Difference between revisions of "Struve function"

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=Properties=
 
=Properties=
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<strong>Theorem:</strong> If $x >0$ and $\nu \geq \dfrac{1}{2}$, then $\mathbf{H}_{\nu}(x) \geq 0$.
 
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<strong>Proof:</strong>  █
 
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[[Relationship between Struve function and hypergeometric pFq]]<br />
 
[[Relationship between Struve function and hypergeometric pFq]]<br />
 
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<strong>Theorem:</strong> The Struve function $H_n$ solves the following nonohomogeneous [[Bessel]] differential equation
 
$$x^2y''(x)+xy'(x)+(x^2-n^2)y(x)=\dfrac{4(\frac{x}{2})^{n+1}}{\sqrt{\pi}\Gamma(n+\frac{1}{2})}.$$
 
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<strong>Proof:</strong> █
 
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[[Relationship between Weber function 0 and Struve function 0]]<br />
 
[[Relationship between Weber function 0 and Struve function 0]]<br />
 
[[Relationship between Weber function 1 and Struve function 1]]<br />
 
[[Relationship between Weber function 1 and Struve function 1]]<br />

Revision as of 13:18, 25 June 2016

The Struve functions are defined by $$\mathbf{H}_{\nu}(z)=\left(\dfrac{z}{2}\right)^{\nu+1} \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k\left(\frac{z}{2}\right)^{2k}}{\Gamma(k+\frac{3}{2})\Gamma(k+\nu+\frac{3}{2})}$$


Properties

Relationship between Struve function and hypergeometric pFq
Relationship between Weber function 0 and Struve function 0
Relationship between Weber function 1 and Struve function 1

References

Struve functions in Abramowitz&Stegun