§11.5 Integral Representations

Referenced by:: §11.13(iii)
Permalink:: http://dlmf.nist.gov/11.5

§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

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 $[0,1]$ .
Keywords:: Struve functions and modified Struve functions
Referenced by:: §11.13(iii)
Permalink:: http://dlmf.nist.gov/11.5.i

11.5.1 $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{2(\tfrac{1}{2}z)^{\nu}}{\sqrt{\pi}\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{1}{2}\right)}\int _{0}^{1}(1-t^{2})^{{\nu-\frac{1}{2}}}\mathop{\sin\/}\nolimits\!\left(zt\right)dt=\frac{2(\tfrac{1}{2}z)^{\nu}}{\sqrt{\pi}\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{1}{2}\right)}\int _{0}^{{\pi/2}}\mathop{\sin\/}\nolimits\!\left(z\mathop{\cos\/}\nolimits\theta\right)(\mathop{\sin\/}\nolimits\theta)^{{2\nu}}d\theta,$ $\realpart{\nu}>-\tfrac{1}{2}$ ,

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, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : 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
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11.5.2 $\mathop{\mathbf{K}_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{2(\tfrac{1}{2}z)^{\nu}}{\sqrt{\pi}\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{1}{2}\right)}\int _{0}^{\infty}e^{{-zt}}(1+t^{2})^{{\nu-\frac{1}{2}}}dt,$ $\realpart{z}>0$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathbf{K}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : 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
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11.5.3 $\mathop{\mathbf{K}_{{0}}\/}\nolimits\!\left(z\right)=\frac{2}{\pi}\int _{0}^{\infty}e^{{-z\mathop{\sinh\/}\nolimits t}}dt,$ $\realpart{z}>0$ ,

Symbols:: $\mathop{\mathbf{K}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\realpart{}$ : real part and $z$ : complex variable
Referenced by:: §11.5(i)
Permalink:: http://dlmf.nist.gov/11.5.E3
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11.5.4 $\mathop{\mathbf{M}_{{\nu}}\/}\nolimits\!\left(z\right)=-\frac{2(\tfrac{1}{2}z)^{\nu}}{\sqrt{\pi}\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{1}{2}\right)}\int _{0}^{1}e^{{-zt}}(1-t^{2})^{{\nu-\frac{1}{2}}}dt,$ $\realpart{\nu}>-\tfrac{1}{2}$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathbf{M}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Struve function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : 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
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11.5.5 $\mathop{\mathbf{M}_{{0}}\/}\nolimits\!\left(z\right)=-\frac{2}{\pi}\int _{0}^{{\pi/2}}e^{{-z\mathop{\cos\/}\nolimits\theta}}d\theta,$

Symbols:: $\mathop{\mathbf{M}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Struve function, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral and $z$ : complex variable
Referenced by:: §11.13(iii), §11.5(i)
Permalink:: http://dlmf.nist.gov/11.5.E5
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11.5.6 $\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{2(\tfrac{1}{2}z)^{\nu}}{\sqrt{\pi}\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{1}{2}\right)}\int _{0}^{{\pi/2}}\mathop{\sinh\/}\nolimits\!\left(z\mathop{\cos\/}\nolimits\theta\right)(\mathop{\sin\/}\nolimits\theta)^{{2\nu}}d\theta,$ $\realpart{\nu}>-\tfrac{1}{2}$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Struve function, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : sine function, $z$ : complex variable and $\nu$ : real or complex order
Referenced by:: §11.5(i)
Permalink:: http://dlmf.nist.gov/11.5.E6
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11.5.7 $\mathop{I_{{-\nu}}\/}\nolimits\!\left(x\right)-\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(x\right)=\frac{2(\tfrac{1}{2}x)^{\nu}}{\sqrt{\pi}\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{1}{2}\right)}\int _{0}^{\infty}(1+t^{2})^{{\nu-\frac{1}{2}}}\mathop{\sin\/}\nolimits\!\left(xt\right)dt,$ $x>0$ , $\realpart{\nu}<\tfrac{1}{2}$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Struve function, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : 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
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§11.5(ii) Contour Integrals

Keywords:: Struve functions and modified Struve functions
Permalink:: http://dlmf.nist.gov/11.5.ii

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

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).
Keywords:: Struve functions and modified Struve functions

11.5.8 $(\tfrac{1}{2}x)^{{-\nu-1}}\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(x\right)=-\frac{1}{2\pi i}\int _{{-i\infty}}^{{i\infty}}\frac{\pi\mathop{\csc\/}\nolimits\!\left(\pi s\right)}{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{3}{2}+s\right)\mathop{\Gamma\/}\nolimits\!\left(\tfrac{3}{2}+\nu+s\right)}(\tfrac{1}{4}x^{2})^{s}ds,$ $x>0$ , $\realpart{\nu}>-1$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, $\mathop{\csc\/}\nolimits z$ : cosecant function, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part, $x$ : real variable and $\nu$ : real or complex order
Referenced by:: ¶ ‣ §11.5(ii), ¶ ‣ §11.5(ii)
Permalink:: http://dlmf.nist.gov/11.5.E8
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11.5.9 $(\tfrac{1}{2}z)^{{-\nu-1}}\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{1}{2\pi i}\int _{{\infty}}^{{(0+)}}\frac{\pi\mathop{\csc\/}\nolimits\!\left(\pi s\right)}{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{3}{2}+s\right)\mathop{\Gamma\/}\nolimits\!\left(\tfrac{3}{2}+\nu+s\right)}(-\tfrac{1}{4}z^{2})^{s}ds.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\csc\/}\nolimits z$ : cosecant function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Struve function, $z$ : complex variable and $\nu$ : real or complex order
Referenced by:: ¶ ‣ §11.5(ii), ¶ ‣ §11.5(ii)
Permalink:: http://dlmf.nist.gov/11.5.E9
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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

Keywords:: Struve functions and modified Struve functions
Permalink:: http://dlmf.nist.gov/11.5.iii

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).

§11.5 Integral Representations

Contents

§11.5(i) Integrals Along the Real Line

§11.5(ii) Contour Integrals

¶ Mellin–Barnes Integrals

§11.5(iii) Compendia