Struve function

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

x 2 d 2 y d x 2 + x d y d x + ( x 2 − α 2 ) y = 4 ( x 2 ) α + 1 π Γ ( α + 1 2 ) {\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 K α ( x ) {\displaystyle \mathbf {K} _{\alpha }(x)} as K α ( x ) = H α ( x ) − Y α ( x ) {\displaystyle \mathbf {K} _{\alpha }(x)=\mathbf {H} _{\alpha }(x)-Y_{\alpha }(x)} , where Y α ( x ) {\displaystyle Y_{\alpha }(x)} is the Neumann function. The modified Struve functions Lα(x) are equal to −ie−iαπ / 2Hα(ix) and are solutions y(x) of the non-homogeneous Bessel's differential equation:

x 2 d 2 y d x 2 + x d y d x − ( x 2 + α 2 ) y = 4 ( x 2 ) α + 1 π Γ ( α + 1 2 ) {\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 M α ( x ) {\displaystyle \mathbf {M} _{\alpha }(x)} as M α ( x ) = L α ( x ) − I α ( x ) {\displaystyle \mathbf {M} _{\alpha }(x)=\mathbf {L} _{\alpha }(x)-I_{\alpha }(x)} , where I α ( x ) {\displaystyle I_{\alpha }(x)} is the modified Bessel function of the first kind.

Definitions Since this is a non-homogeneous equation, solutions can be constructed from a single particular solution by adding the solutions of the homogeneous problem. In this case, the homogeneous solutions are the Bessel functions, and the particular solution may be chosen as the corresponding Struve function.

Power series expansion Struve functions, denoted as Hα(z) have the power series form

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