10 Bessel Functions Bessel and Hankel Functions 10.7 Limiting Forms 10.9 Integral Representations

§10.8 Power Series

ⓘ

Keywords:: Bessel functions, Hankel functions, power series
Notes:: See Olver (1997b, p. 243) and Watson (1944, p. 147).
Referenced by:: §10.74(i), Erratum (V1.0.17) for Section 10.8
Permalink:: http://dlmf.nist.gov/10.8
Addition (effective with 1.0.17):: A sentence was added just below (10.8.3) indicating that it is just a rewriting of (16.12.1).
Suggested by Tom Koornwinder
See also:: Annotations for Ch.10

For $J_{\nu}\left(z\right)$ see (10.2.2) and (10.4.1). When $\nu$ is not an integer the corresponding expansions for $Y_{\nu}\left(z\right)$ , ${H^{(1)}_{\nu}}\left(z\right)$ , and ${H^{(2)}_{\nu}}\left(z\right)$ are obtained by combining (10.2.2) with (10.2.3), (10.4.7), and (10.4.8).

When $n=0,1,2,\dotsc$ ,

10.8.1		$Y_{n}\left(z\right)=-\frac{(\tfrac{1}{2}z)^{-n}}{\pi}\sum_{k=0}^{n-1}\frac{(n-% k-1)!}{k!}\left(\tfrac{1}{4}z^{2}\right)^{k}+\frac{2}{\pi}\ln\left(\tfrac{1}{2% }z\right)J_{n}\left(z\right)-\frac{(\tfrac{1}{2}z)^{n}}{\pi}\sum_{k=0}^{\infty% }(\psi\left(k+1\right)+\psi\left(n+k+1\right))\frac{(-\tfrac{1}{4}z^{2})^{k}}{% k!(n+k)!},$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : integer, $k$ : nonnegative integer and $z$ : complex variable A&S Ref: 9.1.11 Referenced by: §10.19(i), §10.7(i) Permalink: http://dlmf.nist.gov/10.8.E1 Encodings: TeX, pMML, png See also: Annotations for §10.8 and Ch.10

where $\psi\left(x\right)=\Gamma'\left(x\right)/\Gamma\left(x\right)$ (§5.2(i)). In particular,

10.8.2		$Y_{0}\left(z\right)=\frac{2}{\pi}\left(\ln\left(\tfrac{1}{2}z\right)+\gamma% \right)J_{0}\left(z\right)+\frac{2}{\pi}\left(\frac{\tfrac{1}{4}z^{2}}{(1!)^{2% }}-(1+\tfrac{1}{2})\frac{(\tfrac{1}{4}z^{2})^{2}}{(2!)^{2}}+(1+\tfrac{1}{2}+% \tfrac{1}{3})\frac{(\tfrac{1}{4}z^{2})^{3}}{(3!)^{2}}-\dotsi\right),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\gamma$ : Euler’s constant, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function and $z$ : complex variable A&S Ref: 9.1.13 Referenced by: §10.22(iii), §10.24, §10.7(i) Permalink: http://dlmf.nist.gov/10.8.E2 Encodings: TeX, pMML, png See also: Annotations for §10.8 and Ch.10

where $\gamma$ is Euler’s constant (§5.2(ii)).

For negative values of $n$ use (10.4.1).

The corresponding results for ${H^{(1)}_{n}}\left(z\right)$ and ${H^{(2)}_{n}}\left(z\right)$ are obtained via (10.4.3) with $\nu=n$ .

10.8.3		$J_{\nu}\left(z\right)J_{\mu}\left(z\right)=(\tfrac{1}{2}z)^{\nu+\mu}\sum_{k=0}% ^{\infty}\frac{{\left(\nu+\mu+k+1\right)_{k}}(-\tfrac{1}{4}z^{2})^{k}}{k!% \Gamma\left(\nu+k+1\right)\Gamma\left(\mu+k+1\right)}.$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $k$ : nonnegative integer, $z$ : complex variable and $\nu$ : complex parameter Proof sketch: Rewrite (16.12.1). A&S Ref: 9.1.14 Referenced by: §10.22(ii), §10.31, §10.8, §10.8, §18.17(v), Erratum (V1.0.17) for Section 10.8 Permalink: http://dlmf.nist.gov/10.8.E3 Encodings: TeX, pMML, png Correction (effective with 1.1.2): The Pochhammer symbol now links to its definition. See also: Annotations for §10.8 and Ch.10

Note that (10.8.3) is just a rewriting of (16.12.1).

10.7 Limiting Forms 10.9 Integral Representations

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