# §11.4 Basic Properties

## §11.4(i) Half-Integer Orders

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

 11.4.1 $\mathbf{K}_{n+\frac{1}{2}}\left(z\right)=\left(\frac{2}{\pi z}\right)^{\frac{1% }{2}}\sum_{m=0}^{n}\frac{(2m)!\,2^{-2m}}{m!\,(n-m)!}\,(\tfrac{1}{2}z)^{n-2m},$
 11.4.2 $\mathbf{L}_{n+\frac{1}{2}}\left(z\right)=I_{-n-\frac{1}{2}}\left(z\right)-% \left(\frac{2}{\pi z}\right)^{\frac{1}{2}}\sum_{m=0}^{n}\frac{(-1)^{m}(2m)!\,2% ^{-2m}}{m!\,(n-m)!}\,(\tfrac{1}{2}z)^{n-2m},$
 11.4.3 $\displaystyle\mathbf{H}_{-n-\frac{1}{2}}\left(z\right)$ $\displaystyle=(-1)^{n}J_{n+\frac{1}{2}}\left(z\right),$ ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$: 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 $\displaystyle\mathbf{L}_{-n-\frac{1}{2}}\left(z\right)$ $\displaystyle=I_{n+\frac{1}{2}}\left(z\right).$ ⓘ 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 $n$: 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 $\displaystyle\mathbf{H}_{\frac{1}{2}}\left(z\right)$ $\displaystyle=\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}(1-\cos z),$ ⓘ Symbols: $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: 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 $\displaystyle\mathbf{H}_{-\frac{1}{2}}\left(z\right)$ $\displaystyle=\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}\sin z,$
 11.4.7 $\displaystyle\mathbf{L}_{\frac{1}{2}}\left(z\right)$ $\displaystyle=\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}(\cosh z-1),$ 11.4.8 $\displaystyle\mathbf{L}_{-\frac{1}{2}}\left(z\right)$ $\displaystyle=\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}\sinh z,$
 11.4.9 $\mathbf{H}_{\frac{3}{2}}\left(z\right)=\left(\frac{z}{2\pi}\right)^{\frac{1}{2% }}\left(1+\frac{2}{z^{2}}\right)-\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}% \left(\sin z+\frac{\cos z}{z}\right),$
 11.4.10 $\mathbf{H}_{-\frac{3}{2}}\left(z\right)=\left(\frac{2}{\pi z}\right)^{\frac{1}% {2}}\left(\cos z-\frac{\sin z}{z}\right),$
 11.4.11 $\mathbf{L}_{\frac{3}{2}}\left(z\right)=-\left(\frac{z}{2\pi}\right)^{\frac{1}{% 2}}\left(1-\frac{2}{z^{2}}\right)+\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}% \left(\sinh z-\frac{\cosh z}{z}\right),$
 11.4.12 $\mathbf{L}_{-\frac{3}{2}}\left(z\right)=\left(\frac{2}{\pi z}\right)^{\frac{1}% {2}}\left(\cosh z-\frac{\sinh z}{z}\right).$

## §11.4(ii) Inequalities

 11.4.13 $\mathbf{H}_{\nu}\left(x\right)\geq 0,$ $x>0$, $\nu\geq\tfrac{1}{2}$. ⓘ Symbols: $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$: 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 $\mathbf{H}_{\nu}\left(z\right)=\frac{2(\tfrac{1}{2}z)^{\nu+1}}{\sqrt{\pi}% \Gamma\left(\nu+\tfrac{3}{2}\right)}(1+\vartheta),$ $\nu\neq-\tfrac{3}{2},-\tfrac{5}{2},-\tfrac{7}{2},\dots$,

where

 11.4.15 $|\vartheta|<\frac{2}{3}\exp\left(\frac{\tfrac{1}{4}|z|^{2}}{|\nu_{0}+\tfrac{3}% {2}|}-1\right),$ ⓘ Symbols: $\exp\NVar{z}$: 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 $|\nu_{0}+\tfrac{3}{2}|$ is the smallest of the numbers $|\nu+\tfrac{3}{2}|$, $|\nu+\tfrac{5}{2}|$, $|\nu+\tfrac{9}{2}|,\dots$.

## §11.4(iii) Analytic Continuation

 11.4.16 $\mathbf{H}_{\nu}\left(ze^{m\pi i}\right)=e^{m\pi i(\nu+1)}\mathbf{H}_{\nu}% \left(z\right),$ $m\in\mathbb{Z}$,
 11.4.17 $\mathbf{L}_{\nu}\left(ze^{m\pi i}\right)=e^{m\pi i(\nu+1)}\mathbf{L}_{\nu}% \left(z\right),$ $m\in\mathbb{Z}$.

