11 Struve and Related Functions Struve and Modified Struve Functions 11.6 Asymptotic Expansions 11.8 Analogs to Kelvin Functions

§11.7 Integrals and Sums

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Permalink:: http://dlmf.nist.gov/11.7
See also:: Annotations for Ch.11

§11.7(i) Indefinite Integrals

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Keywords:: Struve functions and modified Struve functions, indefinite, integrals
Notes:: Use §11.4(v).
Permalink:: http://dlmf.nist.gov/11.7.i
See also:: Annotations for §11.7 and Ch.11

11.7.1		$\int z^{\nu}\mathbf{H}_{\nu-1}\left(z\right)\,\mathrm{d}z=z^{\nu}\mathbf{H}_{% \nu}\left(z\right),$
ⓘ Symbols: $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $z$ : complex variable and $\nu$ : real or complex order Permalink: http://dlmf.nist.gov/11.7.E1 Encodings: TeX, pMML, png See also: Annotations for §11.7(i), §11.7 and Ch.11

11.7.2		$\int z^{-\nu}\mathbf{H}_{\nu+1}\left(z\right)\,\mathrm{d}z=-z^{-\nu}\mathbf{H}% _{\nu}\left(z\right)+\frac{2^{-\nu}z}{\sqrt{\pi}\Gamma\left(\nu+\tfrac{3}{2}% \right)},$
ⓘ 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, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $z$ : complex variable and $\nu$ : real or complex order A&S Ref: 12.1.24 Permalink: http://dlmf.nist.gov/11.7.E2 Encodings: TeX, pMML, png See also: Annotations for §11.7(i), §11.7 and Ch.11

11.7.3		$\int z^{\nu}\mathbf{L}_{\nu-1}\left(z\right)\,\mathrm{d}z=z^{\nu}\mathbf{L}_{% \nu}\left(z\right),$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\mathbf{L}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function, $z$ : complex variable and $\nu$ : real or complex order Permalink: http://dlmf.nist.gov/11.7.E3 Encodings: TeX, pMML, png See also: Annotations for §11.7(i), §11.7 and Ch.11

11.7.4		$\int z^{-\nu}\mathbf{L}_{\nu+1}\left(z\right)\,\mathrm{d}z=z^{-\nu}\mathbf{L}_% {\nu}\left(z\right)-\frac{2^{-\nu}z}{\sqrt{\pi}\Gamma\left(\nu+\tfrac{3}{2}% \right)}.$
ⓘ 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, $\mathbf{L}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function, $z$ : complex variable and $\nu$ : real or complex order Permalink: http://dlmf.nist.gov/11.7.E4 Encodings: TeX, pMML, png See also: Annotations for §11.7(i), §11.7 and Ch.11

11.7.5		$f_{\nu}(z)=\int_{0}^{z}t^{\nu}\mathbf{H}_{\nu}\left(t\right)\,\mathrm{d}t,$
ⓘ Symbols: $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $z$ : complex variable and $\nu$ : real or complex order Permalink: http://dlmf.nist.gov/11.7.E5 Encodings: TeX, pMML, png See also: Annotations for §11.7(i), §11.7 and Ch.11

then

11.7.6		$f_{\nu+1}(z)=(2\nu+1)f_{\nu}(z)-z^{\nu+1}\mathbf{H}_{\nu}\left(z\right)+\frac{% (\tfrac{1}{2}z^{2})^{\nu+1}}{(\nu+1)\sqrt{\pi}\Gamma\left(\nu+\tfrac{3}{2}% \right)},$
$\Re\nu>-1$ .
ⓘ 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, $\Re$ : real part, $z$ : complex variable and $\nu$ : real or complex order A&S Ref: 12.1.28 (The condition on $\nu$ has been corrected.) Permalink: http://dlmf.nist.gov/11.7.E6 Encodings: TeX, pMML, png See also: Annotations for §11.7(i), §11.7 and Ch.11

§11.7(ii) Definite Integrals

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Keywords:: Struve functions and modified Struve functions, definite, integrals, products
Notes:: See Babister (1967, pp. 68 and 71–72) and Watson (1944, pp. 392 and 397).
Permalink:: http://dlmf.nist.gov/11.7.ii
See also:: Annotations for §11.7 and Ch.11

