§10.8 Power Series

Notes:: See Olver (1997b, p. 243) and Watson (1944, p. 147).
Keywords:: Bessel functions, Hankel functions, power series
Referenced by:: §10.74(i)
Permalink:: http://dlmf.nist.gov/10.8

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

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

10.8.1 $\mathop{Y_{{n}}\/}\nolimits\!\left(z\right)=-\frac{(\tfrac{1}{2}z)^{{-n}}}{\pi}\sum _{{k=0}}^{{n-1}}\frac{(n-k-1)!}{k!}\left(\tfrac{1}{4}z^{2}\right)^{k}+\frac{2}{\pi}\mathop{\ln\/}\nolimits\!\left(\tfrac{1}{2}z\right)\mathop{J_{{n}}\/}\nolimits\!\left(z\right)-\frac{(\tfrac{1}{2}z)^{n}}{\pi}\sum _{{k=0}}^{\infty}(\mathop{\psi\/}\nolimits\!\left(k+1\right)+\mathop{\psi\/}\nolimits\!\left(n+k+1\right))\frac{(-\tfrac{1}{4}z^{2})^{k}}{k!(n+k)!},$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $!$ : $n!$ : factorial, $\mathop{\ln\/}\nolimits z$ : 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

where $\mathop{\psi\/}\nolimits\!\left(x\right)={\mathop{\Gamma\/}\nolimits^{{\prime}}}\!\left(x\right)/\mathop{\Gamma\/}\nolimits\!\left(x\right)$ (§5.2(i)). In particular,

10.8.2 $\mathop{Y_{{0}}\/}\nolimits\!\left(z\right)=\frac{2}{\pi}\left(\mathop{\ln\/}\nolimits\!\left(\tfrac{1}{2}z\right)+\EulerConstant\right)\mathop{J_{{0}}\/}\nolimits\!\left(z\right)+\frac{2}{\pi}\left(\frac{\tfrac{1}{4}z^{2}}{(1!)^{2}}-(1+\tfrac{1}{2})\frac{(\tfrac{1}{4}z^{2})^{2}}{(2!)^{2}}+(1+\tfrac{1}{2}+\tfrac{1}{3})\frac{(\tfrac{1}{4}z^{2})^{3}}{(3!)^{2}}-\cdots\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\EulerConstant$ : Euler’s constant, $!$ : $n!$ : factorial, $\mathop{\ln\/}\nolimits z$ : 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

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

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

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

10.8.3 $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)\mathop{J_{{\mu}}\/}\nolimits\!\left(z\right)=(\tfrac{1}{2}z)^{{\nu+\mu}}\sum _{{k=0}}^{\infty}\frac{(\nu+\mu+k+1)_{k}(-\tfrac{1}{4}z^{2})^{k}}{k!\mathop{\Gamma\/}\nolimits\!\left(\nu+k+1\right)\mathop{\Gamma\/}\nolimits\!\left(\mu+k+1\right)}.$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $!$ : $n!$ : factorial, $k$ : nonnegative integer, $z$ : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.1.14
Referenced by:: §10.22(ii), §10.31, §18.17(v)
Permalink:: http://dlmf.nist.gov/10.8.E3
Encodings:: TeX, pMML, png