Difference between revisions of "Struve function"
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| − | $$ | + | The Struve functions are defined by |
| + | $$\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})}$$ | ||
=Properties= | =Properties= | ||
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</div> | </div> | ||
| − | {{:Relationship between Weber function and Struve function}} | + | {{:Relationship between Weber function 0 and Struve function 0}} |
| + | {{:Relationship between Weber function 1 and Struve function 1}} | ||
=References= | =References= | ||
[http://dualaud.net/specialfunctionswiki/abramowitz_and_stegun-1.03/page_496.htm Struve functions in Abramowitz&Stegun] | [http://dualaud.net/specialfunctionswiki/abramowitz_and_stegun-1.03/page_496.htm Struve functions in Abramowitz&Stegun] | ||
Revision as of 18:19, 28 June 2015
The Struve functions are defined by $$\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})}$$
Contents
Properties
Theorem: The Struve function $H_n$ solves the following nonohomogeneous Bessel differential equation $$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),$$ where $\mathbf{E}_0$ denotes a Weber function and $\mathbf{H}_0$ denotes a Struve function.
Proof
References
Theorem
The following formula holds: $$\mathbf{E}_1(z)=\dfrac{2}{\pi}-\mathbf{H}_1(z),$$ where $\mathbf{E}_1$ denotes a Weber function and $\mathbf{H}_1$ denotes a Struve function.
