11 Struve and Related Functions Struve and Modified Struve Functions 11.3 Graphics 11.5 Integral Representations

§11.4 Basic Properties

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Permalink:: http://dlmf.nist.gov/11.4
See also:: Annotations for Ch.11

§11.4(i) Half-Integer Orders
§11.4(ii) Inequalities
§11.4(iii) Analytic Continuation
§11.4(iv) Expansions in Series of Bessel Functions
§11.4(v) Recurrence Relations and Derivatives
§11.4(vi) Derivatives with Respect to Order
§11.4(vii) Zeros

§11.4(i) Half-Integer Orders

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Keywords:: Struve functions and modified Struve functions, half-integer orders
Notes:: (11.4.1) and (11.4.2) both follow from Erdélyi et al. (1953b, p. 39, Eq. (64)): for (11.4.1) set $\xi=z$ and use (11.2.5); for (11.4.2) replace $Y_{n+\frac{1}{2}}\left(\xi\right)$ by $(-1)^{n+1}J_{-n-\frac{1}{2}}\left(\xi\right)$ in consequence of (10.2.3), then set $\xi=\mathrm{i}z$ and use (11.2.2), (10.27.6). For (11.4.3), (11.4.4) see Babister (1967, pp. 64 and 75). For (11.4.5)–(11.4.12) combine (11.4.3), (11.4.4) with (10.47.3), (10.47.7), §10.49(i), §10.49(ii), (11.4.23), (11.4.25).
Permalink:: http://dlmf.nist.gov/11.4.i
See also:: Annotations for §11.4 and Ch.11

For $n=0,1,2,\dots$ ,

11.4.1		$\mathbf{K}_{n+\frac{1}{2}}\left(z\right)=\left(\frac{2}{\pi z}\right)^{\frac{1% }{2}}\sum_{m=0}^{n}\frac{(2m)!\,2^{-2m}}{m!\,(n-m)!}\,(\tfrac{1}{2}z)^{n-2m},$
ⓘ Symbols: $\mathbf{K}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $z$ : complex variable and $n$ : integer order Referenced by: §11.4(i) Permalink: http://dlmf.nist.gov/11.4.E1 Encodings: TeX, pMML, png See also: Annotations for §11.4(i), §11.4 and Ch.11

11.4.2		$\mathbf{L}_{n+\frac{1}{2}}\left(z\right)=I_{-n-\frac{1}{2}}\left(z\right)-% \left(\frac{2}{\pi z}\right)^{\frac{1}{2}}\sum_{m=0}^{n}\frac{(-1)^{m}(2m)!\,2% ^{-2m}}{m!\,(n-m)!}\,(\tfrac{1}{2}z)^{n-2m},$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\mathbf{L}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function, $z$ : complex variable and $n$ : integer order Referenced by: §11.4(i) Permalink: http://dlmf.nist.gov/11.4.E2 Encodings: TeX, pMML, png See also: Annotations for §11.4(i), §11.4 and Ch.11

11.4.3	$\displaystyle\mathbf{H}_{-n-\frac{1}{2}}\left(z\right)$	$\displaystyle=(-1)^{n}J_{n+\frac{1}{2}}\left(z\right),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $z$ : complex variable and $n$ : integer order A&S Ref: 12.1.15 Referenced by: §11.4(i) Permalink: http://dlmf.nist.gov/11.4.E3 Encodings: TeX, pMML, png See also: Annotations for §11.4(i), §11.4 and Ch.11
11.4.4	$\displaystyle\mathbf{L}_{-n-\frac{1}{2}}\left(z\right)$	$\displaystyle=I_{n+\frac{1}{2}}\left(z\right).$
ⓘ Symbols: $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\mathbf{L}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function, $z$ : complex variable and $n$ : integer order A&S Ref: 12.2.10 Referenced by: §11.4(i) Permalink: http://dlmf.nist.gov/11.4.E4 Encodings: TeX, pMML, png See also: Annotations for §11.4(i), §11.4 and Ch.11

11.4.5	$\displaystyle\mathbf{H}_{\frac{1}{2}}\left(z\right)$	$\displaystyle=\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}(1-\cos z),$
ⓘ Symbols: $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function and $z$ : complex variable A&S Ref: 12.1.16 Referenced by: §11.4(i) Permalink: http://dlmf.nist.gov/11.4.E5 Encodings: TeX, pMML, png See also: Annotations for §11.4(i), §11.4 and Ch.11
11.4.6	$\displaystyle\mathbf{H}_{-\frac{1}{2}}\left(z\right)$	$\displaystyle=\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}\sin z,$
ⓘ Symbols: $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$ : sine function and $z$ : complex variable Permalink: http://dlmf.nist.gov/11.4.E6 Encodings: TeX, pMML, png See also: Annotations for §11.4(i), §11.4 and Ch.11

