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

Permalink:: http://dlmf.nist.gov/11.7

§11.7(i) Indefinite Integrals
§11.7(ii) Definite Integrals
§11.7(iii) Laplace Transforms
§11.7(iv) Integrals with Respect to Order
§11.7(v) Compendia

§11.7(i) Indefinite Integrals

Notes:: Use §11.4(v).
Keywords:: Struve functions and modified Struve functions, integrals
Permalink:: http://dlmf.nist.gov/11.7.i

11.7.1 $\int z^{\nu}\mathop{\mathbf{H}_{{\nu-1}}\/}\nolimits\!\left(z\right)dz=z^{\nu}% \mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, : differential of , $\int$ : integral, : complex variable and $\nu$ : real or complex order
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11.7.2 $\int z^{{-\nu}}\mathop{\mathbf{H}_{{\nu+1}}\/}\nolimits\!\left(z\right)dz=-z^{% {-\nu}}\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)+\frac{2^{{-\nu}}% z}{\sqrt{\pi}\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{3}{2}\right)},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, : differential of , $\int$ : integral, : complex variable and $\nu$ : real or complex order
A&S Ref:: 12.1.24
Permalink:: http://dlmf.nist.gov/11.7.E2
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11.7.3 $\int z^{{\nu}}\mathop{\mathbf{L}_{{\nu-1}}\/}\nolimits\!\left(z\right)dz=z^{% \nu}\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right),$

Symbols:: : differential of , $\int$ : integral, $\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Struve function, : complex variable and $\nu$ : real or complex order
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11.7.4 $\int z^{{-\nu}}\mathop{\mathbf{L}_{{\nu+1}}\/}\nolimits\!\left(z\right)dz=z^{{% -\nu}}\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)-\frac{2^{{-\nu}}z% }{\sqrt{\pi}\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{3}{2}\right)}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : differential of , $\int$ : integral, $\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Struve function, : complex variable and $\nu$ : real or complex order
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11.7.5 $f_{\nu}(z)=\int_{0}^{z}t^{\nu}\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(t% \right)dt,$

Symbols:: $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, : differential of , $\int$ : integral, : complex variable and $\nu$ : real or complex order
Permalink:: http://dlmf.nist.gov/11.7.E5
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then

11.7.6 $f_{{\nu+1}}(z)=(2\nu+1)f_{\nu}(z)-z^{{\nu+1}}\mathop{\mathbf{H}_{{\nu}}\/}% \nolimits\!\left(z\right)+\frac{(\tfrac{1}{2}z^{2})^{{\nu+1}}}{(\nu+1)\sqrt{% \pi}\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{3}{2}\right)},$ $\realpart{\nu}>-1$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, $\realpart{}$ : real part, : 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
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§11.7(ii) Definite Integrals

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

11.7.7 $\int_{0}^{{\pi/2}}\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\mathop{\sin% \/}\nolimits\theta\right)\frac{(\mathop{\sin\/}\nolimits\theta)^{{\nu+1}}}{(% \mathop{\cos\/}\nolimits\theta)^{{2\nu}}}d\theta=\frac{2^{{-\nu}}}{\sqrt{\pi}}% \mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}-\nu\right)z^{{\nu-1}}(1-\mathop% {\cos\/}\nolimits z),$ $-\tfrac{3}{2}<\realpart{\nu}<\tfrac{1}{2}$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\int$ : integral, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : sine function, : complex variable and $\nu$ : real or complex order
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11.7.8

$\int_{0}^{\infty}\mathop{\mathbf{H}_{{0}}\/}\nolimits\!\left(t\right)\,\frac{% dt}{t}=\tfrac{1}{2}\pi,$

$\int_{0}^{\infty}\mathop{\mathbf{H}_{{1}}\/}\nolimits\!\left(t\right)\,\frac{% dt}{t^{2}}=\tfrac{1}{4}\pi,$

Symbols:: $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, : differential of and $\int$ : integral
A&S Ref:: 12.1.22
Permalink:: http://dlmf.nist.gov/11.7.E8
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11.7.9 $\int_{0}^{\infty}\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(t\right)dt=-% \mathop{\cot\/}\nolimits\!\left(\tfrac{1}{2}\pi\nu\right),$ $-2<\realpart{\nu}<0$ ,