11.7.7		$\int_{0}^{\pi/2}\mathbf{H}_{\nu}\left(z\sin\theta\right)\frac{(\sin\theta)^{% \nu+1}}{(\cos\theta)^{2\nu}}\,\mathrm{d}\theta=\frac{2^{-\nu}}{\sqrt{\pi}}% \Gamma\left(\tfrac{1}{2}-\nu\right)z^{\nu-1}(1-\cos z),$
$-\tfrac{3}{2}<\Re\nu<\tfrac{1}{2}$ ,
ⓘ 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 Permalink: http://dlmf.nist.gov/11.7.E7 Encodings: TeX, pMML, png See also: Annotations for §11.7(ii), §11.7 and Ch.11

11.7.8	$\displaystyle\int_{0}^{\infty}\mathbf{H}_{0}\left(t\right)\,\frac{\,\mathrm{d}% t}{t}$	$\displaystyle=\tfrac{1}{2}\pi,$
11.7.8	$\displaystyle\int_{0}^{\infty}\mathbf{H}_{1}\left(t\right)\,\frac{\,\mathrm{d}% t}{t^{2}}$	$\displaystyle=\tfrac{1}{4}\pi,$
ⓘ Symbols: $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ and $\int$ : integral A&S Ref: 12.1.22 Permalink: http://dlmf.nist.gov/11.7.E8 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §11.7(ii), §11.7 and Ch.11

11.7.9		$\int_{0}^{\infty}\mathbf{H}_{\nu}\left(t\right)\,\mathrm{d}t=-\cot\left(\tfrac% {1}{2}\pi\nu\right),$
$-2<\Re\nu<0$ ,
ⓘ Symbols: $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cot\NVar{z}$ : cotangent function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part and $\nu$ : real or complex order Permalink: http://dlmf.nist.gov/11.7.E9 Encodings: TeX, pMML, png See also: Annotations for §11.7(ii), §11.7 and Ch.11

11.7.10		$\int_{0}^{\infty}t^{-\nu-1}\mathbf{H}_{\nu}\left(t\right)\,\mathrm{d}t=\frac{% \pi}{2^{\nu+1}\Gamma\left(\nu+1\right)},$
$\Re\nu>-\tfrac{3}{2}$ ,
ⓘ 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, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part and $\nu$ : real or complex order Keywords: Mellin transform Permalink: http://dlmf.nist.gov/11.7.E10 Encodings: TeX, pMML, png See also: Annotations for §11.7(ii), §11.7 and Ch.11

11.7.11		$\int_{0}^{\infty}t^{\mu-\nu-1}\mathbf{H}_{\nu}\left(t\right)\,\mathrm{d}t=% \frac{\Gamma\left(\tfrac{1}{2}\mu\right)2^{\mu-\nu-1}\tan\left(\tfrac{1}{2}\pi% \mu\right)}{\Gamma\left(\nu-\tfrac{1}{2}\mu+1\right)},$
$\|\Re\mu\|<1$ , $\Re\nu>\Re\mu-\tfrac{3}{2}$ ,
ⓘ 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, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $\tan\NVar{z}$ : tangent function and $\nu$ : real or complex order Keywords: Mellin transform A&S Ref: 12.1.27 Permalink: http://dlmf.nist.gov/11.7.E11 Encodings: TeX, pMML, png See also: Annotations for §11.7(ii), §11.7 and Ch.11

11.7.12		$\int_{0}^{\infty}t^{-\mu-\nu}\mathbf{H}_{\mu}\left(t\right)\mathbf{H}_{\nu}% \left(t\right)\,\mathrm{d}t=\frac{\sqrt{\pi}\Gamma\left(\mu+\nu\right)}{2^{\mu% +\nu}\Gamma\left(\mu+\nu+\tfrac{1}{2}\right)\Gamma\left(\mu+\tfrac{1}{2}\right% )\Gamma\left(\nu+\tfrac{1}{2}\right)},$
$\Re\left(\mu+\nu\right)>0$ .
ⓘ 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, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part and $\nu$ : real or complex order Keywords: Mellin transform Permalink: http://dlmf.nist.gov/11.7.E12 Encodings: TeX, pMML, png See also: Annotations for §11.7(ii), §11.7 and Ch.11

For other integrals involving products of Struve functions see Zanovello (1978, 1995). For integrals involving products of $\mathbf{M}_{\nu}\left(t\right)$ functions, see Paris and Sy (1983, Appendix).