11.4.7	$\displaystyle\mathbf{L}_{\frac{1}{2}}\left(z\right)$	$\displaystyle=\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}(\cosh z-1),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cosh\NVar{z}$ : hyperbolic cosine function, $\mathbf{L}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function and $z$ : complex variable Permalink: http://dlmf.nist.gov/11.4.E7 Encodings: TeX, pMML, png See also: Annotations for §11.4(i), §11.4 and Ch.11
11.4.8	$\displaystyle\mathbf{L}_{-\frac{1}{2}}\left(z\right)$	$\displaystyle=\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}\sinh z,$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathbf{L}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function and $z$ : complex variable Permalink: http://dlmf.nist.gov/11.4.E8 Encodings: TeX, pMML, png See also: Annotations for §11.4(i), §11.4 and Ch.11

11.4.9		$\mathbf{H}_{\frac{3}{2}}\left(z\right)=\left(\frac{z}{2\pi}\right)^{\frac{1}{2% }}\left(1+\frac{2}{z^{2}}\right)-\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}% \left(\sin z+\frac{\cos z}{z}\right),$
ⓘ Symbols: $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function and $z$ : complex variable A&S Ref: 12.1.17 Permalink: http://dlmf.nist.gov/11.4.E9 Encodings: TeX, pMML, png See also: Annotations for §11.4(i), §11.4 and Ch.11

11.4.10		$\mathbf{H}_{-\frac{3}{2}}\left(z\right)=\left(\frac{2}{\pi z}\right)^{\frac{1}% {2}}\left(\cos z-\frac{\sin z}{z}\right),$
ⓘ Symbols: $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function and $z$ : complex variable Permalink: http://dlmf.nist.gov/11.4.E10 Encodings: TeX, pMML, png See also: Annotations for §11.4(i), §11.4 and Ch.11

11.4.11		$\mathbf{L}_{\frac{3}{2}}\left(z\right)=-\left(\frac{z}{2\pi}\right)^{\frac{1}{% 2}}\left(1-\frac{2}{z^{2}}\right)+\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}% \left(\sinh z-\frac{\cosh z}{z}\right),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathbf{L}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function and $z$ : complex variable Permalink: http://dlmf.nist.gov/11.4.E11 Encodings: TeX, pMML, png See also: Annotations for §11.4(i), §11.4 and Ch.11

11.4.12		$\mathbf{L}_{-\frac{3}{2}}\left(z\right)=\left(\frac{2}{\pi z}\right)^{\frac{1}% {2}}\left(\cosh z-\frac{\sinh z}{z}\right).$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathbf{L}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function and $z$ : complex variable Referenced by: §11.4(i) Permalink: http://dlmf.nist.gov/11.4.E12 Encodings: TeX, pMML, png See also: Annotations for §11.4(i), §11.4 and Ch.11

§11.4(ii) Inequalities

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Keywords:: Struve functions and modified Struve functions, inequalities
Notes:: See Watson (1944, §§10.4 and 10.45).
Permalink:: http://dlmf.nist.gov/11.4.ii
See also:: Annotations for §11.4 and Ch.11

11.4.13		$\mathbf{H}_{\nu}\left(x\right)\geq 0,$
$x>0$ , $\nu\geq\tfrac{1}{2}$ .
ⓘ Symbols: $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $x$ : real variable and $\nu$ : real or complex order A&S Ref: 12.1.14 Permalink: http://dlmf.nist.gov/11.4.E13 Encodings: TeX, pMML, png See also: Annotations for §11.4(ii), §11.4 and Ch.11

