10 Bessel Functions Modified Bessel Functions 10.44 Sums 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function

§10.45 Functions of Imaginary Order

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Keywords:: definitions, limiting forms, modified Bessel functions, numerically satisfactory pairs, of imaginary order, uniform asymptotic expansions for large order, zeros
Notes:: Equations (10.45.5)–(10.45.8) follow from (10.25.2), (10.27.4), (10.31.2), (10.40.1), and (10.40.2). The Wronskian (10.45.4) can be verified from (1.13.5) and either (10.45.5) or (10.45.6)–(10.45.8) and their differentiated forms.
Referenced by:: §10.26(iii), §10.75(viii), Erratum (V1.0.10) for References
Permalink:: http://dlmf.nist.gov/10.45
Addition (effective with 1.0.10):: A reference to Booker et al. (2013) has been added at the end of this section.
See also:: Annotations for Ch.10

With , and $\nu$ replaced by $i\nu$ , the modified Bessel’s equation (10.25.1) becomes

10.45.1 $x^{2}\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}+x\frac{\mathrm{d}w}{\mathrm{d% }x}+(\nu^{2}-x^{2})w=0.$

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , : real variable and $\nu$ : complex parameter
Referenced by:: §10.45, §10.45
Permalink:: http://dlmf.nist.gov/10.45.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §10.45 and Ch.10

For $\nu\in\mathbb{R}$ and $\in(0,\infty)$ define

10.45.2 $\displaystyle\widetilde{I}_{\nu}\left(x\right)=\Re\left(I_{i\nu}\left(x\right)% \right),$ $\displaystyle\widetilde{K}_{\nu}\left(x\right)=K_{i\nu}\left(x\right).$

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Defines:: $\widetilde{I}_{\NVar{\nu}}\left(\NVar{x}\right)$ : modified Bessel function of the first kind of imaginary order and $\widetilde{K}_{\NVar{\nu}}\left(\NVar{x}\right)$ : modified Bessel function fo the second kind of imaginary order
Symbols:: $\mathrm{i}$ : imaginary unit, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\Re$ : real part, : real variable and $\nu$ : complex parameter
Referenced by:: §10.30(i), §10.74(viii)
Permalink:: http://dlmf.nist.gov/10.45.E2
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §10.45 and Ch.10

Then

10.45.3 $\displaystyle\widetilde{I}_{-\nu}\left(x\right)=\widetilde{I}_{\nu}\left(x% \right),$ $\displaystyle\widetilde{K}_{-\nu}\left(x\right)=\widetilde{K}_{\nu}\left(x% \right),$

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Symbols:: $\widetilde{I}_{\NVar{\nu}}\left(\NVar{x}\right)$ : modified Bessel function of the first kind of imaginary order, $\widetilde{K}_{\NVar{\nu}}\left(\NVar{x}\right)$ : modified Bessel function fo the second kind of imaginary order, : real variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.45.E3
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §10.45 and Ch.10

and $\widetilde{I}_{\nu}\left(x\right)$ , $\widetilde{K}_{\nu}\left(x\right)$ are real and linearly independent solutions of (10.45.1):

10.45.4 $\mathscr{W}\left\{\widetilde{K}_{\nu}\left(x\right),\widetilde{I}_{\nu}\left(x% \right)\right\}=1/x.$

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Symbols:: $\mathscr{W}$ : Wronskian, $\widetilde{I}_{\NVar{\nu}}\left(\NVar{x}\right)$ : modified Bessel function of the first kind of imaginary order, $\widetilde{K}_{\NVar{\nu}}\left(\NVar{x}\right)$ : modified Bessel function fo the second kind of imaginary order, : real variable and $\nu$ : complex parameter
Referenced by:: §10.45
Permalink:: http://dlmf.nist.gov/10.45.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §10.45 and Ch.10

As $x\to+\infty$

10.45.5

$\widetilde{I}_{\nu}\left(x\right)=(2\pi x)^{-\frac{1}{2}}e^{x}\left(1+O\left(x% ^{-1}\right)\right),$

