Difference between revisions of "Struve function"

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$$H_{\nu}(z)=\left(\dfrac{z}{2}\right)^{\nu+1} \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k\left(\frac{z}{2}\right)^{2k}}{\Gamma(k+\frac{3}{2})\Gamma(k+\nu+\frac{3}{2})}$$
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The Struve functions are defined by
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$$\mathbf{H}_{\nu}(z)=\left(\dfrac{z}{2}\right)^{\nu+1} \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k\left(\frac{z}{2}\right)^{2k}}{\Gamma(k+\frac{3}{2})\Gamma \left(k+\nu+\frac{3}{2} \right)}.$$
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<div align="center">
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<gallery>
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File:Struveh0plot.png|Struve $\mathbf{H}_0$.
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File:Struveplots.png|Various Struve functions.
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File:Struvefunctions(abramowitzandstegun).png|Struve functions from Abramowitz&Stegun.
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</gallery>
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</div>
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=Properties=
 
=Properties=
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<div class="toccolours mw-collapsible mw-collapsed">
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[[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
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[[Relationship between Weber function 0 and Struve function 0]]<br />
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$$x^2y''(x)+xy'(x)+(x^2-n^2)y(x)=\dfrac{4(\frac{x}{2})^{n+1}}{\sqrt{\pi}\Gamma(n+\frac{1}{2})}.$$
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[[Relationship between Weber function 1 and Struve function 1]]<br />
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<div class="mw-collapsible-content">
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[[Integral representation of Struve function]]<br />
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<strong>Proof:</strong> proof goes here █
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[[Integral representation of Struve function (2)]]<br />
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</div>
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[[Integral representation of Struve function (3)]]<br />
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</div>
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[[Recurrence relation for Struve fuction]]<br />
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[[Recurrence relation for Struve function (2)]]<br />
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[[Derivative of Struve H0]]<br />
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[[D/dz(z^(nu)H (nu))=z^(nu)H (nu-1)]]<br />
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[[D/dz(z^(-nu)H (nu))=1/(sqrt(pi)2^(nu)Gamma(nu+3/2))-z^(-nu)H (nu+1)]]<br />
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[[H (nu)(x) geq 0 for x gt 0 and nu geq 1/2]]<br />
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[[H (-(n+1/2))(z)=(-1)^n J (n+1/2)(z) for integer n geq 0]]<br />
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[[H (1/2)(z)=sqrt(2/(pi z))(1-cos(z))]]<br />
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[[H (3/2)(z)=sqrt(z/(2pi))(1+2/z^2)-sqrt(2/(pi z))(sin(z)+cos(z)/z)]]<br />
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=References=
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* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=findme|next=Struve H0}}: $12.1.3$
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[[Category:SpecialFunction]]

Latest revision as of 01:09, 21 December 2017

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

  • Struve $\mathbf{H}_0$.

  • Various Struve functions.

  • Struve functions from Abramowitz&Stegun.


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
Integral representation of Struve function
Integral representation of Struve function (2)
Integral representation of Struve function (3)
Recurrence relation for Struve fuction
Recurrence relation for Struve function (2)
Derivative of Struve H0
D/dz(z^(nu)H (nu))=z^(nu)H (nu-1)
D/dz(z^(-nu)H (nu))=1/(sqrt(pi)2^(nu)Gamma(nu+3/2))-z^(-nu)H (nu+1)
H (nu)(x) geq 0 for x gt 0 and nu geq 1/2
H (-(n+1/2))(z)=(-1)^n J (n+1/2)(z) for integer n geq 0
H (1/2)(z)=sqrt(2/(pi z))(1-cos(z))
H (3/2)(z)=sqrt(z/(2pi))(1+2/z^2)-sqrt(2/(pi z))(sin(z)+cos(z)/z)

References

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