# §11.2 Definitions

## §11.2(i) Power-Series Expansions

 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

### Struve’s Equation

 11.2.7 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\frac{1}{z}\frac{\mathrm{d}w}{% \mathrm{d}z}+\left(1-\frac{\nu^{2}}{z^{2}}\right)w=\frac{(\tfrac{1}{2}z)^{\nu-% 1}}{\sqrt{\pi}\Gamma\left(\nu+\tfrac{1}{2}\right)}.$

Particular solutions:

 11.2.8 $w=\mathbf{H}_{\nu}\left(z\right),\mathbf{K}_{\nu}\left(z\right).$

### Modified Struve’s Equation

 11.2.9 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\frac{1}{z}\frac{\mathrm{d}w}{% \mathrm{d}z}-\left(1+\frac{\nu^{2}}{z^{2}}\right)w=\frac{(\tfrac{1}{2}z)^{\nu-% 1}}{\sqrt{\pi}\Gamma\left(\nu+\tfrac{1}{2}\right)}.$

Particular solutions:

 11.2.10 $w=\mathbf{L}_{\nu}\left(z\right),\mathbf{M}_{\nu}\left(z\right).$

## §11.2(iii) Numerically Satisfactory Solutions

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

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

 11.2.11 $\displaystyle w$ $\displaystyle=\mathbf{H}_{\nu}\left(x\right)+AJ_{\nu}\left(x\right)+BY_{\nu}% \left(x\right),$ 11.2.12 $\displaystyle w$ $\displaystyle=\mathbf{K}_{\nu}\left(x\right)+AJ_{\nu}\left(x\right)+BY_{\nu}% \left(x\right).$

(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 $\displaystyle w$ $\displaystyle=\mathbf{H}_{\nu}\left(z\right)+AJ_{\nu}\left(z\right)+B{H^{(1)}_% {\nu}}\left(z\right),$ 11.2.14 $\displaystyle w$ $\displaystyle=\mathbf{H}_{\nu}\left(z\right)+AJ_{\nu}\left(z\right)+B{H^{(2)}_% {\nu}}\left(z\right),$ 11.2.15 $\displaystyle w$ $\displaystyle=\mathbf{K}_{\nu}\left(z\right)+A{H^{(1)}_{\nu}}\left(z\right)+B{% H^{(2)}_{\nu}}\left(z\right).$

(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 $\displaystyle w$ $\displaystyle=\mathbf{L}_{\nu}\left(z\right)+AK_{\nu}\left(z\right)+BI_{\nu}% \left(z\right),$ 11.2.17 $\displaystyle w$ $\displaystyle=\mathbf{M}_{\nu}\left(z\right)+AK_{\nu}\left(z\right)+BI_{\nu}% \left(z\right).$

(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.