§11.4 Basic Properties

Permalink:: http://dlmf.nist.gov/11.4

§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

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 $\mathop{Y_{{n+\frac{1}{2}}}\/}\nolimits\!\left(\xi\right)$ by $(-1)^{{n+1}}\mathop{J_{{-n-\frac{1}{2}}}\/}\nolimits\!\left(\xi\right)$ in consequence of (10.2.3), then set $\xi=iz$ 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).
Keywords:: Struve functions and modified Struve functions
Permalink:: http://dlmf.nist.gov/11.4.i

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

11.4.1 $\mathop{\mathbf{K}_{{n+\frac{1}{2}}}\/}\nolimits\!\left(z\right)=\left(\frac{2}{\pi z}\right)^{{\frac{1}{2}}}\sum _{{m=0}}^{n}\frac{(2m)!\, 2^{{-2m}}}{m!\,(n-m)!}\,(\tfrac{1}{2}z)^{{n-2m}},$

Symbols:: $\mathop{\mathbf{K}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, $!$ : $n!$ : factorial, $z$ : complex variable and $n$ : integer order
Referenced by:: §11.4(i)
Permalink:: http://dlmf.nist.gov/11.4.E1
Encodings:: TeX, pMML, png

11.4.2 $\mathop{\mathbf{L}_{{n+\frac{1}{2}}}\/}\nolimits\!\left(z\right)=\mathop{I_{{-n-\frac{1}{2}}}\/}\nolimits\!\left(z\right)-\left(\frac{2}{\pi z}\right)^{{\frac{1}{2}}}\sum _{{m=0}}^{n}\frac{(-1)^{m}(2m)!\, 2^{{-2m}}}{m!\,(n-m)!}\,(\tfrac{1}{2}z)^{{n-2m}},$

Symbols:: $!$ : $n!$ : factorial, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)$ : 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

11.4.3 $\mathop{\mathbf{H}_{{-n-\frac{1}{2}}}\/}\nolimits\!\left(z\right)=(-1)^{n}\mathop{J_{{n+\frac{1}{2}}}\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : 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

11.4.4 $\mathop{\mathbf{L}_{{-n-\frac{1}{2}}}\/}\nolimits\!\left(z\right)=\mathop{I_{{n+\frac{1}{2}}}\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)$ : 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

11.4.5 $\mathop{\mathbf{H}_{{\frac{1}{2}}}\/}\nolimits\!\left(z\right)=\left(\frac{2}{\pi z}\right)^{{\frac{1}{2}}}(1-\mathop{\cos\/}\nolimits z),$

Symbols:: $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, $\mathop{\cos\/}\nolimits z$ : 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

11.4.6 $\mathop{\mathbf{H}_{{-\frac{1}{2}}}\/}\nolimits\!\left(z\right)=\left(\frac{2}{\pi z}\right)^{{\frac{1}{2}}}\mathop{\sin\/}\nolimits z,$

Symbols:: $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, $\mathop{\sin\/}\nolimits z$ : sine function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/11.4.E6
Encodings:: TeX, pMML, png

11.4.7 $\mathop{\mathbf{L}_{{\frac{1}{2}}}\/}\nolimits\!\left(z\right)=\left(\frac{2}{\pi z}\right)^{{\frac{1}{2}}}(\mathop{\cosh\/}\nolimits z-1),$

Symbols:: $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Struve function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/11.4.E7
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11.4.8 $\mathop{\mathbf{L}_{{-\frac{1}{2}}}\/}\nolimits\!\left(z\right)=\left(\frac{2}{\pi z}\right)^{{\frac{1}{2}}}\mathop{\sinh\/}\nolimits z,$

Symbols:: $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Struve function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/11.4.E8
Encodings:: TeX, pMML, png

11.4.9 $\mathop{\mathbf{H}_{{\frac{3}{2}}}\/}\nolimits\!\left(z\right)=\left(\frac{z}{2\pi}\right)^{{\frac{1}{2}}}\left(1+\frac{2}{z^{2}}\right)-\left(\frac{2}{\pi z}\right)^{{\frac{1}{2}}}\left(\mathop{\sin\/}\nolimits z+\frac{\mathop{\cos\/}\nolimits z}{z}\right),$