11.4.14		$\mathbf{H}_{\nu}\left(z\right)=\frac{2(\tfrac{1}{2}z)^{\nu+1}}{\sqrt{\pi}% \Gamma\left(\nu+\tfrac{3}{2}\right)}(1+\vartheta),$
$\nu\neq-\tfrac{3}{2},-\tfrac{5}{2},-\tfrac{7}{2},\dots$ ,
ⓘ 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, $z$ : complex variable and $\nu$ : real or complex order Permalink: http://dlmf.nist.gov/11.4.E14 Encodings: TeX, pMML, png See also: Annotations for §11.4(ii), §11.4 and Ch.11

where

11.4.15		$\|\vartheta\|<\frac{2}{3}\exp\left(\frac{\tfrac{1}{4}\|z\|^{2}}{\|\nu_{0}+\tfrac{3}% {2}\|}-1\right),$
ⓘ Symbols: $\exp\NVar{z}$ : exponential function, $z$ : complex variable and $\nu$ : real or complex order Permalink: http://dlmf.nist.gov/11.4.E15 Encodings: TeX, pMML, png See also: Annotations for §11.4(ii), §11.4 and Ch.11

and $|\nu_{0}+\tfrac{3}{2}|$ is the smallest of the numbers $|\nu+\tfrac{3}{2}|$ , $|\nu+\tfrac{5}{2}|$ , $|\nu+\tfrac{9}{2}|,\dots$ .

§11.4(iii) Analytic Continuation

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Keywords:: Struve functions and modified Struve functions, analytic continuation
Notes:: These results follow from (11.2.1) and (11.2.2).
Permalink:: http://dlmf.nist.gov/11.4.iii
See also:: Annotations for §11.4 and Ch.11

11.4.16		$\mathbf{H}_{\nu}\left(ze^{m\pi i}\right)=e^{m\pi i(\nu+1)}\mathbf{H}_{\nu}% \left(z\right),$
$m\in\mathbb{Z}$ ,
ⓘ Symbols: $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\mathbb{Z}$ : set of all integers, $z$ : complex variable and $\nu$ : real or complex order A&S Ref: 12.1.18 Permalink: http://dlmf.nist.gov/11.4.E16 Encodings: TeX, pMML, png See also: Annotations for §11.4(iii), §11.4 and Ch.11

11.4.17		$\mathbf{L}_{\nu}\left(ze^{m\pi i}\right)=e^{m\pi i(\nu+1)}\mathbf{L}_{\nu}% \left(z\right),$
$m\in\mathbb{Z}$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\mathbb{Z}$ : set of all integers, $\mathbf{L}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function, $z$ : complex variable and $\nu$ : real or complex order Permalink: http://dlmf.nist.gov/11.4.E17 Encodings: TeX, pMML, png See also: Annotations for §11.4(iii), §11.4 and Ch.11

§11.4(iv) Expansions in Series of Bessel Functions

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Keywords:: Bessel functions, Struve functions and modified Struve functions, series expansions
Referenced by:: §11.13(ii)
Permalink:: http://dlmf.nist.gov/11.4.iv
See also:: Annotations for §11.4 and Ch.11

11.4.18		$\mathbf{H}_{\nu}\left(z\right)=\frac{4}{\pi^{1/2}\Gamma\left(\nu+\tfrac{1}{2}% \right)}\*\sum_{k=0}^{\infty}\frac{(2k+\nu+1)\Gamma\left(k+\nu+1\right)}{k!(2k% +1)(2k+2\nu+1)}J_{2k+\nu+1}\left(z\right),$
$\nu\neq-1,-2,-3,\dots$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $z$ : complex variable, $\nu$ : real or complex order and $k$ : nonnegative integer Permalink: http://dlmf.nist.gov/11.4.E18 Encodings: TeX, pMML, png See also: Annotations for §11.4(iv), §11.4 and Ch.11

11.4.19		$\mathbf{H}_{\nu}\left(z\right)=\left(\frac{z}{2\pi}\right)^{1/2}\sum_{k=0}^{% \infty}\frac{(\tfrac{1}{2}z)^{k}}{k!(k+\tfrac{1}{2})}J_{k+\nu+\frac{1}{2}}% \left(z\right),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $z$ : complex variable, $\nu$ : real or complex order and $k$ : nonnegative integer Permalink: http://dlmf.nist.gov/11.4.E19 Encodings: TeX, pMML, png See also: Annotations for §11.4(iv), §11.4 and Ch.11

11.4.20		$\mathbf{H}_{\nu}\left(z\right)=\frac{(\tfrac{1}{2}z)^{\nu+\frac{1}{2}}}{\Gamma% \left(\nu+\tfrac{1}{2}\right)}\sum_{k=0}^{\infty}\frac{(\tfrac{1}{2}z)^{k}}{k!% (k+\nu+\tfrac{1}{2})}J_{k+\frac{1}{2}}\left(z\right),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $!$ : factorial (as in $n!$ ), $z$ : complex variable, $\nu$ : real or complex order and $k$ : nonnegative integer Permalink: http://dlmf.nist.gov/11.4.E20 Encodings: TeX, pMML, png See also: Annotations for §11.4(iv), §11.4 and Ch.11

