§11.4 Basic Properties

For $n=0,1,2,\mathrm{\dots }$,

11.4.1		$${\mathbf{K}}_{n+\frac{1}{2}}\left(z\right)={\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}\sum _{m=0}^n\frac{(2m)!{\mathrm{\hspace{0.17em}2}}^{-2m}}{m!(n-m)!}{(\frac{1}{2}z)}^{n-2m},$$
ⓘ Symbols: ${\mathbf{K}}_{\nu }\left(z\right)$: 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		$${\mathbf{L}}_{n+\frac{1}{2}}\left(z\right)=I_{-n-\frac{1}{2}}\left(z\right)-{\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}\sum _{m=0}^n\frac{{(-1)}^m(2m)!{\mathrm{\hspace{0.17em}2}}^{-2m}}{m!(n-m)!}{(\frac{1}{2}z)}^{n-2m},$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, ${\mathbf{L}}_{\nu }\left(z\right)$: 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 See also: Annotations for §11.4(i), §11.4 and Ch.11

11.4.3		${\mathbf{H}}_{-n-\frac{1}{2}}\left(z\right)$	$={(-1)}^n{J}_{n+\frac{1}{2}}\left(z\right),$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, ${\mathbf{H}}_{\nu }\left(z\right)$: 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 See also: Annotations for §11.4(i), §11.4 and Ch.11
11.4.4		${\mathbf{L}}_{-n-\frac{1}{2}}\left(z\right)$	$=I_{n+\frac{1}{2}}\left(z\right).$
ⓘ Symbols: $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, ${\mathbf{L}}_{\nu }\left(z\right)$: 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 See also: Annotations for §11.4(i), §11.4 and Ch.11

11.4.5		${\mathbf{H}}_{\frac{1}{2}}\left(z\right)$	$={\left({\displaystyle \frac{2}{\pi z}}\right)}^{\frac{1}{2}}(1-\cos z),$
ⓘ Symbols: ${\mathbf{H}}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: 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		${\mathbf{H}}_{-\frac{1}{2}}\left(z\right)$	$={\left({\displaystyle \frac{2}{\pi z}}\right)}^{\frac{1}{2}}\sin z,$
ⓘ Symbols: ${\mathbf{H}}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: 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		${\mathbf{L}}_{\frac{1}{2}}\left(z\right)$	$={\left({\displaystyle \frac{2}{\pi z}}\right)}^{\frac{1}{2}}(\cosh z-1),$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cosh z$: hyperbolic cosine function, ${\mathbf{L}}_{\nu }\left(z\right)$: 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		${\mathbf{L}}_{-\frac{1}{2}}\left(z\right)$	$={\left({\displaystyle \frac{2}{\pi z}}\right)}^{\frac{1}{2}}\sinh z,$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\sinh z$: hyperbolic sine function, ${\mathbf{L}}_{\nu }\left(z\right)$: 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: ${\mathbf{H}}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: 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: ${\mathbf{H}}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: 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, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, ${\mathbf{L}}_{\nu }\left(z\right)$: 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, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, ${\mathbf{L}}_{\nu }\left(z\right)$: 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

11.4.13		$${\mathbf{H}}_{\nu }\left(x\right)\ge 0,$$
$x>0$, $\nu \ge \frac{1}{2}$.
ⓘ Symbols: ${\mathbf{H}}_{\nu }\left(z\right)$: 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 Encodings: TeX, pMML, png See also: Annotations for §11.4(ii), §11.4 and Ch.11

11.4.14		$${\mathbf{H}}_{\nu }\left(z\right)=\frac{2{(\frac{1}{2}z)}^{\nu +1}}{\sqrt{\pi }\mathrm{\Gamma }\left(\nu +\frac{3}{2}\right)}(1+\vartheta ),$$
$\nu \ne -\frac{3}{2},-\frac{5}{2},-\frac{7}{2},\mathrm{\dots }$,
ⓘ 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, $z$: 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
ⓘ Symbols: $\exp z$: 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 $|{\nu }_0+\frac{3}{2}|$ is the smallest of the numbers $|\nu +\frac{3}{2}|$, $|\nu +\frac{5}{2}|$, $|\nu +\frac{9}{2}|,\mathrm{\dots }$.

11.4.16		$${\mathbf{H}}_{\nu }\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={\mathrm{e}}^{m\pi \mathrm{i}(\nu +1)}{\mathbf{H}}_{\nu }\left(z\right),$$
$m\in \mathbb{Z}$,
ⓘ Symbols: ${\mathbf{H}}_{\nu }\left(z\right)$: 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		$${\mathbf{L}}_{\nu }\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={\mathrm{e}}^{m\pi \mathrm{i}(\nu +1)}{\mathbf{L}}_{\nu }\left(z\right),$$
$m\in \mathbb{Z}$.
ⓘ 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, ${\mathbf{L}}_{\nu }\left(z\right)$: modified Struve function, $z$: complex variable and $\nu$: real or complex order Permalink: http://dlmf.nist.gov/11.4.E17 Encodings: TeX, pMML, png See also: Annotations for §11.4(iii), §11.4 and Ch.11