Symbols:: $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function and $z$ : complex variable
A&S Ref:: 12.1.17
Permalink:: http://dlmf.nist.gov/11.4.E9
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11.4.10 $\mathop{\mathbf{H}_{{-\frac{3}{2}}}\/}\nolimits\!\left(z\right)=\left(\frac{2}{\pi z}\right)^{{\frac{1}{2}}}\left(\mathop{\cos\/}\nolimits z-\frac{\mathop{\sin\/}\nolimits z}{z}\right),$

Symbols:: $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/11.4.E10
Encodings:: TeX, pMML, png

11.4.11 $\mathop{\mathbf{L}_{{\frac{3}{2}}}\/}\nolimits\!\left(z\right)=-\left(\frac{z}{2\pi}\right)^{{\frac{1}{2}}}\left(1-\frac{2}{z^{2}}\right)+\left(\frac{2}{\pi z}\right)^{{\frac{1}{2}}}\left(\mathop{\sinh\/}\nolimits z-\frac{\mathop{\cosh\/}\nolimits z}{z}\right),$

Symbols:: $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Struve function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/11.4.E11
Encodings:: TeX, pMML, png

11.4.12 $\mathop{\mathbf{L}_{{-\frac{3}{2}}}\/}\nolimits\!\left(z\right)=\left(\frac{2}{\pi z}\right)^{{\frac{1}{2}}}\left(\mathop{\cosh\/}\nolimits z-\frac{\mathop{\sinh\/}\nolimits z}{z}\right).$

Symbols:: $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Struve function and $z$ : complex variable
Referenced by:: §11.4(i)
Permalink:: http://dlmf.nist.gov/11.4.E12
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§11.4(ii) Inequalities

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

11.4.13 $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(x\right)\geq 0,$ $x>0$ , $\nu\geq\tfrac{1}{2}$ .

Symbols:: $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : 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
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11.4.14 $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{2(\tfrac{1}{2}z)^{{\nu+1}}}{\sqrt{\pi}\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{3}{2}\right)}(1+\vartheta),$ $\nu\neq-\tfrac{3}{2},-\tfrac{5}{2},-\tfrac{7}{2},\dots$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, $z$ : complex variable and $\nu$ : real or complex order
Permalink:: http://dlmf.nist.gov/11.4.E14
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where

11.4.15 $|\vartheta|<\frac{2}{3}\mathop{\exp\/}\nolimits\!\left(\frac{\tfrac{1}{4}|z|^{2}}{|\nu _{0}+\tfrac{3}{2}|}-1\right),$

Symbols:: $\mathop{\exp\/}\nolimits z$ : exponential function, $z$ : complex variable and $\nu$ : real or complex order
Permalink:: http://dlmf.nist.gov/11.4.E15
Encodings:: TeX, pMML, png

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

Notes:: These results follow from (11.2.1) and (11.2.2).
Keywords:: Struve functions and modified Struve functions, analytic continuation
Permalink:: http://dlmf.nist.gov/11.4.iii

11.4.16 $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(ze^{{m\pi i}}\right)=e^{{m\pi i(\nu+1)}}\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right),$ $m\in\Integer$ ,

Symbols:: $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, $\in$ : element of, $e$ : base of exponential function, $\Integer$ : 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

11.4.17 $\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(ze^{{m\pi i}}\right)=e^{{m\pi i(\nu+1)}}\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right),$ $m\in\Integer$ .