Symbols:: $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, $\mathop{\cot\/}\nolimits z$ : cotangent function, : differential of , $\int$ : integral, $\realpart{}$ : real part and $\nu$ : real or complex order
Permalink:: http://dlmf.nist.gov/11.7.E9
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11.7.10 $\int_{0}^{\infty}t^{{-\nu-1}}\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(t% \right)dt=\frac{\pi}{2^{{\nu+1}}\mathop{\Gamma\/}\nolimits\!\left(\nu+1\right)},$ $\realpart{\nu}>-\tfrac{3}{2}$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, : differential of , $\int$ : integral, $\realpart{}$ : real part and $\nu$ : real or complex order
Permalink:: http://dlmf.nist.gov/11.7.E10
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11.7.11 $\int_{0}^{\infty}t^{{\mu-\nu-1}}\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(% t\right)dt=\frac{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}\mu\right)2^{{% \mu-\nu-1}}\mathop{\tan\/}\nolimits\!\left(\tfrac{1}{2}\pi\mu\right)}{\mathop{% \Gamma\/}\nolimits\!\left(\nu-\tfrac{1}{2}\mu+1\right)},$ $|\realpart{\mu}|<1$ , $\realpart{\nu}>\realpart{\mu}-\tfrac{3}{2}$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, : differential of , $\int$ : integral, $\realpart{}$ : real part, $\mathop{\tan\/}\nolimits z$ : tangent function and $\nu$ : real or complex order
A&S Ref:: 12.1.27
Permalink:: http://dlmf.nist.gov/11.7.E11
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11.7.12 $\int_{0}^{\infty}t^{{-\mu-\nu}}\mathop{\mathbf{H}_{{\mu}}\/}\nolimits\!\left(t% \right)\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(t\right)dt=\frac{\sqrt{% \pi}\mathop{\Gamma\/}\nolimits\!\left(\mu+\nu\right)}{2^{{\mu+\nu}}\mathop{% \Gamma\/}\nolimits\!\left(\mu+\nu+\tfrac{1}{2}\right)\mathop{\Gamma\/}% \nolimits\!\left(\mu+\tfrac{1}{2}\right)\mathop{\Gamma\/}\nolimits\!\left(\nu+% \tfrac{1}{2}\right)},$ $\realpart{(\mu+\nu)}>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, : differential of , $\int$ : integral, $\realpart{}$ : real part and $\nu$ : real or complex order
Permalink:: http://dlmf.nist.gov/11.7.E12
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For other integrals involving products of Struve functions see Zanovello (1978, 1995). For integrals involving products of $\mathop{\mathbf{M}_{{\nu}}\/}\nolimits\!\left(t\right)$ functions, see Paris and Sy (1983, Appendix).

§11.7(iii) Laplace Transforms

Notes:: See Babister (1967, §§ 3.13, 3.15).
Keywords:: Struve functions and modified Struve functions, integrals
Permalink:: http://dlmf.nist.gov/11.7.iii

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

11.7.13 $\int_{0}^{\infty}e^{{-at}}\mathop{\mathbf{H}_{{0}}\/}\nolimits\!\left(t\right)% dt=\frac{2}{\pi\sqrt{1+a^{2}}}\mathop{\ln\/}\nolimits\!\left(\frac{1+\sqrt{1+a% ^{2}}}{a}\right),$

Symbols:: $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, : differential of , : base of exponential function, $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and : parameter
Permalink:: http://dlmf.nist.gov/11.7.E13
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11.7.14 $\int_{0}^{\infty}e^{{-at}}\mathop{\mathbf{H}_{{1}}\/}\nolimits\!\left(t\right)% dt=\frac{2}{\pi a}-\frac{2a}{\pi\sqrt{1+a^{2}}}\mathop{\ln\/}\nolimits\!\left(% \frac{1+\sqrt{1+a^{2}}}{a}\right),$

Symbols:: $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, : differential of , : base of exponential function, $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and : parameter
Permalink:: http://dlmf.nist.gov/11.7.E14
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11.7.15 $\int_{0}^{\infty}e^{{-at}}\mathop{\mathbf{L}_{{0}}\/}\nolimits\!\left(t\right)% dt=\frac{2}{\pi\sqrt{a^{2}\!-\!1}}\mathop{\mathrm{arcsin}\/}\nolimits\!\left(% \frac{1}{a}\right),$

Symbols:: : differential of , : base of exponential function, $\int$ : integral, $\mathop{\mathrm{arcsin}\/}\nolimits z$ : arcsine function, $\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Struve function and : parameter
Permalink:: http://dlmf.nist.gov/11.7.E15
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11.7.16

$\int_{0}^{\infty}e^{{-at}}\mathop{\mathbf{L}_{{1}}\/}\nolimits\!\left(t\right)dt$

$=\frac{2a}{\pi\sqrt{a^{2}\!-\!1}}\mathop{\mathrm{arctan}\/}\nolimits\!\left(% \frac{1}{\sqrt{a^{2}\!-\!1}}\right)-\frac{2}{\pi a}.$

Symbols:: : differential of , : base of exponential function, $\int$ : integral, $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function, $\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Struve function and : parameter
Permalink:: http://dlmf.nist.gov/11.7.E16
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§11.7(iv) Integrals with Respect to Order

Keywords:: Struve functions and modified Struve functions, integrals
Permalink:: http://dlmf.nist.gov/11.7.iv

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

§11.7(v) Compendia

Keywords:: Struve functions and modified Struve functions, integrals
Permalink:: http://dlmf.nist.gov/11.7.v

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 (2000, §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).

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