11.4.21		$\mathbf{H}_{0}\left(z\right)=\frac{4}{\pi}\sum_{k=0}^{\infty}\frac{J_{2k+1}% \left(z\right)}{2k+1}=2\sum_{k=0}^{\infty}(-1)^{k}{J_{k+\frac{1}{2}}}^{2}\left% (\tfrac{1}{2}z\right),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\pi$ : the ratio of the circumference of a circle to its diameter, $z$ : complex variable and $k$ : nonnegative integer A&S Ref: 12.1.19 (First part) Permalink: http://dlmf.nist.gov/11.4.E21 Encodings: TeX, pMML, png See also: Annotations for §11.4(iv), §11.4 and Ch.11

11.4.22		$\mathbf{H}_{1}\left(z\right)=\frac{2}{\pi}(1-J_{0}\left(z\right))+\frac{4}{\pi% }\sum_{k=1}^{\infty}\frac{J_{2k}\left(z\right)}{4k^{2}-1}=4\sum_{k=0}^{\infty}% J_{2k+\frac{1}{2}}\left(\tfrac{1}{2}z\right)J_{2k+\frac{3}{2}}\left(\tfrac{1}{% 2}z\right).$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\pi$ : the ratio of the circumference of a circle to its diameter, $z$ : complex variable and $k$ : nonnegative integer A&S Ref: 12.1.20 (First part) Permalink: http://dlmf.nist.gov/11.4.E22 Encodings: TeX, pMML, png See also: Annotations for §11.4(iv), §11.4 and Ch.11

For these and further results see Luke (1969b, §9.4.5), and §10.23(iii).

§11.4(v) Recurrence Relations and Derivatives

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Keywords:: Struve functions and modified Struve functions, derivatives, recurrence relations
A&S Ref:: 12.1.9--12.1.13 12.2.4--12.2.5
Notes:: See Watson (1944, p. 329). For (11.4.31) see Babister (1967, pp. 60, 74).
Referenced by:: §11.7(i)
Permalink:: http://dlmf.nist.gov/11.4.v
See also:: Annotations for §11.4 and Ch.11

11.4.23	$\displaystyle\mathbf{H}_{\nu-1}\left(z\right)+\mathbf{H}_{\nu+1}\left(z\right)$	$\displaystyle=\frac{2\nu}{z}\mathbf{H}_{\nu}\left(z\right)+\frac{(\tfrac{1}{2}% z)^{\nu}}{\sqrt{\pi}\Gamma\left(\nu+\tfrac{3}{2}\right)},$
ⓘ 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, $z$ : complex variable and $\nu$ : real or complex order Referenced by: §11.13(v), §11.4(i) Permalink: http://dlmf.nist.gov/11.4.E23 Encodings: TeX, pMML, png See also: Annotations for §11.4(v), §11.4 and Ch.11
11.4.24	$\displaystyle\mathbf{H}_{\nu-1}\left(z\right)-\mathbf{H}_{\nu+1}\left(z\right)$	$\displaystyle=2\mathbf{H}_{\nu}'\left(z\right)-\frac{(\tfrac{1}{2}z)^{\nu}}{% \sqrt{\pi}\Gamma\left(\nu+\tfrac{3}{2}\right)},$
ⓘ 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, $z$ : complex variable and $\nu$ : real or complex order Permalink: http://dlmf.nist.gov/11.4.E24 Encodings: TeX, pMML, png See also: Annotations for §11.4(v), §11.4 and Ch.11
11.4.25	$\displaystyle\mathbf{L}_{\nu-1}\left(z\right)-\mathbf{L}_{\nu+1}\left(z\right)$	$\displaystyle=\frac{2\nu}{z}\mathbf{L}_{\nu}\left(z\right)+\frac{(\tfrac{1}{2}% z)^{\nu}}{\sqrt{\pi}\Gamma\left(\nu+\tfrac{3}{2}\right)},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathbf{L}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function, $z$ : complex variable and $\nu$ : real or complex order Referenced by: §11.13(v), §11.4(i) Permalink: http://dlmf.nist.gov/11.4.E25 Encodings: TeX, pMML, png See also: Annotations for §11.4(v), §11.4 and Ch.11
11.4.26	$\displaystyle\mathbf{L}_{\nu-1}\left(z\right)+\mathbf{L}_{\nu+1}\left(z\right)$	$\displaystyle=2\mathbf{L}_{\nu}'\left(z\right)-\frac{(\tfrac{1}{2}z)^{\nu}}{% \sqrt{\pi}\Gamma\left(\nu+\tfrac{3}{2}\right)}.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathbf{L}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function, $z$ : complex variable and $\nu$ : real or complex order Permalink: http://dlmf.nist.gov/11.4.E26 Encodings: TeX, pMML, png See also: Annotations for §11.4(v), §11.4 and Ch.11

