11 Struve and Related Functions Struve and Modified Struve Functions 11.1 Special Notation 11.3 Graphics

§11.2 Definitions

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Keywords:: Struve functions and modified Struve functions, definitions
Permalink:: http://dlmf.nist.gov/11.2
See also:: Annotations for Ch.11

§11.2(i) Power-Series Expansions
§11.2(ii) Differential Equations
§11.2(iii) Numerically Satisfactory Solutions

§11.2(i) Power-Series Expansions

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Keywords:: Struve functions and modified Struve functions, power series, principal values, series expansions
Notes:: See Watson (1944, pp. 328–329) and Olver (1997b, pp. 274–276). The notation $\mathbf{M}_{\nu}\left(z\right)$ is new and this function has been introduced to play a similar role to $\mathbf{L}_{\nu}\left(z\right)$ that $\mathbf{K}_{\nu}\left(z\right)$ does to $\mathbf{H}_{\nu}\left(z\right)$ .
Referenced by:: §10.22(i), §10.43(i), §11.13(iv)
Permalink:: http://dlmf.nist.gov/11.2.i
See also:: Annotations for §11.2 and Ch.11

11.2.1	$\displaystyle\mathbf{H}_{\nu}\left(z\right)$	$\displaystyle=(\tfrac{1}{2}z)^{\nu+1}\sum_{n=0}^{\infty}\frac{(-1)^{n}(\tfrac{% 1}{2}z)^{2n}}{\Gamma\left(n+\tfrac{3}{2}\right)\Gamma\left(n+\nu+\tfrac{3}{2}% \right)},$
ⓘ Defines: $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function Symbols: $\Gamma\left(\NVar{z}\right)$ : 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	$\displaystyle\mathbf{L}_{\nu}\left(z\right)$	$\displaystyle=-ie^{-\frac{1}{2}\pi i\nu}\mathbf{H}_{\nu}\left(iz\right)=(% \tfrac{1}{2}z)^{\nu+1}\sum_{n=0}^{\infty}\frac{(\tfrac{1}{2}z)^{2n}}{\Gamma% \left(n+\tfrac{3}{2}\right)\Gamma\left(n+\nu+\tfrac{3}{2}\right)}.$
ⓘ Defines: $\mathbf{L}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function 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, $\mathrm{e}$ : 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 $(\tfrac{1}{2}z)^{\nu+1}$ ; 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	$\displaystyle\mathbf{H}_{0}\left(z\right)$	$\displaystyle=\frac{2}{\pi}\left(z-\frac{z^{3}}{1^{2}\cdot 3^{2}}+\frac{z^{5}}% {1^{2}\cdot 3^{2}\cdot 5^{2}}-\cdots\right),$
ⓘ Symbols: $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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	$\displaystyle\mathbf{L}_{0}\left(z\right)$	$\displaystyle=\frac{2}{\pi}\left(z+\frac{z^{3}}{1^{2}\cdot 3^{2}}+\frac{z^{5}}% {1^{2}\cdot 3^{2}\cdot 5^{2}}+\cdots\right).$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathbf{L}_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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	$\displaystyle\mathbf{K}_{\nu}\left(z\right)$	$\displaystyle=\mathbf{H}_{\nu}\left(z\right)-Y_{\nu}\left(z\right),$
ⓘ Defines: $\mathbf{K}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function Symbols: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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	$\displaystyle\mathbf{M}_{\nu}\left(z\right)$	$\displaystyle=\mathbf{L}_{\nu}\left(z\right)-I_{\nu}\left(z\right).$
ⓘ Defines: $\mathbf{M}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function 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 $\nu$ : 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.

§11.2(ii) Differential Equations

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Keywords:: Bessel’s equation, Struve functions and modified Struve functions, differential equations, inhomogeneous forms
Notes:: See Watson (1944, pp. 328–329) and Olver (1997b, pp. 274–276).
Permalink:: http://dlmf.nist.gov/11.2.ii
See also:: Annotations for §11.2 and Ch.11

Struve’s Equation

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Keywords:: Struve functions and modified Struve functions, differential equations, particular solutions
See also:: Annotations for §11.2(ii), §11.2 and Ch.11

11.2.7		$\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\frac{1}{z}\frac{\mathrm{d}w}{% \mathrm{d}z}+\left(1-\frac{\nu^{2}}{z^{2}}\right)w=\frac{(\tfrac{1}{2}z)^{\nu-% 1}}{\sqrt{\pi}\Gamma\left(\nu+\tfrac{1}{2}\right)}.$
ⓘ 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 $x$ , $z$ : complex variable and $\nu$ : real or complex order Referenced by: §11.13(iv), §11.2(iii), §11.2(iii), §11.9(i) Permalink: http://dlmf.nist.gov/11.2.E7 Encodings: TeX, pMML, png See also: Annotations for §11.2(ii), §11.2(ii), §11.2 and Ch.11

Particular solutions:

