10 Bessel Functions Bessel and Hankel Functions 10.23 Sums 10.25 Definitions

§10.24 Functions of Imaginary Order

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Keywords:: Bessel functions, applications, definitions, limiting forms, numerically satisfactory pairs, of imaginary order, uniform asymptotic expansions for large order, zeros
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.
Referenced by:: §10.3(iii), §10.45, §10.7(i)
Permalink:: http://dlmf.nist.gov/10.24
See also:: Annotations for Ch.10

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

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

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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:: §10.24, §10.24
Permalink:: http://dlmf.nist.gov/10.24.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §10.24 and Ch.10

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

10.24.2

$\widetilde{J}_{\nu}\left(x\right)=\operatorname{sech}\left(\tfrac{1}{2}\pi\nu% \right)\Re\left(J_{i\nu}\left(x\right)\right),$

$\widetilde{Y}_{\nu}\left(x\right)=\operatorname{sech}\left(\tfrac{1}{2}\pi\nu% \right)\Re\left(Y_{i\nu}\left(x\right)\right),$

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Defines:: $\widetilde{J}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Bessel function of imaginary order and $\widetilde{Y}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Bessel function of imaginary order
Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, $\mathrm{i}$ : imaginary unit, $\Re$ : real part, : real variable and $\nu$ : complex parameter
Referenced by:: §10.24
Permalink:: http://dlmf.nist.gov/10.24.E2
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §10.24 and Ch.10

and

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

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\nu$ : complex parameter and $\gamma_{\nu}$ : parameter
Permalink:: http://dlmf.nist.gov/10.24.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §10.24 and Ch.10

where $\gamma_{\nu}$ is real and continuous with $\gamma_{0}=0$ ; compare (5.4.3). Then

10.24.4

$\widetilde{J}_{-\nu}\left(x\right)=\widetilde{J}_{\nu}\left(x\right),$

$\widetilde{Y}_{-\nu}\left(x\right)=\widetilde{Y}_{\nu}\left(x\right),$

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Symbols:: $\widetilde{J}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Bessel function of imaginary order, $\widetilde{Y}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Bessel function of imaginary order, : real variable and $\nu$ : complex parameter
Referenced by:: §10.24
Permalink:: http://dlmf.nist.gov/10.24.E4
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §10.24 and Ch.10

and $\widetilde{J}_{\nu}\left(x\right)$ , $\widetilde{Y}_{\nu}\left(x\right)$ are linearly independent solutions of (10.24.1):

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

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Symbols:: $\widetilde{J}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Bessel function of imaginary order, $\widetilde{Y}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Bessel function of imaginary order, $\mathscr{W}$ : Wronskian, $\pi$ : the ratio of the circumference of a circle to its diameter, : real variable and $\nu$ : complex parameter
Referenced by:: §10.24
Permalink:: http://dlmf.nist.gov/10.24.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §10.24 and Ch.10

As $x\to+\infty$ , with $\nu$ fixed,

10.24.6

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

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

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Symbols:: $\widetilde{J}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Bessel function of imaginary order, $\widetilde{Y}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Bessel function of imaginary order, $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, : real variable and $\nu$ : complex parameter
Referenced by:: §10.24, §10.24
Permalink:: http://dlmf.nist.gov/10.24.E6
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §10.24 and Ch.10

As $x\to 0+$ , with $\nu$ fixed,

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

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Symbols:: $\widetilde{J}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Bessel function of imaginary order, $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\tanh\NVar{z}$ : hyperbolic tangent function, $\ln\NVar{z}$ : 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
See also:: Annotations for §10.24 and Ch.10

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

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Symbols:: $\widetilde{Y}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Bessel function of imaginary order, $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $\coth\NVar{z}$ : hyperbolic cotangent function, $\ln\NVar{z}$ : principal branch of logarithm function, $\sin\NVar{z}$ : sine function, : real variable, $\nu$ : complex parameter and $\gamma_{\nu}$ : parameter
Permalink:: http://dlmf.nist.gov/10.24.E8
Encodings:: TeX, pMML, png
See also:: Annotations for §10.24 and Ch.10

and

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

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Symbols:: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\widetilde{Y}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Bessel function of imaginary order, $O\left(\NVar{x}\right)$ : order not exceeding, $\gamma$ : Euler’s constant, $\pi$ : the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$ : principal branch of logarithm function and : real variable
Referenced by:: §10.24, §10.24
Permalink:: http://dlmf.nist.gov/10.24.E9
Encodings:: TeX, pMML, png
See also:: Annotations for §10.24 and Ch.10

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

In consequence of (10.24.6), when is large $\widetilde{J}_{\nu}\left(x\right)$ and $\widetilde{Y}_{\nu}\left(x\right)$ 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 $\widetilde{J}_{\nu}\left(x\right)$ and $\tanh\left(\tfrac{1}{2}\pi\nu\right)\widetilde{Y}_{\nu}\left(x\right)$ or $\widetilde{J}_{\nu}\left(x\right)$ and $\widetilde{Y}_{\nu}\left(x\right)$ comprise a numerically satisfactory pair depending whether $\nu\neq 0$ or $\nu=0$ .

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

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

10.23 Sums 10.25 Definitions

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