11.4.27		$\frac{\mathrm{d}}{\mathrm{d}z}\left(z^{\nu}\mathbf{H}_{\nu}\left(z\right)% \right)=z^{\nu}\mathbf{H}_{\nu-1}\left(z\right),$
ⓘ Symbols: $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable and $\nu$ : real or complex order Referenced by: §10.22(i) Permalink: http://dlmf.nist.gov/11.4.E27 Encodings: TeX, pMML, png See also: Annotations for §11.4(v), §11.4 and Ch.11

11.4.28		$\frac{\mathrm{d}}{\mathrm{d}z}\left(z^{-\nu}\mathbf{H}_{\nu}\left(z\right)% \right)=\frac{2^{-\nu}}{\sqrt{\pi}\Gamma\left(\nu+\tfrac{3}{2}\right)}-z^{-\nu% }\mathbf{H}_{\nu+1}\left(z\right),$
ⓘ 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, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable and $\nu$ : real or complex order Referenced by: §10.22(i) Permalink: http://dlmf.nist.gov/11.4.E28 Encodings: TeX, pMML, png See also: Annotations for §11.4(v), §11.4 and Ch.11

11.4.29		$\frac{\mathrm{d}}{\mathrm{d}z}\left(z^{\nu}\mathbf{L}_{\nu}\left(z\right)% \right)=z^{\nu}\mathbf{L}_{\nu-1}\left(z\right),$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathbf{L}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function, $z$ : complex variable and $\nu$ : real or complex order Referenced by: §10.43(i) Permalink: http://dlmf.nist.gov/11.4.E29 Encodings: TeX, pMML, png See also: Annotations for §11.4(v), §11.4 and Ch.11

11.4.30		$\frac{\mathrm{d}}{\mathrm{d}z}\left(z^{-\nu}\mathbf{L}_{\nu}\left(z\right)% \right)=\frac{2^{-\nu}}{\sqrt{\pi}\Gamma\left(\nu+\tfrac{3}{2}\right)}+z^{-\nu% }\mathbf{L}_{\nu+1}\left(z\right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathbf{L}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function, $z$ : complex variable and $\nu$ : real or complex order Referenced by: §10.43(i) Permalink: http://dlmf.nist.gov/11.4.E30 Encodings: TeX, pMML, png See also: Annotations for §11.4(v), §11.4 and Ch.11

11.4.31		${\cal H}_{\nu-m}(z)=z^{m-\nu}\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}% \right)^{m}(z^{\nu}{\cal H}_{\nu}(z)),$
$m=1,2,3,\dots$ ,
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable and $\nu$ : real or complex order Referenced by: §11.4(v) Permalink: http://dlmf.nist.gov/11.4.E31 Encodings: TeX, pMML, png See also: Annotations for §11.4(v), §11.4 and Ch.11

where ${\cal H}_{\nu}(z)$ denotes either $\mathbf{H}_{\nu}\left(z\right)$ or $\mathbf{L}_{\nu}\left(z\right)$ .

11.4.32	$\displaystyle\mathbf{H}_{0}'\left(z\right)$	$\displaystyle=\frac{2}{\pi}-\mathbf{H}_{1}\left(z\right),$
11.4.32	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}(z\mathbf{H}_{1}\left(z\right))$	$\displaystyle=z\mathbf{H}_{0}\left(z\right),$
ⓘ Symbols: $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ and $z$ : complex variable Permalink: http://dlmf.nist.gov/11.4.E32 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §11.4(v), §11.4 and Ch.11

11.4.33	$\displaystyle\mathbf{L}_{0}'\left(z\right)$	$\displaystyle=\frac{2}{\pi}+\mathbf{L}_{1}\left(z\right),$
11.4.33	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}(z\mathbf{L}_{1}\left(z\right))$	$\displaystyle=z\mathbf{L}_{0}\left(z\right).$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathbf{L}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function and $z$ : complex variable Permalink: http://dlmf.nist.gov/11.4.E33 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §11.4(v), §11.4 and Ch.11