Notes on Whittaker & Watson, Chapter XVII, part 3

Whittaker & Watson, A Course of Modern Analysis. Chapter XVII, Bessel Functions.

17.6 The second solution of Bessel’s equation when the order is an integer.

The Bessel functions $J_n$​ and $J_{-n}$​​​​ are two solutions to Bessel’s equation, which are not linearly independent when $n$​​ is an integer. The Bessel function of the second kind is defined by $$ \begin{gather*} Y_n(z)=\frac{J_n(z)\cos n\pi- J_{-n}(z)}{\sin n\pi} & \text{or}\\ \mathsf{Y}_n=2\pi e^{n\pi i}\frac{J_n(z)\cos n\pi- J_{-n}(z)}{\sin 2n\pi} \end{gather*} $$ where $Y_n$​​ is due to Weber and Schläfli, and $\mathsf{Y}_n$​​ is introduced by Hankel. For integer values of $n$​, the function is defined by the limit of $Y_{n+\epsilon}$​​ as $\epsilon\to0$ $$ \mathsf{Y}_n=\left(\frac{\pi}{z}\right)^{\frac12}\left[e^{(\frac12n+\frac34)\pi i}\ W_{0,n}(2iz)+ e^{-(\frac12n+\frac34)\pi i}\ W_{0,n}(-2iz)\right]. $$ Asymptotically, $$ Y_n(z)\sim\left(\frac{\pi}{2z}\right)^{\frac12}\left[\sin{\left(z-\frac12n\pi-\frac14\pi\right)}U_n(z) + \cos{\left(z-\frac12n\pi-\frac14\pi\right)}V_n(z)\right] $$ based on the asymptotic expansion of $J_n(z)$​​​ (§17.5).

17.61 The ascending series for $\mathsf{Y}_n(z)$.

For $z$ in the neibourhood of 0​, the series expansion of $\mathsf{Y}_n$​ can be obtained through the double series in powers of $\epsilon$​ and $z$​. From the series definition of $J_n(z)$​​ (§17.2), we have $$ \begin{align*} \mathsf{Y}_n(z) & =\lim_{\epsilon\to0}\epsilon^{-1}\Big(J_{n+\epsilon}(z)-(-1)^{n}J_{-n-\epsilon}(z)\Big) \\ & = \lim_{\epsilon\to0}\epsilon^{-1}\left(\sum_{r=0}^\infty\frac{(-1)^r\left(\frac z2\right)^{n+\epsilon+2r}}{r!\ \Gamma(n+\epsilon+r+1)}- (-1)^n\sum_{r=0}^\infty\frac{(-1)^r\left(\frac z2\right)^{-n-\epsilon+2r}}{r!\ \Gamma(-n-\epsilon+r+1)}\right) \end{align*} $$ Expanding $\left(\frac z2\right)^{n+\epsilon+2r}$, $\frac1{\Gamma(s+\epsilon+1)}$, $\frac1{\Gamma(t+\epsilon+1)}$ with $s,t\in\mathbb{Z}$ and $s\ge0, t<0$ in power series of $\epsilon$, we have $$ \begin{align*}\mathsf{Y}_n= & \sum_{r=0}^\infty\frac{(-1)^r\left(\frac z2\right)^{n+r}}{r!\ (n+r)!}\left(2\log\left(\frac z2\right)+2\gamma-\sum_{m=1}^{n+r}m^{-1}-\sum_{m=1}^{r}m^{-1}\right) \\ &- \sum_{r=0}^{n-1}\frac{\left(\frac z2\right)^{-n+2r}(n-r-1)!}{r!}. \end{align*} $$ 17.7 Bessel functions with purely imaginary argument.

Defined as $$ I_n(z)=i^{-n}J_n(iz)=\sum_{r=0}^{\infty}\frac{\left(\frac z2\right)^{n+2r}}{r!(n+r)!} $$ the function satisfies the differential equation $$ \frac{d^2I_n(z)}{dz^2}+\frac1z\frac{dI_n(z)}{dz}-\left(1+\frac{n^2}{z^2}\right)I_n(z)=0 $$ with the reccurrence relations $$ \begin{gather*} I_{n-1}(z)-I_{n+1}(z)=\frac{2n}{z}I_n(z),\\ \frac{d}{dz}z^nI_n(z) = z^nI_{n-1}(z),\\ \frac{d}{dz}z^{-n}I_{n}(z) = z^{-n}I_{n+1}(z), \end{gather*} $$ the integral representation for $\Re(n+\frac12)>0$​ $$ I_n(z)=\frac{z^n}{2^n\Gamma(\frac12)\Gamma(n+\frac12)}\int_0^\pi\cosh(z\cos\phi)\sin^{2n}\phi\ d\phi, $$ and the asymptotic expansion for $-\frac32\pi<\arg z<\frac12\pi$​ $$ \begin{align*}I_n(z)\sim & \frac{e^z}{(2\pi z)^\frac12}\left[1+\sum_{r=1}^\infty(-1)^r\frac{(4n^2-1)(4n^2-3)\cdots(4n^2-(2r-1)^2)}{r!2^{3r}z^r}\right]\\ & \frac{e^{-(n+\frac12)\pi i}e^{-z}}{(2\pi z)^\frac12}\left[1+\sum_{r=1}^\infty\frac{(4n^2-1)(4n^2-3)\cdots(4n^2-(2r-1)^2)}{r!2^{3r}z^r}\right]. \end{align*} $$ 17.71 Modified Bessel functions of the second kind.

