# Struve functions and modified Struve functions

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##### 1: 11.2 Definitions
###### §11.2(i) Power-Series Expansions
11.2.2 $\mathbf{L}_{\nu}\left(z\right)=-ie^{-\frac{1}{2}\pi i\nu}\mathbf{H}_{\nu}\left% (iz\right)=(\tfrac{1}{2}z)^{\nu+1}\sum_{n=0}^{\infty}\frac{(\tfrac{1}{2}z)^{2n% }}{\Gamma\left(n+\tfrac{3}{2}\right)\Gamma\left(n+\nu+\tfrac{3}{2}\right)}.$
11.2.6 $\mathbf{M}_{\nu}\left(z\right)=\mathbf{L}_{\nu}\left(z\right)-I_{\nu}\left(z% \right).$
##### 4: 11.1 Special Notation
###### §11.1 Special Notation
For the functions $J_{\nu}\left(z\right)$, $Y_{\nu}\left(z\right)$, ${H^{(1)}_{\nu}}\left(z\right)$, ${H^{(2)}_{\nu}}\left(z\right)$, $I_{\nu}\left(z\right)$, and $K_{\nu}\left(z\right)$ see §§10.2(ii), 10.25(ii). The functions treated in this chapter are the Struve functions $\mathbf{H}_{\nu}\left(z\right)$ and $\mathbf{K}_{\nu}\left(z\right)$, the modified Struve functions $\mathbf{L}_{\nu}\left(z\right)$ and $\mathbf{M}_{\nu}\left(z\right)$, the Lommel functions $s_{{\mu},{\nu}}\left(z\right)$ and $S_{{\mu},{\nu}}\left(z\right)$, the Anger function $\mathbf{J}_{\nu}\left(z\right)$, the Weber function $\mathbf{E}_{\nu}\left(z\right)$, and the associated Anger–Weber function $\mathbf{A}_{\nu}\left(z\right)$.
##### 10: 11.5 Integral Representations
###### §11.5(i) Integrals Along the Real Line
11.5.4 $\mathbf{M}_{\nu}\left(z\right)=-\frac{2(\tfrac{1}{2}z)^{\nu}}{\sqrt{\pi}\Gamma% \left(\nu+\tfrac{1}{2}\right)}\int_{0}^{1}e^{-zt}(1-t^{2})^{\nu-\frac{1}{2}}% \mathrm{d}t,$ $\Re\nu>-\tfrac{1}{2}$,