Struve function

Remarks

It is a particular solution of the inhomogeneous Bessel equation $$z^2 w'' + z w' + (z^2 - \nu^2)\, w = \frac{4 \left( z/2 \right)^{\nu + 1}}{\sqrt{\pi}\, \Gamma\left( \nu + \frac{1}{2} \right)}.$$ The general solution adds to $\mathbf{H}_\nu$ any solution of Bessel's equation, such as a combination of the Bessel functions $J_\nu$ and $Y_\nu$. Example: for $\nu = \frac{1}{2}$ and $x > 0$, $\mathbf{H}_{1/2}(x) = \sqrt{\frac{2}{\pi x}}\,(1 - \cos x)$; the first term of the series is $\frac{(x/2)^{3/2}}{\Gamma(3/2)\Gamma(2)} = \frac{x^{3/2}}{\sqrt{2\pi}}$, which matches $\sqrt{\frac{2}{\pi x}} \cdot \frac{x^2}{2}$. A non-example: $J_{1/2}(x) = \sqrt{\frac{2}{\pi x}} \sin x$ is not a Struve function; it solves the equation with right-hand side $0$. The modified Struve function is $\mathbf{L}_\nu(z) = -i e^{-i\nu\pi/2}\, \mathbf{H}_\nu(iz)$.