§11.7 Integrals and Sums

§11.7(i) Indefinite Integrals

 11.7.1 $\int z^{\nu}\mathop{\mathbf{H}_{\nu-1}\/}\nolimits\!\left(z\right)dz=z^{\nu}% \mathop{\mathbf{H}_{\nu}\/}\nolimits\!\left(z\right),$
 11.7.2 $\int z^{-\nu}\mathop{\mathbf{H}_{\nu+1}\/}\nolimits\!\left(z\right)dz=-z^{-\nu% }\mathop{\mathbf{H}_{\nu}\/}\nolimits\!\left(z\right)+\frac{2^{-\nu}z}{\sqrt{% \pi}\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{3}{2}\right)},$
 11.7.3 $\int z^{\nu}\mathop{\mathbf{L}_{\nu-1}\/}\nolimits\!\left(z\right)dz=z^{\nu}% \mathop{\mathbf{L}_{\nu}\/}\nolimits\!\left(z\right),$
 11.7.4 $\int z^{-\nu}\mathop{\mathbf{L}_{\nu+1}\/}\nolimits\!\left(z\right)dz=z^{-\nu}% \mathop{\mathbf{L}_{\nu}\/}\nolimits\!\left(z\right)-\frac{2^{-\nu}z}{\sqrt{% \pi}\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{3}{2}\right)}.$

If

 11.7.5 $f_{\nu}(z)=\int_{0}^{z}t^{\nu}\mathop{\mathbf{H}_{\nu}\/}\nolimits\!\left(t% \right)dt,$

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: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\mathop{\mathbf{H}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: Struve function, $\realpart{}$: 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: info for 11.7(i)

§11.7(ii) Definite Integrals

 11.7.7 $\int_{0}^{\pi/2}\mathop{\mathbf{H}_{\nu}\/}\nolimits\!\left(z\mathop{\sin\/}% \nolimits\theta\right)\frac{(\mathop{\sin\/}\nolimits\theta)^{\nu+1}}{(\mathop% {\cos\/}\nolimits\theta)^{2\nu}}d\theta=\frac{2^{-\nu}}{\sqrt{\pi}}\mathop{% \Gamma\/}\nolimits\!\left(\tfrac{1}{2}-\nu\right)z^{\nu-1}(1-\mathop{\cos\/}% \nolimits z),$ $-\tfrac{3}{2}<\realpart{\nu}<\tfrac{1}{2}$,
 11.7.8 $\displaystyle\int_{0}^{\infty}\mathop{\mathbf{H}_{0}\/}\nolimits\!\left(t% \right)\,\frac{dt}{t}$ $\displaystyle=\tfrac{1}{2}\pi,$ $\displaystyle\int_{0}^{\infty}\mathop{\mathbf{H}_{1}\/}\nolimits\!\left(t% \right)\,\frac{dt}{t^{2}}$ $\displaystyle=\tfrac{1}{4}\pi,$ Symbols: $\mathop{\mathbf{H}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: Struve function, $d\NVar{x}$: 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: info for 11.7(ii)
 11.7.9 $\int_{0}^{\infty}\mathop{\mathbf{H}_{\nu}\/}\nolimits\!\left(t\right)dt=-% \mathop{\cot\/}\nolimits\!\left(\tfrac{1}{2}\pi\nu\right),$ $-2<\realpart{\nu}<0$,
 11.7.10 $\int_{0}^{\infty}t^{-\nu-1}\mathop{\mathbf{H}_{\nu}\/}\nolimits\!\left(t\right% )dt=\frac{\pi}{2^{\nu+1}\mathop{\Gamma\/}\nolimits\!\left(\nu+1\right)},$ $\realpart{\nu}>-\tfrac{3}{2}$,
 11.7.11 $\int_{0}^{\infty}t^{\mu-\nu-1}\mathop{\mathbf{H}_{\nu}\/}\nolimits\!\left(t% \right)dt=\frac{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}\mu\right)2^{\mu% -\nu-1}\mathop{\tan\/}\nolimits\!\left(\tfrac{1}{2}\pi\mu\right)}{\mathop{% \Gamma\/}\nolimits\!\left(\nu-\tfrac{1}{2}\mu+1\right)},$ $|\realpart{\mu}|<1$, $\realpart{\nu}>\realpart{\mu}-\tfrac{3}{2}$,
 11.7.12 $\int_{0}^{\infty}t^{-\mu-\nu}\mathop{\mathbf{H}_{\mu}\/}\nolimits\!\left(t% \right)\mathop{\mathbf{H}_{\nu}\/}\nolimits\!\left(t\right)dt=\frac{\sqrt{\pi}% \mathop{\Gamma\/}\nolimits\!\left(\mu+\nu\right)}{2^{\mu+\nu}\mathop{\Gamma\/}% \nolimits\!\left(\mu+\nu+\tfrac{1}{2}\right)\mathop{\Gamma\/}\nolimits\!\left(% \mu+\tfrac{1}{2}\right)\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{1}{2}% \right)},$ $\realpart{(\mu+\nu)}>0$.

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

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 $\int_{0}^{\infty}e^{-at}\mathop{\mathbf{H}_{0}\/}\nolimits\!\left(t\right)dt=% \frac{2}{\pi\sqrt{1+a^{2}}}\mathop{\ln\/}\nolimits\!\left(\frac{1+\sqrt{1+a^{2% }}}{a}\right),$
 11.7.14 $\int_{0}^{\infty}e^{-at}\mathop{\mathbf{H}_{1}\/}\nolimits\!\left(t\right)dt=% \frac{2}{\pi a}-\frac{2a}{\pi\sqrt{1+a^{2}}}\mathop{\ln\/}\nolimits\!\left(% \frac{1+\sqrt{1+a^{2}}}{a}\right),$
 11.7.15 $\int_{0}^{\infty}e^{-at}\mathop{\mathbf{L}_{0}\/}\nolimits\!\left(t\right)dt=% \frac{2}{\pi\sqrt{a^{2}\!-\!1}}\mathop{\mathrm{arcsin}\/}\nolimits\!\left(% \frac{1}{a}\right),$
 11.7.16 $\int_{0}^{\infty}e^{-at}\mathop{\mathbf{L}_{1}\/}\nolimits\!\left(t\right)dt$ $\displaystyle=\frac{2a}{\pi\sqrt{a^{2}\!-\!1}}\mathop{\mathrm{arctan}\/}% \nolimits\!\left(\frac{1}{\sqrt{a^{2}\!-\!1}}\right)-\frac{2}{\pi a}.$

§11.7(iv) Integrals with Respect to Order

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

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). See also Baricz and Pogány (2013)