Symbols:: $\in$ : element of, $e$ : base of exponential function, $\Integer$ : set of all integers, $\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Struve function, $z$ : complex variable and $\nu$ : real or complex order
Permalink:: http://dlmf.nist.gov/11.4.E17
Encodings:: TeX, pMML, png

§11.4(iv) Expansions in Series of Bessel Functions

Keywords:: Bessel functions, Struve functions and modified Struve functions
Referenced by:: §11.13(ii)
Permalink:: http://dlmf.nist.gov/11.4.iv

11.4.18 $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{4}{\pi^{{1/2}}\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{1}{2}\right)}\*\sum _{{k=0}}^{\infty}\frac{(2k+\nu+1)\mathop{\Gamma\/}\nolimits\!\left(k+\nu+1\right)}{k!(2k+1)(2k+2\nu+1)}\mathop{J_{{2k+\nu+1}}\/}\nolimits\!\left(z\right),$ $\nu\neq-1,-2,-3,\dots$ ,

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, $!$ : $n!$ : factorial, $z$ : complex variable, $\nu$ : real or complex order and $k$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/11.4.E18
Encodings:: TeX, pMML, png

11.4.19 $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)=\left(\frac{z}{2\pi}\right)^{{1/2}}\sum _{{k=0}}^{\infty}\frac{(\tfrac{1}{2}z)^{k}}{k!(k+\tfrac{1}{2})}\mathop{J_{{k+\nu+\frac{1}{2}}}\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, $!$ : $n!$ : factorial, $z$ : complex variable, $\nu$ : real or complex order and $k$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/11.4.E19
Encodings:: TeX, pMML, png

11.4.20 $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{(\tfrac{1}{2}z)^{{\nu+\frac{1}{2}}}}{\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{1}{2}\right)}\sum _{{k=0}}^{\infty}\frac{(\tfrac{1}{2}z)^{k}}{k!(k+\nu+\tfrac{1}{2})}\mathop{J_{{k+\frac{1}{2}}}\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, $!$ : $n!$ : factorial, $z$ : complex variable, $\nu$ : real or complex order and $k$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/11.4.E20
Encodings:: TeX, pMML, png

11.4.21 $\mathop{\mathbf{H}_{{0}}\/}\nolimits\!\left(z\right)=\frac{4}{\pi}\sum _{{k=0}}^{\infty}\frac{\mathop{J_{{2k+1}}\/}\nolimits\!\left(z\right)}{2k+1}=2\sum _{{k=0}}^{\infty}(-1)^{k}{\mathop{J_{{k+\frac{1}{2}}}\/}\nolimits^{{2}}}\!\left(\tfrac{1}{2}z\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, $z$ : 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

11.4.22 $\mathop{\mathbf{H}_{{1}}\/}\nolimits\!\left(z\right)=\frac{2}{\pi}(1-\mathop{J_{{0}}\/}\nolimits\!\left(z\right))+\frac{4}{\pi}\sum _{{k=1}}^{\infty}\frac{\mathop{J_{{2k}}\/}\nolimits\!\left(z\right)}{4k^{2}-1}=4\sum _{{k=0}}^{\infty}\mathop{J_{{2k+\frac{1}{2}}}\/}\nolimits\!\left(\tfrac{1}{2}z\right)\mathop{J_{{2k+\frac{3}{2}}}\/}\nolimits\!\left(\tfrac{1}{2}z\right).$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, $z$ : 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

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

§11.4(v) Recurrence Relations and Derivatives

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

11.4.23 $\mathop{\mathbf{H}_{{\nu-1}}\/}\nolimits\!\left(z\right)+\mathop{\mathbf{H}_{{\nu+1}}\/}\nolimits\!\left(z\right)=\frac{2\nu}{z}\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)+\frac{(\tfrac{1}{2}z)^{\nu}}{\sqrt{\pi}\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{3}{2}\right)},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, $z$ : 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

11.4.24 $\mathop{\mathbf{H}_{{\nu-1}}\/}\nolimits\!\left(z\right)-\mathop{\mathbf{H}_{{\nu+1}}\/}\nolimits\!\left(z\right)=2\!{\mathop{\mathbf{H}_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(z\right)-\frac{(\tfrac{1}{2}z)^{\nu}}{\sqrt{\pi}\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{3}{2}\right)},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, $z$ : complex variable and $\nu$ : real or complex order
Permalink:: http://dlmf.nist.gov/11.4.E24
Encodings:: TeX, pMML, png

