10 Bessel Functions Bessel and Hankel Functions 10.23 Sums 10.25 Definitions

§10.24 Functions of Imaginary Order

Notes:: (10.24.6)–(10.24.9) follow from (10.24.2)–(10.24.4) combined with (10.2.2), (10.2.3), (10.8.2), (10.17.3), and (10.17.4). (10.24.5) can be verified from (1.13.5) and either (10.24.6) or (10.24.7)–(10.24.9) and their differentiated forms.
Keywords:: Bessel functions
Referenced by:: §10.3(iii), §10.45, §10.7(i)
Permalink:: http://dlmf.nist.gov/10.24

With and $\nu$ replaced by $i\nu$ , Bessel’s equation (10.2.1) becomes

10.24.1 $x^{2}\frac{{d}^{2}w}{{dx}^{2}}+x\frac{dw}{dx}+(x^{2}+\nu^{2})w=0.$

Symbols:: $\frac{df}{dx}$ : derivative of with respect to , : real variable and $\nu$ : complex parameter
Referenced by:: §10.24, §10.24
Permalink:: http://dlmf.nist.gov/10.24.E1
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For $\nu\in\Real$ and $\in(0,\infty)$ define

10.24.2

$\mathop{\widetilde{J}_{{\nu}}\/}\nolimits\!\left(x\right)=\mathop{\mathrm{sech% }\/}\nolimits\left(\tfrac{1}{2}\pi\nu\right)\realpart{(\mathop{J_{{i\nu}}\/}% \nolimits\!\left(x\right))},$

$\mathop{\widetilde{Y}_{{\nu}}\/}\nolimits\!\left(x\right)=\mathop{\mathrm{sech% }\/}\nolimits\left(\tfrac{1}{2}\pi\nu\right)\realpart{(\mathop{Y_{{i\nu}}\/}% \nolimits\!\left(x\right))},$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\widetilde{J}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Bessel function of imaginary order, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{\widetilde{Y}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Bessel function of imaginary order, $\mathop{\mathrm{sech}\/}\nolimits z$ : hyperbolic secant function, $\realpart{}$ : real part, : real variable and $\nu$ : complex parameter
Referenced by:: §10.24
Permalink:: http://dlmf.nist.gov/10.24.E2
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and

10.24.3 $\mathop{\Gamma\/}\nolimits\!\left(1+i\nu\right)=\left(\frac{\pi\nu}{\mathop{% \sinh\/}\nolimits(\pi\nu)}\right)^{{\frac{1}{2}}}e^{{i\gamma_{\nu}}},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : base of exponential function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\nu$ : complex parameter and $\gamma_{\nu}$ : parameter
Permalink:: http://dlmf.nist.gov/10.24.E3
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where $\gamma_{\nu}$ is real and continuous with $\gamma_{0}=0$ ; compare (5.4.3). Then

10.24.4

$\mathop{\widetilde{J}_{{-\nu}}\/}\nolimits\!\left(x\right)=\mathop{\widetilde{% J}_{{\nu}}\/}\nolimits\!\left(x\right),$

$\mathop{\widetilde{Y}_{{-\nu}}\/}\nolimits\!\left(x\right)=\mathop{\widetilde{% Y}_{{\nu}}\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{\widetilde{J}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Bessel function of imaginary order, $\mathop{\widetilde{Y}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Bessel function of imaginary order, : real variable and $\nu$ : complex parameter
Referenced by:: §10.24
Permalink:: http://dlmf.nist.gov/10.24.E4
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and $\mathop{\widetilde{J}_{{\nu}}\/}\nolimits\!\left(x\right)$ , $\mathop{\widetilde{Y}_{{\nu}}\/}\nolimits\!\left(x\right)$ are linearly independent solutions of (10.24.1):

10.24.5 $\mathop{\mathscr{W}\/}\nolimits\{\mathop{\widetilde{J}_{{\nu}}\/}\nolimits\!% \left(x\right),\mathop{\widetilde{Y}_{{\nu}}\/}\nolimits\!\left(x\right)\}=2/(% \pi x).$

Symbols:: $\mathop{\widetilde{J}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Bessel function of imaginary order, $\mathop{\widetilde{Y}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Bessel function of imaginary order, $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, : real variable and $\nu$ : complex parameter
Referenced by:: §10.24
Permalink:: http://dlmf.nist.gov/10.24.E5
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As $x\to+\infty$ , with $\nu$ fixed,

10.24.6

$\mathop{\widetilde{J}_{{\nu}}\/}\nolimits\!\left(x\right)=\sqrt{2/(\pi x)}% \mathop{\cos\/}\nolimits\left(x-\tfrac{1}{4}\pi\right)+\mathop{O\/}\nolimits% \left(x^{{-\frac{3}{2}}}\right),$

$\mathop{\widetilde{Y}_{{\nu}}\/}\nolimits\!\left(x\right)=\sqrt{2/(\pi x)}% \mathop{\sin\/}\nolimits\left(x-\tfrac{1}{4}\pi\right)+\mathop{O\/}\nolimits% \left(x^{{-\frac{3}{2}}}\right).$

