10 Bessel Functions Modified Bessel Functions 10.30 Limiting Forms 10.32 Integral Representations

§10.31 Power Series

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Keywords:: modified Bessel functions, power series
Notes:: See Olver (1997b, p. 253) or Watson (1944, p. 80). For (10.31.3) combine (10.8.3) and (10.27.6).
Permalink:: http://dlmf.nist.gov/10.31
See also:: Annotations for Ch.10

For $I_{\nu}\left(z\right)$ see (10.25.2) and (10.27.1). When $\nu$ is not an integer the corresponding expansion for $K_{\nu}\left(z\right)$ is obtained from (10.25.2) and (10.27.4).

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

10.31.1 $K_{n}\left(z\right)=\tfrac{1}{2}(\tfrac{1}{2}z)^{-n}\sum_{k=0}^{n-1}\frac{(n-k% -1)!}{k!}(-\tfrac{1}{4}z^{2})^{k}+(-1)^{n+1}\ln\left(\tfrac{1}{2}z\right)I_{n}% \left(z\right)+(-1)^{n}\tfrac{1}{2}(\tfrac{1}{2}z)^{n}\sum_{k=0}^{\infty}\left% (\psi\left(k+1\right)+\psi\left(n+k+1\right)\right)\frac{(\tfrac{1}{4}z^{2})^{% k}}{k!(n+k)!},$

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Symbols:: $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, : factorial (as in ), $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\ln\NVar{z}$ : principal branch of logarithm function, : integer, : nonnegative integer and : complex variable
A&S Ref:: 9.6.11
Referenced by:: §10.30(i), §10.65(ii)
Permalink:: http://dlmf.nist.gov/10.31.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §10.31 and Ch.10

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

10.31.2 $K_{0}\left(z\right)=-\left(\ln\left(\tfrac{1}{2}z\right)+\gamma\right)I_{0}% \left(z\right)+\frac{\tfrac{1}{4}z^{2}}{(1!)^{2}}+(1+\tfrac{1}{2})\frac{(% \tfrac{1}{4}z^{2})^{2}}{(2!)^{2}}+(1+\tfrac{1}{2}+\tfrac{1}{3})\frac{(\tfrac{1% }{4}z^{2})^{3}}{(3!)^{2}}+\dotsi.$

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Symbols:: $\gamma$ : Euler’s constant, : factorial (as in ), $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\ln\NVar{z}$ : principal branch of logarithm function and : complex variable
A&S Ref:: 9.6.13
Referenced by:: §10.45
Permalink:: http://dlmf.nist.gov/10.31.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §10.31 and Ch.10

For negative values of use (10.27.3).

10.31.3 $I_{\nu}\left(z\right)I_{\mu}\left(z\right)=(\tfrac{1}{2}z)^{\nu+\mu}\sum_{k=0}% ^{\infty}\frac{{\left(\nu+\mu+k+1\right)_{k}}(\tfrac{1}{4}z^{2})^{k}}{k!\Gamma% \left(\nu+k+1\right)\Gamma\left(\mu+k+1\right)}.$

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), : factorial (as in ), $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, : nonnegative integer, : complex variable and $\nu$ : complex parameter
Referenced by:: §10.31
Permalink:: http://dlmf.nist.gov/10.31.E3
Encodings:: TeX, pMML, png
Correction (effective with 1.1.2):: The Pochhammer symbol now links to its definition.
See also:: Annotations for §10.31 and Ch.10

10.30 Limiting Forms 10.32 Integral Representations

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