11.4.18		$${\mathbf{H}}_{\nu }\left(z\right)=\frac{4}{{\pi }^{1/2}\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{(2k+\nu +1)\mathrm{\Gamma }\left(k+\nu +1\right)}{k!(2k+1)(2k+2\nu +1)}{J}_{2k+\nu +1}\left(z\right),$$
$\nu \ne -1,-2,-3,\mathrm{\dots }$,
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathbf{H}}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: 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		$${\mathbf{H}}_{\nu }\left(z\right)={\left(\frac{z}{2\pi }\right)}^{1/2}\sum _{k=0}^{\mathrm{\infty }}\frac{{(\frac{1}{2}z)}^k}{k!(k+\frac{1}{2})}{J}_{k+\nu +\frac{1}{2}}\left(z\right),$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, ${\mathbf{H}}_{\nu }\left(z\right)$: 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		$${\mathbf{H}}_{\nu }\left(z\right)=\frac{{(\frac{1}{2}z)}^{\nu +\frac{1}{2}}}{\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{{(\frac{1}{2}z)}^k}{k!(k+\nu +\frac{1}{2})}{J}_{k+\frac{1}{2}}\left(z\right),$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathbf{H}}_{\nu }\left(z\right)$: 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		$${\mathbf{H}}_0\left(z\right)=\frac{4}{\pi }\sum _{k=0}^{\mathrm{\infty }}\frac{J_{2k+1}\left(z\right)}{2k+1}=2\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k{J}_{k+\frac{1}{2}}^2\left(\frac{1}{2}z\right),$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, ${\mathbf{H}}_{\nu }\left(z\right)$: 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		$${\mathbf{H}}_1\left(z\right)=\frac{2}{\pi }(1-J_0\left(z\right))+\frac{4}{\pi }\sum _{k=1}^{\mathrm{\infty }}\frac{J_{2k}\left(z\right)}{4k^2-1}=4\sum _{k=0}^{\mathrm{\infty }}{J}_{2k+\frac{1}{2}}\left(\frac{1}{2}z\right)J_{2k+\frac{3}{2}}\left(\frac{1}{2}z\right).$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, ${\mathbf{H}}_{\nu }\left(z\right)$: 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).

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

11.4.27		$$\frac{d}{dz}\left(z^{\nu }{\mathbf{H}}_{\nu }\left(z\right)\right)=z^{\nu }{\mathbf{H}}_{\nu -1}\left(z\right),$$
ⓘ Symbols: ${\mathbf{H}}_{\nu }\left(z\right)$: 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 See also: Annotations for §11.4(v), §11.4 and Ch.11

11.4.28		$$\frac{d}{dz}\left(z^{-\nu }{\mathbf{H}}_{\nu }\left(z\right)\right)=\frac{2^{-\nu }}{\sqrt{\pi }\mathrm{\Gamma }\left(\nu +\frac{3}{2}\right)}-z^{-\nu }{\mathbf{H}}_{\nu +1}\left(z\right),$$
ⓘ 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, $\frac{df}{dx}$: 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 See also: Annotations for §11.4(v), §11.4 and Ch.11

11.4.29		$$\frac{d}{dz}\left(z^{\nu }{\mathbf{L}}_{\nu }\left(z\right)\right)=z^{\nu }{\mathbf{L}}_{\nu -1}\left(z\right),$$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, ${\mathbf{L}}_{\nu }\left(z\right)$: 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 See also: Annotations for §11.4(v), §11.4 and Ch.11

11.4.30		$$\frac{d}{dz}\left(z^{-\nu }{\mathbf{L}}_{\nu }\left(z\right)\right)=\frac{2^{-\nu }}{\sqrt{\pi }\mathrm{\Gamma }\left(\nu +\frac{3}{2}\right)}+z^{-\nu }{\mathbf{L}}_{\nu +1}\left(z\right).$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{df}{dx}$: derivative of $f$ with respect to $x$, ${\mathbf{L}}_{\nu }\left(z\right)$: 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 See also: Annotations for §11.4(v), §11.4 and Ch.11

11.4.31		$${\mathscr{H}}_{\nu -m}(z)=z^{m-\nu }{\left(\frac{1}{z}\frac{d}{dz}\right)}^m(z^{\nu }{\mathscr{H}}_{\nu }(z)),$$
$m=1,2,3,\mathrm{\dots }$,
ⓘ 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 See also: Annotations for §11.4(v), §11.4 and Ch.11

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

11.4.32		${\mathbf{H}}_0^{\prime }\left(z\right)$	$={\displaystyle \frac{2}{\pi }}-{\mathbf{H}}_1\left(z\right),$
		${\displaystyle \frac{d}{dz}}(z{\mathbf{H}}_1\left(z\right))$	$=z{\mathbf{H}}_0\left(z\right),$
ⓘ Symbols: ${\mathbf{H}}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{df}{dx}$: 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 See also: Annotations for §11.4(v), §11.4 and Ch.11

11.4.33		${\mathbf{L}}_0^{\prime }\left(z\right)$	$={\displaystyle \frac{2}{\pi }}+{\mathbf{L}}_1\left(z\right),$
		${\displaystyle \frac{d}{dz}}(z{\mathbf{L}}_1\left(z\right))$	$=z{\mathbf{L}}_0\left(z\right).$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{df}{dx}$: derivative of $f$ with respect to $x$, ${\mathbf{L}}_{\nu }\left(z\right)$: modified Struve function and $z$: complex variable Permalink: http://dlmf.nist.gov/11.4.E33 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §11.4(v), §11.4 and Ch.11

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

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

For asymptotic expansions of zeros of ${\mathbf{H}}_0\left(x\right)$ see MacLeod (2002a).