Symbols:: $\mathop{\widetilde{J}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Bessel function of imaginary order, $\mathop{\widetilde{Y}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Bessel function of imaginary order, $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, : real variable and $\nu$ : complex parameter
Referenced by:: §10.24, §10.24
Permalink:: http://dlmf.nist.gov/10.24.E6
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As $x\to 0+$ , with $\nu$ fixed,

10.24.7 $\mathop{\widetilde{J}_{{\nu}}\/}\nolimits\!\left(x\right)=\left(\frac{2\mathop% {\tanh\/}\nolimits(\frac{1}{2}\pi\nu)}{\pi\nu}\right)^{{\frac{1}{2}}}\mathop{% \cos\/}\nolimits\left(\nu\mathop{\ln\/}\nolimits(\tfrac{1}{2}x)-\gamma_{\nu}% \right)+\mathop{O\/}\nolimits(x^{2}),$

Symbols:: $\mathop{\widetilde{J}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Bessel function of imaginary order, $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\tanh\/}\nolimits z$ : hyperbolic tangent function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : real variable, $\nu$ : complex parameter and $\gamma_{\nu}$ : parameter
Referenced by:: §10.24, §10.24
Permalink:: http://dlmf.nist.gov/10.24.E7
Encodings:: TeX, pMML, png

10.24.8 $\mathop{\widetilde{Y}_{{\nu}}\/}\nolimits\!\left(x\right)=\left(\frac{2\mathop% {\coth\/}\nolimits(\frac{1}{2}\pi\nu)}{\pi\nu}\right)^{{\frac{1}{2}}}\*\mathop% {\sin\/}\nolimits\left(\nu\mathop{\ln\/}\nolimits(\tfrac{1}{2}x)-\gamma_{\nu}% \right)+\mathop{O\/}\nolimits(x^{2}),$ $\nu>0$ ,

Symbols:: $\mathop{\widetilde{Y}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Bessel function of imaginary order, $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\coth\/}\nolimits z$ : hyperbolic cotangent function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\sin\/}\nolimits z$ : sine function, : real variable, $\nu$ : complex parameter and $\gamma_{\nu}$ : parameter
Permalink:: http://dlmf.nist.gov/10.24.E8
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and

10.24.9 $\mathop{\widetilde{Y}_{{0}}\/}\nolimits\!\left(x\right)=\mathop{Y_{{0}}\/}% \nolimits\!\left(x\right)=\frac{2}{\pi}\left(\mathop{\ln\/}\nolimits(\tfrac{1}% {2}x)+\EulerConstant\right)+\mathop{O\/}\nolimits(x^{2}\mathop{\ln\/}\nolimits x),$

Symbols:: $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{\widetilde{Y}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Bessel function of imaginary order, $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\EulerConstant$ : Euler’s constant, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and : real variable
Referenced by:: §10.24, §10.24
Permalink:: http://dlmf.nist.gov/10.24.E9
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where $\EulerConstant$ denotes Euler’s constant §5.2(ii).

In consequence of (10.24.6), when is large $\mathop{\widetilde{J}_{{\nu}}\/}\nolimits\!\left(x\right)$ and $\mathop{\widetilde{Y}_{{\nu}}\/}\nolimits\!\left(x\right)$ comprise a numerically satisfactory pair of solutions of (10.24.1); compare §2.7(iv). Also, in consequence of (10.24.7)–(10.24.9), when is small either $\mathop{\widetilde{J}_{{\nu}}\/}\nolimits\!\left(x\right)$ and $\mathop{\tanh\/}\nolimits(\tfrac{1}{2}\pi\nu)\mathop{\widetilde{Y}_{{\nu}}\/}% \nolimits\!\left(x\right)$ or $\mathop{\widetilde{J}_{{\nu}}\/}\nolimits\!\left(x\right)$ and $\mathop{\widetilde{Y}_{{\nu}}\/}\nolimits\!\left(x\right)$ comprise a numerically satisfactory pair depending whether $\nu\neq 0$ or $\nu=0$ .

For graphs of $\mathop{\widetilde{J}_{{\nu}}\/}\nolimits\!\left(x\right)$ and $\mathop{\widetilde{Y}_{{\nu}}\/}\nolimits\!\left(x\right)$ see §10.3(iii).

For mathematical properties and applications of $\mathop{\widetilde{J}_{{\nu}}\/}\nolimits\!\left(x\right)$ and $\mathop{\widetilde{Y}_{{\nu}}\/}\nolimits\!\left(x\right)$ , including zeros and uniform asymptotic expansions for large $\nu$ , see Dunster (1990a). In this reference $\mathop{\widetilde{J}_{{\nu}}\/}\nolimits\!\left(x\right)$ and $\mathop{\widetilde{Y}_{{\nu}}\/}\nolimits\!\left(x\right)$ are denoted respectively by $F_{{i\nu}}{(x)}$ and $G_{{i\nu}}{(x)}$ .