## §11.4(iv) Expansions in Series of Bessel Functions

 11.4.18 $\mathbf{H}_{\nu}\left(z\right)=\frac{4}{\pi^{1/2}\Gamma\left(\nu+\tfrac{1}{2}% \right)}\*\sum_{k=0}^{\infty}\frac{(2k+\nu+1)\Gamma\left(k+\nu+1\right)}{k!(2k% +1)(2k+2\nu+1)}J_{2k+\nu+1}\left(z\right),$ $\nu\neq-1,-2,-3,\dots$,
 11.4.19 $\mathbf{H}_{\nu}\left(z\right)=\left(\frac{z}{2\pi}\right)^{1/2}\sum_{k=0}^{% \infty}\frac{(\tfrac{1}{2}z)^{k}}{k!(k+\tfrac{1}{2})}J_{k+\nu+\frac{1}{2}}% \left(z\right),$
 11.4.20 $\mathbf{H}_{\nu}\left(z\right)=\frac{(\tfrac{1}{2}z)^{\nu+\frac{1}{2}}}{\Gamma% \left(\nu+\tfrac{1}{2}\right)}\sum_{k=0}^{\infty}\frac{(\tfrac{1}{2}z)^{k}}{k!% (k+\nu+\tfrac{1}{2})}J_{k+\frac{1}{2}}\left(z\right),$
 11.4.21 $\mathbf{H}_{0}\left(z\right)=\frac{4}{\pi}\sum_{k=0}^{\infty}\frac{J_{2k+1}% \left(z\right)}{2k+1}=2\sum_{k=0}^{\infty}(-1)^{k}{J_{k+\frac{1}{2}}}^{2}\left% (\tfrac{1}{2}z\right),$
 11.4.22 $\mathbf{H}_{1}\left(z\right)=\frac{2}{\pi}(1-J_{0}\left(z\right))+\frac{4}{\pi% }\sum_{k=1}^{\infty}\frac{J_{2k}\left(z\right)}{4k^{2}-1}=4\sum_{k=0}^{\infty}% J_{2k+\frac{1}{2}}\left(\tfrac{1}{2}z\right)J_{2k+\frac{3}{2}}\left(\tfrac{1}{% 2}z\right).$

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

## §11.4(v) Recurrence Relations and Derivatives

 11.4.23 $\displaystyle\mathbf{H}_{\nu-1}\left(z\right)+\mathbf{H}_{\nu+1}\left(z\right)$ $\displaystyle=\frac{2\nu}{z}\mathbf{H}_{\nu}\left(z\right)+\frac{(\tfrac{1}{2}% z)^{\nu}}{\sqrt{\pi}\Gamma\left(\nu+\tfrac{3}{2}\right)},$ 11.4.24 $\displaystyle\mathbf{H}_{\nu-1}\left(z\right)-\mathbf{H}_{\nu+1}\left(z\right)$ $\displaystyle=2\mathbf{H}_{\nu}'\left(z\right)-\frac{(\tfrac{1}{2}z)^{\nu}}{% \sqrt{\pi}\Gamma\left(\nu+\tfrac{3}{2}\right)},$ 11.4.25 $\displaystyle\mathbf{L}_{\nu-1}\left(z\right)-\mathbf{L}_{\nu+1}\left(z\right)$ $\displaystyle=\frac{2\nu}{z}\mathbf{L}_{\nu}\left(z\right)+\frac{(\tfrac{1}{2}% z)^{\nu}}{\sqrt{\pi}\Gamma\left(\nu+\tfrac{3}{2}\right)},$ 11.4.26 $\displaystyle\mathbf{L}_{\nu-1}\left(z\right)+\mathbf{L}_{\nu+1}\left(z\right)$ $\displaystyle=2\mathbf{L}_{\nu}'\left(z\right)-\frac{(\tfrac{1}{2}z)^{\nu}}{% \sqrt{\pi}\Gamma\left(\nu+\tfrac{3}{2}\right)}.$
 11.4.27 $\frac{\mathrm{d}}{\mathrm{d}z}\left(z^{\nu}\mathbf{H}_{\nu}\left(z\right)% \right)=z^{\nu}\mathbf{H}_{\nu-1}\left(z\right),$
 11.4.28 $\frac{\mathrm{d}}{\mathrm{d}z}\left(z^{-\nu}\mathbf{H}_{\nu}\left(z\right)% \right)=\frac{2^{-\nu}}{\sqrt{\pi}\Gamma\left(\nu+\tfrac{3}{2}\right)}-z^{-\nu% }\mathbf{H}_{\nu+1}\left(z\right),$
 11.4.29 $\frac{\mathrm{d}}{\mathrm{d}z}\left(z^{\nu}\mathbf{L}_{\nu}\left(z\right)% \right)=z^{\nu}\mathbf{L}_{\nu-1}\left(z\right),$
 11.4.30 $\frac{\mathrm{d}}{\mathrm{d}z}\left(z^{-\nu}\mathbf{L}_{\nu}\left(z\right)% \right)=\frac{2^{-\nu}}{\sqrt{\pi}\Gamma\left(\nu+\tfrac{3}{2}\right)}+z^{-\nu% }\mathbf{L}_{\nu+1}\left(z\right).$
 11.4.31 ${\cal H}_{\nu-m}(z)=z^{m-\nu}\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}z}% \right)^{m}(z^{\nu}{\cal H}_{\nu}(z)),$ $m=1,2,3,\dots$, ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$, $z$: complex variable and $\nu$: real or complex order Referenced by: §11.4(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 ${\cal H}_{\nu}(z)$ denotes either $\mathbf{H}_{\nu}\left(z\right)$ or $\mathbf{L}_{\nu}\left(z\right)$.

 11.4.32 $\displaystyle\mathbf{H}_{0}'\left(z\right)$ $\displaystyle=\frac{2}{\pi}-\mathbf{H}_{1}\left(z\right),$ $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}(z\mathbf{H}_{1}\left(z\right))$ $\displaystyle=z\mathbf{H}_{0}\left(z\right),$
 11.4.33 $\displaystyle\mathbf{L}_{0}'\left(z\right)$ $\displaystyle=\frac{2}{\pi}+\mathbf{L}_{1}\left(z\right),$ $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}(z\mathbf{L}_{1}\left(z\right))$ $\displaystyle=z\mathbf{L}_{0}\left(z\right).$

## §11.4(vi) Derivatives with Respect to Order

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

## §11.4(vii) Zeros

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).