Difference between revisions of "Struve function"
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[http://dualaud.net/specialfunctionswiki/abramowitz_and_stegun-1.03/page_496.htm Struve functions in Abramowitz&Stegun] | [http://dualaud.net/specialfunctionswiki/abramowitz_and_stegun-1.03/page_496.htm Struve functions in Abramowitz&Stegun] | ||
Revision as of 18:18, 28 June 2015
$$H_{\nu}(z)=\left(\dfrac{z}{2}\right)^{\nu+1} \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k\left(\frac{z}{2}\right)^{2k}}{\Gamma(k+\frac{3}{2})\Gamma(k+\nu+\frac{3}{2})}$$
Properties
Theorem: The Struve function $H_n$ solves the following nonohomogeneous Bessel differential equation $$x^2y(x)+xy'(x)+(x^2-n^2)y(x)=\dfrac{4(\frac{x}{2})^{n+1}}{\sqrt{\pi}\Gamma(n+\frac{1}{2})}.$$
Proof: █
Theorem
The following theorem holds: $$\mathbf{E}_0(z)=-\mathbf{H}_0(z),$$ where $\mathbf{E}_0$ denotes a Weber function and $\mathbf{H}_0$ denotes a Struve function.
