Difference between revisions of "Struve function"

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(Properties)
(Properties)
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=Properties=
 
=Properties=
 
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<strong>Theorem:</strong> The Struve solve the following nonohomogeneous [[Bessel]] differential equation  
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<strong>Theorem:</strong> The Struve function $H_n$ solves the following nonohomogeneous [[Bessel]] differential equation  
 
$$x^2y''(x)+xy'(x)+(x^2-n^2)y(x)=\dfrac{4(\frac{x}{2})^{n+1}}{\sqrt{\pi}\Gamma(n+\frac{1}{2})}.$$
 
$$x^2y''(x)+xy'(x)+(x^2-n^2)y(x)=\dfrac{4(\frac{x}{2})^{n+1}}{\sqrt{\pi}\Gamma(n+\frac{1}{2})}.$$
 
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Revision as of 19:16, 7 March 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: proof goes here █