Struve function

In mathematics, the Struve functions H_α(x), are solutions y(x) of the non-homogeneous Bessel's differential equation:

${\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}+\left(x^{2}-\alpha ^{2}\right)y={\frac {4\left({\frac {x}{2}}\right)^{\alpha +1}}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}}$

introduced by Hermann Struve (1882). The complex number α is the order of the Struve function, and is often an integer.

And further defined its second-kind version ${\displaystyle \mathbf {K} _{\alpha }(x)}$ as ${\displaystyle \mathbf {K} _{\alpha }(x)=\mathbf {H} _{\alpha }(x)-Y_{\alpha }(x)}$, where ${\displaystyle Y_{\alpha }(x)}$ is the Neumann function.

The modified Struve functions L_α(x) are equal to −ie^{−iαπ / 2}H_α(ix) and are solutions y(x) of the non-homogeneous Bessel's differential equation:

${\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}-\left(x^{2}+\alpha ^{2}\right)y={\frac {4\left({\frac {x}{2}}\right)^{\alpha +1}}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}}$

And further defined its second-kind version ${\displaystyle \mathbf {M} _{\alpha }(x)}$ as ${\displaystyle \mathbf {M} _{\alpha }(x)=\mathbf {L} _{\alpha }(x)-I_{\alpha }(x)}$, where ${\displaystyle I_{\alpha }(x)}$ is the modified Bessel function.