Notes on Whittaker & Watson, Chapter XVII, part 1

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

17.1 The Bessel coefficients.

The Bessel functions are confluent hypergeometric functions with one regular and one irregular singularity.

Particular cases of Bessels functions for integer $n$ are the coefficients $J_n(z)$ of $t^n$ in the Laurent series of the generating function $$ e^{\frac12z\left(t-\frac1t\right)} $$ which can be calculated as $$ \begin{align*} J_n(z)&=\frac{1}{2\pi i}\int^{(0+)}u^{-n-1}e^{\frac12z\left(u-\frac1u\right)}du \\ &=\frac{1}{2\pi i}\left(\frac z2\right)^n\int^{(0+)}t^{-n-1}e^{t-\frac{z^2}{4t}}du & u=\frac{2t}{z} \\ &= \frac{1}{2\pi i}\sum_{r=0}^{\infty}\frac{(-1)^r}{r!}\left(\frac z2\right)^{n+2r}\int^{(0+)}t^{-n-r-1}e^{t}du \\ &=\begin{cases} \sum_{r=0}^\infty \frac{(-1)^r}{r!(n+r)!}\left(\frac z2\right)^{n+2r} & n\ge0,\\ \sum_{r=m}^\infty \frac{(-1)^r}{r!(r-m)!}\left(\frac z2\right)^{2r-m}= \sum_{s=0}^\infty\frac{(-1)^{m+s}}{s!(m+s)!}\left(\frac z2\right)^{m+2s}& n=-m<0. \end{cases} \end{align*} $$ We have $J_n(z)=(-1)^mJ_m(z)$.

17.11 Bessel’s differential equation $$ \frac{d^2y}{dz^2}-\frac1z\frac{dy}{dz}+\left(1-\frac{n^2}{z^2}\right)y=0. $$ 17.2 The solution of Bessel’s equation when $n$ is not necessarily an integer.

For non-integer values $n$​​​, Bessel’s equation can still be solved by the contour integral $z^n\int_Ct^{-n-1}\exp(t-\frac{z^2}{4t})du$​​​​​. Thus we can define the Bessel function of the first kind $$ J_n(z)=\frac{z^n}{2^{n+1}\pi i}\int_{-\infty}^{(0+)}t^{-n-1}\exp(t-\frac{z^2}{4t})dt $$ with $|\arg(t)|\le\pi$​, and $\arg(z)$ having its principal value, or in series expansion $$ J_n(z)=\sum_{r=0}^\infty\frac{(-1)^rz^{n+2r}}{2^{n+r}r!\Gamma(n+r+1)}. $$ Since Bessel’s equation is symmetric in $\pm n$, both $J_{\pm n}(z)$ are solutions to the equation, which are independent when $n$ is not integer.

17.21 The recurrence formulae for the Bessel functions.

By differentiation under the integral sign with respect to $t$, we have $$ J_{n-1}(z)+J_{n+1}(z)=\frac{2n}{z}J_n(z). $$ By differentiation with respect to $z$, we have $\frac{d}{dz}\Big(z^{-n}J_n(z)\Big) = -z^{-n}J_{n+1}(z)$​, thus $$ J_n'(z)=\frac{n}{z}J_n(z)-J_{n+1}(z). $$ Combining the above relations, we can derive that $$ \begin{gather*} J_n'(z)=\frac12\Big(J_{n-1}(z)-J_{n+1}(z)\Big),\\ J_n'(z)=J_{n-1}(z) - \frac{n}{z}J_n(z). \end{gather*} $$ 17.211 Relation between two Bessel functions whose orders differ by an integer.

With $r\in\mathbb{N}$ $$ z^{-n-r}J_{n+r}(z)=(-1)^r\frac{d^r}{(zdz)^r}\Big(z^{-n}J_n(z)\Big). $$ 17.212 The connexion between $J_n(z)$ and $W_{k,m}$ functions.

With the change of variables $y=z^{-\frac12}v$​​ and $z=\frac x{2i}$​, Bessels differential equation becomes the differential equation satisified by $W_{0,n}(x)$​. By comparing the coefficients of $z^{\pm n}$, we have $$ J_n(z)=\frac{z^{-\frac12}}{2^{2n+\frac12}i^{n+\frac12}\Gamma(n+1)}M_{0,n}(2iz). $$ 17.22 The zeros of Bessel functions whose order $n$​ is real.

Theorem: Between any two consecutive real zeros of $z^{-n}J_n(z)$, there lies one and only one zero of $z^{-n}J_{n+1}(z)$.

17.23 Bessel’s integral for the Bessel coefficients.

When $n$ is integer, with the contour chosen as the circle $|u|=1$, we have $$ \begin{align*} J_n(z) &=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{-ni\theta+iz\sin\theta}d\theta\\ &= \frac1\pi\int_0^\pi\cos(n\theta-z\sin\theta)d\theta. \end{align*} $$ 17.231 The modification of Bessel’s integral when $n$​ is not an integer.

For non-integer values of $n$ $$ J_n(z)=\frac1\pi\int_0^\pi\cos(n\theta-z\sin\theta)d\theta-\frac{\sin n\pi}{\pi}\int_0^\infty e^{-n\theta-z\sinh\theta}d\theta $$ which is obtained from the contour integral definition of $J_n(z)$​ (§17.2) with the contour chosen as $(-\infty,-1), |u|=1, (-1,-\infty)$​, where $t=\frac12uz$.​

17.24 Bessel functions whose order is half an odd integer.

When $n$​ is a half-integer, $J_n(z)$​ can be expressed by elementary functions. With $n=\frac12$, $$ J_{\frac12}(z)=\frac{2^\frac12z^\frac12}{\pi^\frac12}\left(1-\frac{z^2}{2\cdot 3}+\frac{z^4}{2\cdot 3\cdot 4\cdot 5}-\cdots\right)=\left(\frac{2}{\pi z}\right)^\frac12\sin z. $$ By reccurrence formula, we have $$ J_{k+\frac12}(z)=\frac{(-1)^k(2z)^{k+\frac12}}{\pi^\frac12}\frac{d^k}{d(z^2)^k}\left(\frac{\sin z}{z}\right)=P_k\sin z+Q_k\cos z $$ where $P_k,Q_k$ are polynomials in $z^\frac12$.