§11.2 Definitions

11.2.1		${\mathbf{H}}_{\nu }\left(z\right)$	$={(\frac{1}{2}z)}^{\nu +1}{\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\displaystyle \frac{{(-1)}^n{(\frac{1}{2}z)}^{2n}}{\mathrm{\Gamma }\left(n+\frac{3}{2}\right)\mathrm{\Gamma }\left(n+\nu +\frac{3}{2}\right)}},$
ⓘ Defines: ${\mathbf{H}}_{\nu }\left(z\right)$: Struve function Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $z$: complex variable, $\nu$: real or complex order and $n$: integer order A&S Ref: 12.1.3 Referenced by: §11.10(vi), §11.13(ii), §11.2(i), §11.4(iii), §11.6(ii), §11.6(ii) Permalink: http://dlmf.nist.gov/11.2.E1 Encodings: TeX, pMML, png See also: Annotations for §11.2(i), §11.2 and Ch.11
11.2.2		${\mathbf{L}}_{\nu }\left(z\right)$	$=-\mathrm{i}{\mathrm{e}}^{-\frac{1}{2}\pi \mathrm{i}\nu }{\mathbf{H}}_{\nu }\left(\mathrm{i}z\right)={(\frac{1}{2}z)}^{\nu +1}{\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\displaystyle \frac{{(\frac{1}{2}z)}^{2n}}{\mathrm{\Gamma }\left(n+\frac{3}{2}\right)\mathrm{\Gamma }\left(n+\nu +\frac{3}{2}\right)}}.$
ⓘ Defines: ${\mathbf{L}}_{\nu }\left(z\right)$: modified Struve function 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, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $\nu$: real or complex order and $n$: integer order A&S Ref: 12.2.1 Referenced by: §11.13(ii), §11.2(i), §11.4(i), §11.4(iii), §11.5(ii), §11.5(i), §11.6(ii), §11.6(ii) Permalink: http://dlmf.nist.gov/11.2.E2 Encodings: TeX, pMML, png See also: Annotations for §11.2(i), §11.2 and Ch.11

Principal values correspond to principal values of ${(\frac{1}{2}z)}^{\nu +1}$; compare §4.2(i).

The expansions (11.2.1) and (11.2.2) are absolutely convergent for all finite values of $z$. The functions $z^{-\nu -1}{\mathbf{H}}_{\nu }\left(z\right)$ and $z^{-\nu -1}{\mathbf{L}}_{\nu }\left(z\right)$ are entire functions of $z$ and $\nu$.

11.2.3		${\mathbf{H}}_0\left(z\right)$	$={\displaystyle \frac{2}{\pi }}\left(z-{\displaystyle \frac{z^3}{1^2\cdot 3^2}}+{\displaystyle \frac{z^5}{1^2\cdot 3^2\cdot 5^2}}-\mathrm{\cdots }\right),$
ⓘ Symbols: ${\mathbf{H}}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter and $z$: complex variable A&S Ref: 12.1.4 Permalink: http://dlmf.nist.gov/11.2.E3 Encodings: TeX, pMML, png See also: Annotations for §11.2(i), §11.2 and Ch.11
11.2.4		${\mathbf{L}}_0\left(z\right)$	$={\displaystyle \frac{2}{\pi }}\left(z+{\displaystyle \frac{z^3}{1^2\cdot 3^2}}+{\displaystyle \frac{z^5}{1^2\cdot 3^2\cdot 5^2}}+\mathrm{\cdots }\right).$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, ${\mathbf{L}}_{\nu }\left(z\right)$: modified Struve function and $z$: complex variable Permalink: http://dlmf.nist.gov/11.2.E4 Encodings: TeX, pMML, png See also: Annotations for §11.2(i), §11.2 and Ch.11

11.2.5		${\mathbf{K}}_{\nu }\left(z\right)$	$={\mathbf{H}}_{\nu }\left(z\right)-Y_{\nu }\left(z\right),$
ⓘ Defines: ${\mathbf{K}}_{\nu }\left(z\right)$: Struve function Symbols: $Y_{\nu }\left(z\right)$: Bessel function of the second kind, ${\mathbf{H}}_{\nu }\left(z\right)$: Struve function, $z$: complex variable and $\nu$: real or complex order Referenced by: §11.13(iii), §11.13(iv), §11.2(i), §11.4(i), §11.5(i), §11.6(i), §11.6(i), §11.6(iii) Permalink: http://dlmf.nist.gov/11.2.E5 Encodings: TeX, pMML, png See also: Annotations for §11.2(i), §11.2 and Ch.11
11.2.6		${\mathbf{M}}_{\nu }\left(z\right)$	$={\mathbf{L}}_{\nu }\left(z\right)-I_{\nu }\left(z\right).$
ⓘ Defines: ${\mathbf{M}}_{\nu }\left(z\right)$: modified Struve function 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 $\nu$: real or complex order Referenced by: §11.13(iii), §11.13(iv), §11.2(i), §11.6(i), §11.6(i) Permalink: http://dlmf.nist.gov/11.2.E6 Encodings: TeX, pMML, png See also: Annotations for §11.2(i), §11.2 and Ch.11

Principal values of ${\mathbf{K}}_{\nu }\left(z\right)$ and ${\mathbf{M}}_{\nu }\left(z\right)$ correspond to principal values of the functions on the right-hand sides of (11.2.5) and (11.2.6).