Both $I_n$ and $I_{-n}$ satisfies the differential equation for $I_n$ which are not linearly independent when $n$ is integer. Analogous to the Bessel function of the second kind, we define $$ \begin{align*} K_n(z) &=\frac\pi2\big(I_{-n}(z)-I_n(z)\big)\cot n\pi\\ &= \left(\frac{\pi}{2z}\right)^\frac12\cos n\pi\ W_{0,n}(2z). \end{align*} $$ which has the asymptotic expansion $$ K_n(z)\sim\left(\frac{\pi}{2z}\right)^\frac12e^{-z}\cos n\pi \left[1+\sum_{r=1}^\infty\frac{(4n^2-1)(4n^2-3)\cdots(4n^2-(2r-1)^2)}{r!2^{3r}z^r}\right] $$ and the ascending series $$ \begin{align*} K_n(z)=&-\sum_{r=0}^{\infty}\frac{(\frac z2)^{n+2r}}{r!(n+r)!}\left[\log\frac z2+\gamma-\frac12\sum_{m=1}^{n+r}m^{-1}-\frac12\sum_{m=1}^{r}m^{-1}\right]\\ &+\frac12\sum_{r=0}^{n-1}\left(\frac z2\right)^{-n+2r}\frac{(-1)^{n-r}(n-r-1)!}{r!}. \end{align*} $$ 17.8 Neumann’s expansion of an analytic function in a series of Bessel coefficients.

A function $f(z)$​ analytic in a domain including $z=0$ can be expanded as $$ f(z)=\alpha_0J_0(z)+\alpha_1J_1(z)+\alpha_2J_2(z)+\cdots. $$ In particular, consider the expansion for $\frac{1}{t-z}$ $$ \frac{1}{t-z}=O_0(t)J_0(z)+2O_1(t)J_1(z)+2O_2(t)J_2(z)+\cdots. $$ Using the equation $\left(\frac{\partial}{\partial t}+\frac{\partial}{\partial z}\right)\frac{1}{t-z}=0$​ and recurrence relation that $2J_n'(z)=J_{n-1}(z)-J_{n+1}(z)$, we have $$ \begin{gather*} O_0(t)=\frac1t\\ O_1(t)=-O_0'(t) \\ O_{n+1}(t)=O_{n-1}(t)-2O_n'(t) \end{gather*} $$ and obtain by induction that $O_n(t)=\frac12\int_0^\infty e^{-tu}\left((u+\sqrt{u^2+1})^n+(u-\sqrt{u^2+1})^n\right)du$​​​ when $\Re(t)>0$​. For $n\ge1$​ we have the expression of $O_n$ as a polynomial in $\frac1t$ $$ O_n(t)=\frac{2^{n-1}n!}{t^{n+1}}\left(1+\frac{t^2}{2(2n-2)}+\frac{t^4}{2\cdot4\cdot(2n-2)(2n-4)}+\cdots\right). $$ 17.81 Proof of Neumann’s expansion.

For funtion $f(z)$​​ in general, we use the expansion of $\frac{1}{t-z}$ to obtain $$ f(z)=\frac{1}{2\pi i}\int\frac{f(t)}{t-z}dt=J_0(z)f(0)+\sum_{n=1}^\infty\frac{J_n(z)}{\pi i}\int O_n(t)f(t)dt. $$ 17.82 Schlömilch’s expansion of an arbitrary function in series of Bessel coefficients of order 0.

For $f(x)$​​ with continuous derivatives and finite total variation in the interval $(0,\pi)$​, we have the expansion $$ f(x)=\alpha_0+\alpha_1J_0(z)+\alpha_2J_0(2z)+\alpha_3J_0(3z)+\cdots $$ where the coefficients $$ \begin{gather*} \alpha_0=f(0)+\frac1\pi\int_0^\pi u\int_0^{\frac\pi2}f'(u\sin\theta)\ d\theta\ du\\ \alpha_n=\frac2\pi\int_0^\pi u\cos nu\int_0^{\frac\pi2}f'(u\sin\theta)\ d\theta\ du & n>0 \end{gather*} $$ is obtained from the integral equation $f(x)=\frac2\pi\int_0^{\frac\pi2}F(x\sin\phi)\ d\phi$, which has the solution $F(x)=f(0)+x\int_0^\frac\pi2f'(x\sin\theta)\ d\theta$. The function $f(x)$ can be recovered by integrating the Fourier series $$ F(x\sin\phi)=\frac1\pi \int_0^\pi F(u)du+\frac2\pi\sum_{n=1}^\infty\int_0^\pi \cos nu \cos(nx\sin\phi)F(u)du $$ and calculation of Bessel’s integrals (§17.23).