (9.61) rearranges into a separate equation for ρ:

(9.63) $\rho \frac{d}{d\rho }\left(\rho \frac{dP}{d\rho }\right)+(n^2{\rho }^2-m^2)P=0.$

Typically, Eq. (9.62) will be subject to the boundary condition that Φ have periodicity 2π and will therefore have solutions

$e^{\pm im\phi }\text{\hspace{0.17em}or,\hspace{0.17em}equivalently}\sin m\phi ,\cos m\phi ,\text{\hspace{0.17em}with\hspace{0.17em}integer}m\text{.}$

The ρ equation, Eq. (9.63), is Bessel's differential equation (in the independent variable nρ), originally encountered in Chapter 7. Because of its occurrence here (and in many other places relevant to physics), it warrants extensive study and is the topic of Chapter 14. The separation of variables of Laplace's equation in parabolic coordinates also gives rise to Bessel's equation. It may be noted that the Bessel equation is notorious for the variety of disguises it may assume. For an extensive tabulation of possible forms the reader is referred to Tables of Functions by Jahnke and Emde. ⁵

Summarizing, we have found that the original Helmholtz equation, a 3-D PDE, can be replaced by three ODEs, Eqs. (9.58), (9.62), and (9.63). Noting that the ODE for ρ contains the separation constants from the z and φ equations, the solutions we have obtained for the Helmholtz equation can be written, with labels, as

(9.64) ${\psi }_{lm}(\rho ,\phi ,z)=P_{lm}(\rho ){\mathrm{\Phi }}_m(\phi )Z_l(z),$

where we probably should recall that the n in Eq. (9.63) for P is a function of l (specifically, n ² = l ² + k ²). The most general solution of the Helmholtz equation can now be constructed as a linear combination of the product solutions:

(9.65) $\mathrm{\Psi }(\rho ,\phi ,z)=\sum _{l,m}{a}_{lm}{P}_{lm}(\rho ){\mathrm{\Phi }}_m(\phi )Z_l(z).$

Reviewing what we have done, we note that the separation could still have been achieved if k ² had been replaced by any additive function of the form

$k^2\to f(r)+\frac{g(\phi )}{{\rho }^2}+h(z).$

Example 9.4.3

Cylindrical Eigenvalue Problem

In this example we regard Eq. (9.53) as an eigenvalue problem, with Dirichlet boundary conditions ψ = 0 on all boundaries of a finite cylinder, with k ² initially unknown and to be determined. Our region of interest will be a cylinder with curved boundaries at ρ = R and with end caps at z = ±L/2, as shown in Fig. 9.2. To emphasize that k ² is an eigenvalue, we rename it λ, and our eigenvalue equation is, symbolically,

(9.66) $-\nabla {}^2\psi =\mathrm{\lambda }\psi ,$

with boundary conditions ψ = 0 at ρ = R and at z = ±L/2. Apart from constants, this is the time-independent Schrödinger equation for a particle in a cylindrical cavity. We limit the present example to the determination of the smallest eigenvalue (the ground state). This will be the solution to the PDE with the smallest number of oscillations, so we seek a solution without zeros (nodes) in the interior of the cylindrical region.

Figure 9.2. Cylindrical region for solution of the Helmholtz equation.

Again, we seek separated solutions of the form given in Eq. (9.55). The ODEs for Z and Φ, Eqs. (9.58) and (9.62), have the simple forms

$Z^{″}=l^{2}Z,{\mathrm{\Phi }}^{″}=-m^2\mathrm{\Phi },$

with general solutions

$Z=A{e}^{lz}+B{e}^{-lz},\mathrm{\Phi }=A^{\mathfrak{\prime }}\sin m\phi +B^{\mathfrak{\prime }}\cos m\phi .$

We now need to specialize these solutions to satisfy the boundary conditions. The condition on Φ is simply that it be periodic in φ with period 2π; this result will be obtained if m is any integer (including m = 0, which corresponds to the simple solution Φ = constant). Since our objective here is to obtain the least oscillatory solution, we choose that form, Φ = constant, for Φ.

Looking next at Z, we note that the arbitrary choice of sign for the separation constant l ² has led to a form of solution that appears not to be optimum for fulfilling conditions requiring Z = 0 at the boundaries. But, writing l ² = −ω ², l = iω, Z becomes a linear combination of sin ωz and cos ωz; the least oscillatory solution with Z(±L/2) = 0 is Z = cos(πz/L), so ω = π/L, and l ² = −π ²/L ².

The functions Z(z) and Φ(φ) that we have found satisfy the boundary conditions in z and φ but it remains to choose P(ρ) in a way that produces P = 0 at ρ = R with the least oscillation in P. The equation governing P, Eq. (9.63), is

(9.67) ${\rho }^2{P}^{″}+\rho P^{\mathfrak{\prime }}+n^2{\rho }^{2}P=0,$

where n was introduced as satisfying (in the current notation) n ² = λ + l ², see Eq. (9.60). Continuing now with Eq. (9.67), we identify as the Bessel equation of order zero in x = nρ. As we learned in Chapter 7, this ODE has two linearly independent solutions, of which only the one designated J ₀ is nonsingular at the origin. Since we need here a solution that is regular over the entire range 0 ≤ x ≤ nR, the solution we must choose is J ₀(nρ).

We can now see what is necessary to satisfy the boundary condition at ρ = R, namely that J ₀(nR) vanish. This is a condition on the parameter n. Remembering that we want the least oscillatory function P, we need for n to be such that nR will be the location of the smallest zero of J ₀. Giving this point the name α (which by numerical methods can be found to be approximately 2.4048), our boundary condition takes the form nR = α, or n = α/R, and our complete solution to the Helmholtz equation can be written

(9.68) $\psi (\rho ,\phi ,z)=J_0\left(\frac{\alpha \rho }{R}\right)\cos \left(\frac{\pi z}{L}\right).$

To complete our analysis, we must figure out how to arrange that n = α/R. Since the condition connecting n, l, and λ rearranges to

(9.69) $\mathrm{\lambda }=n^2-l^2,$

we see that the condition on n translates into one on λ. Our PDE has a unique ground-state solution consistent with the boundary conditions, namely an eigenfunction whose eigenvalue can be computed from Eq. (9.69), yielding

$\mathrm{\lambda }=\frac{{\alpha }^2}{R^2}+\frac{{\pi }^2}{L^2}.$

If we had not restricted consideration to the ground state (by choosing the least oscillatory solution), we would have (in principle) been able to obtain a complete set of eigenfunctions, each with its own eigenvalue.

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Find the General Solution of the Bessel Equation

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