§11.5 Integral Representations

11.5.1		$$\begin{array}{ll}{\mathbf{H}}_{\nu }\left(z\right)& =\frac{2{(\frac{1}{2}z)}^{\nu }}{\sqrt{\pi }\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}{\int }_0^1{(1-t^2)}^{\nu -\frac{1}{2}}\sin \left(zt\right)dt\\ & =\frac{2{(\frac{1}{2}z)}^{\nu }}{\sqrt{\pi }\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}{\int }_0^{\pi /2}\sin \left(z\cos \theta \right){(\sin \theta )}^{2\nu }d\theta ,\end{array}$$
$\mathrm{\Re }\nu >-\frac{1}{2}$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathbf{H}}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\int$: integral, $\mathrm{\Re }$: real part, $\sin z$: 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		$${\mathbf{K}}_{\nu }\left(z\right)=\frac{2{(\frac{1}{2}z)}^{\nu }}{\sqrt{\pi }\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{(1+t^2)}^{\nu -\frac{1}{2}}dt,$$
$\mathrm{\Re }z>0$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathbf{K}}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: 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		$${\mathbf{K}}_0\left(z\right)=\frac{2}{\pi }{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-z\sinh t}dt,$$
$\mathrm{\Re }z>0$,
ⓘ Symbols: ${\mathbf{K}}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\int$: integral, $\mathrm{\Re }$: 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		$${\mathbf{M}}_{\nu }\left(z\right)=-\frac{2{(\frac{1}{2}z)}^{\nu }}{\sqrt{\pi }\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}{\int }_0^1{\mathrm{e}}^{-zt}{(1-t^2)}^{\nu -\frac{1}{2}}dt,$$
$\mathrm{\Re }\nu >-\frac{1}{2}$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathbf{M}}_{\nu }\left(z\right)$: modified Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: 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		$${\mathbf{M}}_0\left(z\right)=-\frac{2}{\pi }{\int }_0^{\pi /2}{\mathrm{e}}^{-z\cos \theta }d\theta ,$$
ⓘ Symbols: ${\mathbf{M}}_{\nu }\left(z\right)$: modified Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: 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		$${\mathbf{L}}_{\nu }\left(z\right)=\frac{2{(\frac{1}{2}z)}^{\nu }}{\sqrt{\pi }\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}{\int }_0^{\pi /2}\sinh \left(z\cos \theta \right){(\sin \theta )}^{2\nu }d\theta ,$$
$\mathrm{\Re }\nu >-\frac{1}{2}$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\sinh z$: hyperbolic sine function, $\int$: integral, ${\mathbf{L}}_{\nu }\left(z\right)$: modified Struve function, $\mathrm{\Re }$: real part, $\sin z$: 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		$$I_{-\nu }\left(x\right)-{\mathbf{L}}_{\nu }\left(x\right)=\frac{2{(\frac{1}{2}x)}^{\nu }}{\sqrt{\pi }\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}{\int }_0^{\mathrm{\infty }}{(1+t^2)}^{\nu -\frac{1}{2}}\sin \left(xt\right)dt,$$
$x>0$, .
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, ${\mathbf{L}}_{\nu }\left(z\right)$: modified Struve function, $\mathrm{\Re }$: real part, $\sin z$: 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

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).

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).