11 Struve and Related Functions Struve and Modified Struve Functions 11.4 Basic Properties 11.6 Asymptotic Expansions

§11.5 Integral Representations

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Referenced by:: §11.13(iii)
Permalink:: http://dlmf.nist.gov/11.5
See also:: Annotations for Ch.11

§11.5(i) Integrals Along the Real Line
§11.5(ii) Contour Integrals
§11.5(iii) Compendia

§11.5(i) Integrals Along the Real Line

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Keywords:: Struve functions and modified Struve functions, along real line, integral representations
Notes:: For (11.5.1)–(11.5.3), (11.5.6), and (11.5.7) see Watson (1944, pp. 328,331,332), together with (11.2.5) in the case of (11.5.2) and (11.5.3), and (11.2.2) in the case of (11.5.6). For (11.5.4) and (11.5.5) see Babister (1967, Eq. (3.102)) and collapse the integration path onto the interval $[0,1]$ .
Referenced by:: §11.13(iii)
Permalink:: http://dlmf.nist.gov/11.5.i
See also:: Annotations for §11.5 and Ch.11

11.5.1		$\mathbf{H}_{\nu}\left(z\right)=\frac{2(\tfrac{1}{2}z)^{\nu}}{\sqrt{\pi}\Gamma% \left(\nu+\tfrac{1}{2}\right)}\int_{0}^{1}(1-t^{2})^{\nu-\frac{1}{2}}\sin\left% (zt\right)\,\mathrm{d}t=\frac{2(\tfrac{1}{2}z)^{\nu}}{\sqrt{\pi}\Gamma\left(% \nu+\tfrac{1}{2}\right)}\int_{0}^{\pi/2}\sin\left(z\cos\theta\right)(\sin% \theta)^{2\nu}\,\mathrm{d}\theta,$
$\Re\nu>-\tfrac{1}{2}$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $\nu$ : real or complex order A&S Ref: 12.1.6--12.1.7 Referenced by: §11.10(vii), §11.5(i), §11.5(ii) Permalink: http://dlmf.nist.gov/11.5.E1 Encodings: TeX, pMML, png See also: Annotations for §11.5(i), §11.5 and Ch.11

11.5.2		$\mathbf{K}_{\nu}\left(z\right)=\frac{2(\tfrac{1}{2}z)^{\nu}}{\sqrt{\pi}\Gamma% \left(\nu+\tfrac{1}{2}\right)}\int_{0}^{\infty}e^{-zt}(1+t^{2})^{\nu-\frac{1}{% 2}}\,\mathrm{d}t,$
$\Re z>0$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathbf{K}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part, $z$ : complex variable and $\nu$ : real or complex order A&S Ref: 12.1.8 Referenced by: §11.13(iii), §11.5(i), §11.5(ii), §11.6(i) Permalink: http://dlmf.nist.gov/11.5.E2 Encodings: TeX, pMML, png See also: Annotations for §11.5(i), §11.5 and Ch.11

11.5.3		$\mathbf{K}_{0}\left(z\right)=\frac{2}{\pi}\int_{0}^{\infty}e^{-z\sinh t}\,% \mathrm{d}t,$
$\Re z>0$ ,
ⓘ Symbols: $\mathbf{K}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $\Re$ : real part and $z$ : complex variable Referenced by: §11.5(i) Permalink: http://dlmf.nist.gov/11.5.E3 Encodings: TeX, pMML, png See also: Annotations for §11.5(i), §11.5 and Ch.11

11.5.4		$\mathbf{M}_{\nu}\left(z\right)=-\frac{2(\tfrac{1}{2}z)^{\nu}}{\sqrt{\pi}\Gamma% \left(\nu+\tfrac{1}{2}\right)}\int_{0}^{1}e^{-zt}(1-t^{2})^{\nu-\frac{1}{2}}\,% \mathrm{d}t,$
$\Re\nu>-\tfrac{1}{2}$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathbf{M}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part, $z$ : complex variable and $\nu$ : real or complex order Referenced by: §11.5(i), §11.5(ii), §11.6(i), §11.6(iii) Permalink: http://dlmf.nist.gov/11.5.E4 Encodings: TeX, pMML, png See also: Annotations for §11.5(i), §11.5 and Ch.11

11.5.5		$\mathbf{M}_{0}\left(z\right)=-\frac{2}{\pi}\int_{0}^{\pi/2}e^{-z\cos\theta}\,% \mathrm{d}\theta,$
ⓘ Symbols: $\mathbf{M}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral and $z$ : complex variable Referenced by: §11.13(iii), §11.5(i) Permalink: http://dlmf.nist.gov/11.5.E5 Encodings: TeX, pMML, png See also: Annotations for §11.5(i), §11.5 and Ch.11

