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

§10.31 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).
Keywords:: modified Bessel functions, power series
Permalink:: http://dlmf.nist.gov/10.31

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

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

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

Symbols:: $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, : : factorial, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\ln\/}\nolimits z$ : 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

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,

10.31.2 $\mathop{K_{{0}}\/}\nolimits\!\left(z\right)=-\left(\mathop{\ln\/}\nolimits\!% \left(\tfrac{1}{2}z\right)+\EulerConstant\right)\mathop{I_{{0}}\/}\nolimits\!% \left(z\right)+\frac{\tfrac{1}{4}z^{2}}{(1!)^{2}}+(1+\tfrac{1}{2})\frac{(% \tfrac{1}{4}z^{2})^{2}}{(2!)^{2}}+(1+\tfrac{1}{2}+\tfrac{1}{3})\frac{(\tfrac{1% }{4}z^{2})^{3}}{(3!)^{2}}+\cdots.$

Symbols:: $\EulerConstant$ : Euler’s constant, : : factorial, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\ln\/}\nolimits z$ : 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

For negative values of use (10.27.3).

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : : factorial, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, : nonnegative integer, : complex variable and $\nu$ : complex parameter
Referenced by:: §10.31
Permalink:: http://dlmf.nist.gov/10.31.E3
Encodings:: TeX, pMML, png

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