Unless indicated otherwise, ${\mathbf{H}}_{\nu }\left(z\right)$, ${\mathbf{K}}_{\nu }\left(z\right)$, ${\mathbf{L}}_{\nu }\left(z\right)$, and ${\mathbf{M}}_{\nu }\left(z\right)$ assume their principal values throughout the DLMF.

In this subsection $A$ and $B$ are arbitrary constants.

When $z=x$, , and $\mathrm{\Re }\nu \ge 0$, numerically satisfactory general solutions of (11.2.7) are given by

11.2.11		$w$	$={\mathbf{H}}_{\nu }\left(x\right)+A{J}_{\nu }\left(x\right)+B{Y}_{\nu }\left(x\right),$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, ${\mathbf{H}}_{\nu }\left(z\right)$: Struve function, $x$: real variable, $\nu$: real or complex order, $A$: arbitrary constant and $B$: arbitrary constant Referenced by: §11.2(iii) Permalink: http://dlmf.nist.gov/11.2.E11 Encodings: TeX, pMML, png See also: Annotations for §11.2(iii), §11.2 and Ch.11
11.2.12		$w$	$={\mathbf{K}}_{\nu }\left(x\right)+A{J}_{\nu }\left(x\right)+B{Y}_{\nu }\left(x\right).$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, ${\mathbf{K}}_{\nu }\left(z\right)$: Struve function, $x$: real variable, $\nu$: real or complex order, $A$: arbitrary constant and $B$: arbitrary constant Referenced by: §11.2(iii) Permalink: http://dlmf.nist.gov/11.2.E12 Encodings: TeX, pMML, png See also: Annotations for §11.2(iii), §11.2 and Ch.11

(11.2.11) applies when $x$ is bounded, and (11.2.12) applies when $x$ is bounded away from the origin.

When $z\in ℂ$ and $\mathrm{\Re }\nu \ge 0$, numerically satisfactory general solutions of (11.2.7) are given by

11.2.13		$w$	$={\mathbf{H}}_{\nu }\left(z\right)+A{J}_{\nu }\left(z\right)+B{H}_{\nu }^{(1)}\left(z\right),$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), ${\mathbf{H}}_{\nu }\left(z\right)$: Struve function, $z$: complex variable, $\nu$: real or complex order, $A$: arbitrary constant and $B$: arbitrary constant Referenced by: §11.2(iii) Permalink: http://dlmf.nist.gov/11.2.E13 Encodings: TeX, pMML, png See also: Annotations for §11.2(iii), §11.2 and Ch.11
11.2.14		$w$	$={\mathbf{H}}_{\nu }\left(z\right)+A{J}_{\nu }\left(z\right)+B{H}_{\nu }^{(2)}\left(z\right),$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), ${\mathbf{H}}_{\nu }\left(z\right)$: Struve function, $z$: complex variable, $\nu$: real or complex order, $A$: arbitrary constant and $B$: arbitrary constant Referenced by: §11.2(iii) Permalink: http://dlmf.nist.gov/11.2.E14 Encodings: TeX, pMML, png See also: Annotations for §11.2(iii), §11.2 and Ch.11
11.2.15		$w$	$={\mathbf{K}}_{\nu }\left(z\right)+A{H}_{\nu }^{(1)}\left(z\right)+B{H}_{\nu }^{(2)}\left(z\right).$
ⓘ Symbols: $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), ${\mathbf{K}}_{\nu }\left(z\right)$: Struve function, $z$: complex variable, $\nu$: real or complex order, $A$: arbitrary constant and $B$: arbitrary constant Referenced by: §11.2(iii), §11.2(iii) Permalink: http://dlmf.nist.gov/11.2.E15 Encodings: TeX, pMML, png See also: Annotations for §11.2(iii), §11.2 and Ch.11

(11.2.13) applies when $0\le \mathrm{ph}z\le \pi$ and $|z|$ is bounded. (11.2.14) applies when $-\pi \le \mathrm{ph}z\le 0$ and $|z|$ is bounded. (11.2.15) applies when $\left|\mathrm{ph}z\right|\le \pi$ and $z$ is bounded away from the origin.

When $\mathrm{\Re }\nu \ge 0$, numerically satisfactory general solutions of (11.2.9) are given by

11.2.16		$w$	$={\mathbf{L}}_{\nu }\left(z\right)+A{K}_{\nu }\left(z\right)+B{I}_{\nu }\left(z\right),$
ⓘ Symbols: $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, ${\mathbf{L}}_{\nu }\left(z\right)$: modified Struve function, $z$: complex variable, $\nu$: real or complex order, $A$: arbitrary constant and $B$: arbitrary constant Referenced by: §11.2(iii) Permalink: http://dlmf.nist.gov/11.2.E16 Encodings: TeX, pMML, png See also: Annotations for §11.2(iii), §11.2 and Ch.11
11.2.17		$w$	$={\mathbf{M}}_{\nu }\left(z\right)+A{K}_{\nu }\left(z\right)+B{I}_{\nu }\left(z\right).$
ⓘ Symbols: ${\mathbf{M}}_{\nu }\left(z\right)$: modified Struve function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $z$: complex variable, $\nu$: real or complex order, $A$: arbitrary constant and $B$: arbitrary constant Referenced by: §11.2(iii) Permalink: http://dlmf.nist.gov/11.2.E17 Encodings: TeX, pMML, png See also: Annotations for §11.2(iii), §11.2 and Ch.11

(11.2.16) applies when $|\mathrm{ph}z|\le \frac{1}{2}\pi$ with $|z|$ bounded. (11.2.17) applies when $|\mathrm{ph}z|\le \frac{1}{2}\pi$ with $z$ bounded away from the origin.