11.2.8		$w=\mathbf{H}_{\nu}\left(z\right),\mathbf{K}_{\nu}\left(z\right).$
ⓘ Symbols: $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\mathbf{K}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $z$ : complex variable and $\nu$ : real or complex order Permalink: http://dlmf.nist.gov/11.2.E8 Encodings: TeX, pMML, png See also: Annotations for §11.2(ii), §11.2(ii), §11.2 and Ch.11

Modified Struve’s Equation

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Keywords:: Struve functions and modified Struve functions, differential equations, inhomogeneous forms, modified Bessel’s equation, particular solutions
See also:: Annotations for §11.2(ii), §11.2 and Ch.11

11.2.9		$\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\frac{1}{z}\frac{\mathrm{d}w}{% \mathrm{d}z}-\left(1+\frac{\nu^{2}}{z^{2}}\right)w=\frac{(\tfrac{1}{2}z)^{\nu-% 1}}{\sqrt{\pi}\Gamma\left(\nu+\tfrac{1}{2}\right)}.$
ⓘ 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 $x$ , $z$ : complex variable and $\nu$ : real or complex order Referenced by: §11.13(iv), §11.2(iii) Permalink: http://dlmf.nist.gov/11.2.E9 Encodings: TeX, pMML, png See also: Annotations for §11.2(ii), §11.2(ii), §11.2 and Ch.11

Particular solutions:

11.2.10		$w=\mathbf{L}_{\nu}\left(z\right),\mathbf{M}_{\nu}\left(z\right).$
ⓘ Symbols: $\mathbf{M}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function, $\mathbf{L}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function, $z$ : complex variable and $\nu$ : real or complex order Permalink: http://dlmf.nist.gov/11.2.E10 Encodings: TeX, pMML, png See also: Annotations for §11.2(ii), §11.2(ii), §11.2 and Ch.11

§11.2(iii) Numerically Satisfactory Solutions

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Keywords:: Struve functions and modified Struve functions, differential equations, numerically satisfactory solutions
Notes:: See §2.7(iv) and Olver (1997b, pp. 274–277). The last reference restricts (11.2.15) to the sector $\left|\operatorname{ph}z\right|\leq\tfrac{1}{2}\pi$ , and instead covers the sector $\tfrac{1}{2}\pi\leq\operatorname{ph}z\leq\tfrac{3}{2}\pi$ with another set of solutions. (Similarly for the conjugate sector $-\tfrac{3}{2}\pi\leq\operatorname{ph}z\leq-\tfrac{1}{2}\pi$ .)
Permalink:: http://dlmf.nist.gov/11.2.iii
See also:: Annotations for §11.2 and Ch.11

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

When $z=x$ , $0<x<\infty$ , and $\Re\nu\geq 0$ , numerically satisfactory general solutions of (11.2.7) are given by

11.2.11	$\displaystyle w$	$\displaystyle=\mathbf{H}_{\nu}\left(x\right)+AJ_{\nu}\left(x\right)+BY_{\nu}% \left(x\right),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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	$\displaystyle w$	$\displaystyle=\mathbf{K}_{\nu}\left(x\right)+AJ_{\nu}\left(x\right)+BY_{\nu}% \left(x\right).$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\mathbf{K}_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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 $z\in\mathbb{C}$ and $\Re\nu\geq 0$ , numerically satisfactory general solutions of (11.2.7) are given by

11.2.13	$\displaystyle w$	$\displaystyle=\mathbf{H}_{\nu}\left(z\right)+AJ_{\nu}\left(z\right)+B{H^{(1)}_% {\nu}}\left(z\right),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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	$\displaystyle w$	$\displaystyle=\mathbf{H}_{\nu}\left(z\right)+AJ_{\nu}\left(z\right)+B{H^{(2)}_% {\nu}}\left(z\right),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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	$\displaystyle w$	$\displaystyle=\mathbf{K}_{\nu}\left(z\right)+A{H^{(1)}_{\nu}}\left(z\right)+B{% H^{(2)}_{\nu}}\left(z\right).$
ⓘ Symbols: ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $\mathbf{K}_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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 $0\leq\operatorname{ph}z\leq\pi$ and $|z|$ is bounded. (11.2.14) applies when $-\pi\leq\operatorname{ph}z\leq 0$ and $|z|$ is bounded. (11.2.15) applies when $\left|\operatorname{ph}z\right|\leq\pi$ and $z$ is bounded away from the origin.

When $\Re\nu\geq 0$ , numerically satisfactory general solutions of (11.2.9) are given by

11.2.16	$\displaystyle w$	$\displaystyle=\mathbf{L}_{\nu}\left(z\right)+AK_{\nu}\left(z\right)+BI_{\nu}% \left(z\right),$
ⓘ Symbols: $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\mathbf{L}_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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	$\displaystyle w$	$\displaystyle=\mathbf{M}_{\nu}\left(z\right)+AK_{\nu}\left(z\right)+BI_{\nu}% \left(z\right).$
ⓘ Symbols: $\mathbf{M}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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 $|\operatorname{ph}z|\leq\tfrac{1}{2}\pi$ with $|z|$ bounded. (11.2.17) applies when $|\operatorname{ph}z|\leq\tfrac{1}{2}\pi$ with $z$ bounded away from the origin.

11.1 Special Notation 11.3 Graphics

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