§10.45 Functions of Imaginary Order

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.
Keywords:: modified Bessel functions
Referenced by:: §10.26(iii), §10.75(viii)
Permalink:: http://dlmf.nist.gov/10.45

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

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

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $x$ : real variable and $\nu$ : complex parameter
Referenced by:: §10.45, §10.45
Permalink:: http://dlmf.nist.gov/10.45.E1
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For $\nu\in\Real$ and $x$ $\in(0,\infty)$ define

10.45.2 $\displaystyle \mathop{\widetilde{I}_{{\nu}}\/}\nolimits\!\left(x\right) =\realpart{(\mathop{I_{{i\nu}}\/}\nolimits\!\left(x\right))},$ $\displaystyle \mathop{\widetilde{K}_{{\nu}}\/}\nolimits\!\left(x\right) =\mathop{K_{{i\nu}}\/}\nolimits\!\left(x\right).$

Defines:: $\mathop{\widetilde{I}_{{\nu}}\/}\nolimits\!\left(x\right)$ : modified Bessel function of imaginary order and $\mathop{\widetilde{K}_{{\nu}}\/}\nolimits\!\left(x\right)$ : modified Bessel function of imaginary order
Symbols:: $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\realpart{}$ : real part, $x$ : real variable and $\nu$ : complex parameter
Referenced by:: §10.30(i), §10.74(viii)
Permalink:: http://dlmf.nist.gov/10.45.E2
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Then

10.45.3 $\displaystyle \mathop{\widetilde{I}_{{-\nu}}\/}\nolimits\!\left(x\right) =\mathop{\widetilde{I}_{{\nu}}\/}\nolimits\!\left(x\right),$ $\displaystyle \mathop{\widetilde{K}_{{-\nu}}\/}\nolimits\!\left(x\right) =\mathop{\widetilde{K}_{{\nu}}\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{\widetilde{I}_{{\nu}}\/}\nolimits\!\left(x\right)$ : modified Bessel function of imaginary order, $\mathop{\widetilde{K}_{{\nu}}\/}\nolimits\!\left(x\right)$ : modified Bessel function of imaginary order, $x$ : real variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.45.E3
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and $\mathop{\widetilde{I}_{{\nu}}\/}\nolimits\!\left(x\right)$ , $\mathop{\widetilde{K}_{{\nu}}\/}\nolimits\!\left(x\right)$ are real and linearly independent solutions of (10.45.1):

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

Symbols:: $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, $\mathop{\widetilde{I}_{{\nu}}\/}\nolimits\!\left(x\right)$ : modified Bessel function of imaginary order, $\mathop{\widetilde{K}_{{\nu}}\/}\nolimits\!\left(x\right)$ : modified Bessel function of imaginary order, $x$ : real variable and $\nu$ : complex parameter
Referenced by:: §10.45
Permalink:: http://dlmf.nist.gov/10.45.E4
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As $x\to+\infty$

10.45.5

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

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $e$ : base of exponential function, $\mathop{\widetilde{I}_{{\nu}}\/}\nolimits\!\left(x\right)$ : modified Bessel function of imaginary order, $\mathop{\widetilde{K}_{{\nu}}\/}\nolimits\!\left(x\right)$ : modified Bessel function of imaginary order, $x$ : real variable and $\nu$ : complex parameter
Referenced by:: §10.45, §10.45
Permalink:: http://dlmf.nist.gov/10.45.E5
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As $x\to 0+$

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{\widetilde{I}_{{\nu}}\/}\nolimits\!\left(x\right)$ : modified Bessel function of imaginary order, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $x$ : real variable, $\nu$ : complex parameter and $\gamma _{\nu}$ : parameter
Referenced by:: §10.45
Permalink:: http://dlmf.nist.gov/10.45.E6
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where $\gamma _{\nu}$ is as in §10.24. The corresponding result for $\mathop{\widetilde{K}_{{\nu}}\/}\nolimits\!\left(x\right)$ is given by

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{\widetilde{K}_{{\nu}}\/}\nolimits\!\left(x\right)$ : modified Bessel function of imaginary order, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\sin\/}\nolimits z$ : sine function, $x$ : real variable, $\nu$ : complex parameter and $\gamma _{\nu}$ : parameter
Referenced by:: §10.30(i), §10.45
Permalink:: http://dlmf.nist.gov/10.45.E7
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when $\nu>0$ , and

10.45.8 $\mathop{\widetilde{K}_{{0}}\/}\nolimits\!\left(x\right)=\mathop{K_{{0}}\/}\nolimits\!\left(x\right)=-\mathop{\ln\/}\nolimits(\tfrac{1}{2}x)-\EulerConstant+\mathop{O\/}\nolimits(x^{2}\mathop{\ln\/}\nolimits x),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\EulerConstant$ : Euler’s constant, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\widetilde{K}_{{\nu}}\/}\nolimits\!\left(x\right)$ : modified Bessel function of imaginary order, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $x$ : real variable
Referenced by:: §10.45
Permalink:: http://dlmf.nist.gov/10.45.E8
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where $\EulerConstant$ again denotes Euler’s constant (§5.2(ii)).

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

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

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