$\widetilde{K}_{\nu}\left(x\right)=(\pi/(2x))^{\frac{1}{2}}e^{-x}\left(1+O\left% (x^{-1}\right)\right).$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\widetilde{I}_{\NVar{\nu}}\left(\NVar{x}\right)$ : modified Bessel function of the first kind of imaginary order, $\widetilde{K}_{\NVar{\nu}}\left(\NVar{x}\right)$ : modified Bessel function fo the second kind of imaginary order, : real variable and $\nu$ : complex parameter
Referenced by:: §10.45, §10.45
Permalink:: http://dlmf.nist.gov/10.45.E5
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §10.45 and Ch.10

As $x\to 0+$

10.45.6 $\widetilde{I}_{\nu}\left(x\right)=\left(\frac{\sinh\left(\pi\nu\right)}{\pi\nu% }\right)^{\frac{1}{2}}\cos\left(\nu\ln\left(\tfrac{1}{2}x\right)-\gamma_{\nu}% \right)+O\left(x^{2}\right),$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\widetilde{I}_{\NVar{\nu}}\left(\NVar{x}\right)$ : modified Bessel function of the first kind of imaginary order, $\ln\NVar{z}$ : principal branch of logarithm function, : real variable, $\nu$ : complex parameter and $\gamma_{\nu}$ : parameter
Referenced by:: §10.45
Permalink:: http://dlmf.nist.gov/10.45.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §10.45 and Ch.10

where $\gamma_{\nu}$ is as in §10.24. The corresponding result for $\widetilde{K}_{\nu}\left(x\right)$ is given by

10.45.7 $\widetilde{K}_{\nu}\left(x\right)=-\left(\frac{\pi}{\nu\sinh\left(\pi\nu\right% )}\right)^{\frac{1}{2}}\*\sin\left(\nu\ln\left(\tfrac{1}{2}x\right)-\gamma_{% \nu}\right)+O\left(x^{2}\right),$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $\sinh\NVar{z}$ : hyperbolic sine function, $\widetilde{K}_{\NVar{\nu}}\left(\NVar{x}\right)$ : modified Bessel function fo the second kind of imaginary order, $\ln\NVar{z}$ : principal branch of logarithm function, $\sin\NVar{z}$ : sine function, : real variable, $\nu$ : complex parameter and $\gamma_{\nu}$ : parameter
Referenced by:: §10.30(i), §10.45
Permalink:: http://dlmf.nist.gov/10.45.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §10.45 and Ch.10

when $\nu>0$ , and

10.45.8 $\widetilde{K}_{0}\left(x\right)=K_{0}\left(x\right)=-\ln\left(\tfrac{1}{2}x% \right)-\gamma+O\left(x^{2}\ln x\right),$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\gamma$ : Euler’s constant, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\widetilde{K}_{\NVar{\nu}}\left(\NVar{x}\right)$ : modified Bessel function fo the second kind of imaginary order, $\ln\NVar{z}$ : principal branch of logarithm function and : real variable
Referenced by:: §10.45
Permalink:: http://dlmf.nist.gov/10.45.E8
Encodings:: TeX, pMML, png
See also:: Annotations for §10.45 and Ch.10

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

In consequence of (10.45.5)–(10.45.7), $\widetilde{I}_{\nu}\left(x\right)$ and $\widetilde{K}_{\nu}\left(x\right)$ comprise a numerically satisfactory pair of solutions of (10.45.1) when is large, and either $\widetilde{I}_{\nu}\left(x\right)$ and $(1/\pi)\sinh\left(\pi\nu\right)\widetilde{K}_{\nu}\left(x\right)$ , or $\widetilde{I}_{\nu}\left(x\right)$ and $\widetilde{K}_{\nu}\left(x\right)$ , comprise a numerically satisfactory pair when is small, depending whether $\nu\neq 0$ or $\nu=0$ .

For graphs of $\widetilde{I}_{\nu}\left(x\right)$ and $\widetilde{K}_{\nu}\left(x\right)$ see §10.26(iii).

For properties of $\widetilde{I}_{\nu}\left(x\right)$ and $\widetilde{K}_{\nu}\left(x\right)$ , including uniform asymptotic expansions for large $\nu$ and zeros, see Dunster (1990a). In this reference $\widetilde{I}_{\nu}\left(x\right)$ is denoted by $(1/\pi)\sinh\left(\pi\nu\right)L_{i\nu}(x)$ . See also Gil et al. (2003a), Balogh (1967) and Booker et al. (2013).

10.44 Sums 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function

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