11.5.6		$\mathbf{L}_{\nu}\left(z\right)=\frac{2(\tfrac{1}{2}z)^{\nu}}{\sqrt{\pi}\Gamma% \left(\nu+\tfrac{1}{2}\right)}\int_{0}^{\pi/2}\sinh\left(z\cos\theta\right)(% \sin\theta)^{2\nu}\,\mathrm{d}\theta,$
$\Re\nu>-\tfrac{1}{2}$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $\mathbf{L}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function, $\Re$ : real part, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $\nu$ : real or complex order Referenced by: §11.5(i) Permalink: http://dlmf.nist.gov/11.5.E6 Encodings: TeX, pMML, png See also: Annotations for §11.5(i), §11.5 and Ch.11

11.5.7		$I_{-\nu}\left(x\right)-\mathbf{L}_{\nu}\left(x\right)=\frac{2(\tfrac{1}{2}x)^{% \nu}}{\sqrt{\pi}\Gamma\left(\nu+\tfrac{1}{2}\right)}\int_{0}^{\infty}(1+t^{2})% ^{\nu-\frac{1}{2}}\sin\left(xt\right)\,\mathrm{d}t,$
$x>0$ , $\Re\nu<\tfrac{1}{2}$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\mathbf{L}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function, $\Re$ : real part, $\sin\NVar{z}$ : sine function, $x$ : real variable and $\nu$ : real or complex order Referenced by: §11.5(i), §11.5(ii) Permalink: http://dlmf.nist.gov/11.5.E7 Encodings: TeX, pMML, png See also: Annotations for §11.5(i), §11.5 and Ch.11

§11.5(ii) Contour Integrals

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Keywords:: Struve functions and modified Struve functions, contour integrals, integral representations
Permalink:: http://dlmf.nist.gov/11.5.ii
See also:: Annotations for §11.5 and Ch.11

For loop-integral versions of (11.5.1), (11.5.2), (11.5.4), and (11.5.7) see Babister (1967, §§3.3 and 3.14).

Mellin–Barnes Integrals

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Keywords:: Mellin–Barnes type, Struve functions and modified Struve functions, integral representations
Notes:: For (11.5.8) see Babister (1967, §3.7) with modified convergence conditions. For (11.5.9) deform the integration path in (11.5.8) into a loop and use (11.2.2).
See also:: Annotations for §11.5(ii), §11.5 and Ch.11

11.5.8		$(\tfrac{1}{2}x)^{-\nu-1}\mathbf{H}_{\nu}\left(x\right)=-\frac{1}{2\pi i}\int_{% -i\infty}^{i\infty}\frac{\pi\csc\left(\pi s\right)}{\Gamma\left(\tfrac{3}{2}+s% \right)\Gamma\left(\tfrac{3}{2}+\nu+s\right)}(\tfrac{1}{4}x^{2})^{s}\,\mathrm{% d}s,$
$x>0$ , $\Re\nu>-1$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\csc\NVar{z}$ : cosecant function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part, $x$ : real variable and $\nu$ : real or complex order Keywords: Mellin transform Referenced by: §11.5(ii), §11.5(ii) Permalink: http://dlmf.nist.gov/11.5.E8 Encodings: TeX, pMML, png See also: Annotations for §11.5(ii), §11.5(ii), §11.5 and Ch.11

11.5.9		$(\tfrac{1}{2}z)^{-\nu-1}\mathbf{L}_{\nu}\left(z\right)=\frac{1}{2\pi i}\int_{% \infty}^{(0+)}\frac{\pi\csc\left(\pi s\right)}{\Gamma\left(\tfrac{3}{2}+s% \right)\Gamma\left(\tfrac{3}{2}+\nu+s\right)}(-\tfrac{1}{4}z^{2})^{s}\,\mathrm% {d}s.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\csc\NVar{z}$ : cosecant function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\mathbf{L}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function, $z$ : complex variable and $\nu$ : real or complex order Keywords: Mellin transform Referenced by: §11.5(ii), §11.5(ii) Permalink: http://dlmf.nist.gov/11.5.E9 Encodings: TeX, pMML, png See also: Annotations for §11.5(ii), §11.5(ii), §11.5 and Ch.11

In (11.5.8) and (11.5.9) the path of integration separates the poles of the integrand at $s=0,1,2,\dots$ from those at $s=-1,-2,-3,\dots$ .

§11.5(iii) Compendia

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Keywords:: Struve functions and modified Struve functions, compendia, integral representations
Permalink:: http://dlmf.nist.gov/11.5.iii
See also:: Annotations for §11.5 and Ch.11

For further integral representations see Babister (1967, §§3.3, 3.14), Erdélyi et al. (1954a, §§5.17, 15.3), Magnus et al. (1966, p. 114), Oberhettinger (1972), Oberhettinger (1974, §2.7), Oberhettinger and Badii (1973, §2.14), and Watson (1944, pp. 330, 374, and 426).

11.4 Basic Properties 11.6 Asymptotic Expansions

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§11.5 Integral Representations

Contents

§11.5(i) Integrals Along the Real Line

§11.5(ii) Contour Integrals

Mellin–Barnes Integrals

§11.5(iii) Compendia