§11.7(iii) Laplace Transforms

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Keywords:: Laplace transforms, Struve functions and modified Struve functions, integrals
Notes:: See Babister (1967, §§ 3.13, 3.15).
Permalink:: http://dlmf.nist.gov/11.7.iii
See also:: Annotations for §11.7 and Ch.11

The following Laplace transforms of $\mathbf{H}_{\nu}\left(t\right)$ require $\Re a>0$ for convergence, while those of $\mathbf{L}_{\nu}\left(t\right)$ require $\Re a>1$ .

11.7.13		$\int_{0}^{\infty}e^{-at}\mathbf{H}_{0}\left(t\right)\,\mathrm{d}t=\frac{2}{\pi% \sqrt{1+a^{2}}}\ln\left(\frac{1+\sqrt{1+a^{2}}}{a}\right),$
ⓘ Symbols: $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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, $\ln\NVar{z}$ : principal branch of logarithm function and $a$ : parameter Permalink: http://dlmf.nist.gov/11.7.E13 Encodings: TeX, pMML, png See also: Annotations for §11.7(iii), §11.7 and Ch.11

11.7.14		$\int_{0}^{\infty}e^{-at}\mathbf{H}_{1}\left(t\right)\,\mathrm{d}t=\frac{2}{\pi a% }-\frac{2a}{\pi\sqrt{1+a^{2}}}\ln\left(\frac{1+\sqrt{1+a^{2}}}{a}\right),$
ⓘ Symbols: $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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, $\ln\NVar{z}$ : principal branch of logarithm function and $a$ : parameter Permalink: http://dlmf.nist.gov/11.7.E14 Encodings: TeX, pMML, png See also: Annotations for §11.7(iii), §11.7 and Ch.11

11.7.15		$\int_{0}^{\infty}e^{-at}\mathbf{L}_{0}\left(t\right)\,\mathrm{d}t=\frac{2}{\pi% \sqrt{a^{2}\!-\!1}}\operatorname{arcsin}\left(\frac{1}{a}\right),$
ⓘ Symbols: $\pi$ : 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, $\operatorname{arcsin}\NVar{z}$ : arcsine function, $\mathbf{L}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function and $a$ : parameter Permalink: http://dlmf.nist.gov/11.7.E15 Encodings: TeX, pMML, png See also: Annotations for §11.7(iii), §11.7 and Ch.11

11.7.16	$\int_{0}^{\infty}e^{-at}\mathbf{L}_{1}\left(t\right)\,\mathrm{d}t$
		$\displaystyle=\frac{2a}{\pi\sqrt{a^{2}\!-\!1}}\operatorname{arctan}\left(\frac% {1}{\sqrt{a^{2}\!-\!1}}\right)-\frac{2}{\pi a}.$
ⓘ Symbols: $\pi$ : 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, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\mathbf{L}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function and $a$ : parameter Permalink: http://dlmf.nist.gov/11.7.E16 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §11.7(iii), §11.7 and Ch.11

§11.7(iv) Integrals with Respect to Order

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Keywords:: Struve functions and modified Struve functions, integrals, with respect to order
Permalink:: http://dlmf.nist.gov/11.7.iv
See also:: Annotations for §11.7 and Ch.11

For integrals of $\mathbf{H}_{\nu}\left(x\right)$ and $\mathbf{L}_{\nu}\left(x\right)$ with respect to the order $\nu$ , see Apelblat (1989).

§11.7(v) Compendia

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Keywords:: Struve functions and modified Struve functions, compendia, integrals, sums
Referenced by:: Erratum (V1.0.10) for References
Permalink:: http://dlmf.nist.gov/11.7.v
Addition (effective with 1.0.10):: A reference to Baricz and Pogány (2013) was added at the end of this section.
See also:: Annotations for §11.7 and Ch.11

For further integrals see Apelblat (1983, §12.16), Babister (1967, Chapter 3), Erdélyi et al. (1954a, §§4.19, 6.8, 8.15, 9.4, 10.3, 11.3, and 15.3), Luke (1962, Chapters 9, 11), Gradshteyn and Ryzhik (2015, §6.8), Marichev (1983, pp. 192–193 and 215–216), Oberhettinger (1972), Oberhettinger (1974, §1.12), Oberhettinger (1990, §§1.21 and 2.21), Oberhettinger and Badii (1973, §1.16), Prudnikov et al. (1990, §§1.4 and 2.7), Prudnikov et al. (1992a, §3.17), and Prudnikov et al. (1992b, §3.17).

For sums of Struve functions see Hansen (1975, p. 456) and Prudnikov et al. (1990, §6.4.1). See also Baricz and Pogány (2013)

11.6 Asymptotic Expansions 11.8 Analogs to Kelvin Functions

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