11.4.25 $\mathop{\mathbf{L}_{{\nu-1}}\/}\nolimits\!\left(z\right)-\mathop{\mathbf{L}_{{\nu+1}}\/}\nolimits\!\left(z\right)=\frac{2\nu}{z}\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)+\frac{(\tfrac{1}{2}z)^{\nu}}{\sqrt{\pi}\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{3}{2}\right)},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)$ : 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

11.4.26 $\mathop{\mathbf{L}_{{\nu-1}}\/}\nolimits\!\left(z\right)+\mathop{\mathbf{L}_{{\nu+1}}\/}\nolimits\!\left(z\right)=2\!{\mathop{\mathbf{L}_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(z\right)-\frac{(\tfrac{1}{2}z)^{\nu}}{\sqrt{\pi}\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{3}{2}\right)}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Struve function, $z$ : complex variable and $\nu$ : real or complex order
Permalink:: http://dlmf.nist.gov/11.4.E26
Encodings:: TeX, pMML, png

11.4.27 $\frac{d}{dz}\left(z^{\nu}\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)\right)=z^{\nu}\mathop{\mathbf{H}_{{\nu-1}}\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, $\frac{df}{dx}$ : 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

11.4.28 $\frac{d}{dz}\left(z^{{-\nu}}\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)\right)=\frac{2^{{-\nu}}}{\sqrt{\pi}\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{3}{2}\right)}-z^{{-\nu}}\mathop{\mathbf{H}_{{\nu+1}}\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, $\frac{df}{dx}$ : 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

11.4.29 $\frac{d}{dz}\left(z^{\nu}\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)\right)=z^{\nu}\mathop{\mathbf{L}_{{\nu-1}}\/}\nolimits\!\left(z\right),$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)$ : 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

11.4.30 $\frac{d}{dz}\left(z^{{-\nu}}\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)\right)=\frac{2^{{-\nu}}}{\sqrt{\pi}\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{3}{2}\right)}+z^{{-\nu}}\mathop{\mathbf{L}_{{\nu+1}}\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)$ : 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

11.4.31 ${\cal H}_{{\nu-m}}(z)=z^{{m-\nu}}\left(\frac{1}{z}\frac{d}{dz}\right)^{m}(z^{\nu}{\cal H}_{\nu}(z)),$ $m=1,2,3,\dots$ ,

Symbols:: $\frac{df}{dx}$ : 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

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

11.4.32

${\mathop{\mathbf{H}_{{0}}\/}\nolimits^{{\prime}}}\!\left(z\right)=\frac{2}{\pi}-\mathop{\mathbf{H}_{{1}}\/}\nolimits\!\left(z\right),$

$\frac{d}{dz}(z\mathop{\mathbf{H}_{{1}}\/}\nolimits\!\left(z\right))=z\mathop{\mathbf{H}_{{0}}\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Struve function, $\frac{df}{dx}$ : 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

11.4.33

${\mathop{\mathbf{L}_{{0}}\/}\nolimits^{{\prime}}}\!\left(z\right)=\frac{2}{\pi}+\mathop{\mathbf{L}_{{1}}\/}\nolimits\!\left(z\right),$

$\frac{d}{dz}(z\mathop{\mathbf{L}_{{1}}\/}\nolimits\!\left(z\right))=z\mathop{\mathbf{L}_{{0}}\/}\nolimits\!\left(z\right).$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\mathbf{L}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Struve function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/11.4.E33
Encodings:: TeX, TeX, pMML, pMML, png, png

§11.4(vi) Derivatives with Respect to Order

Keywords:: Struve functions and modified Struve functions, derivatives
Permalink:: http://dlmf.nist.gov/11.4.vi

For derivatives with respect to the order $\nu$ , see Apelblat (1989) and Brychkov and Geddes (2005).

§11.4(vii) Zeros

Keywords:: Struve functions and modified Struve functions
Permalink:: http://dlmf.nist.gov/11.4.vii

For properties of zeros of $\mathop{\mathbf{H}_{{\nu}}\/}\nolimits\!\left(x\right)$ see Steinig (1970).

For asymptotic expansions of zeros of $\mathop{\mathbf{H}_{{0}}\/}\nolimits\!\left(x\right)$ see MacLeod (2002a).