Difference between revisions of "Weber function"
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{{:Relationship between Anger function and Weber function}} | {{:Relationship between Anger function and Weber function}} | ||
{{:Relationship between Weber function and Struve function}} | {{:Relationship between Weber function and Struve function}} | ||
| + | {{:Relationship between Weber function 1 and Struve function 1}} | ||
=References= | =References= | ||
[http://dualaud.net/specialfunctionswiki/abramowitz_and_stegun-1.03/page_498.htm Abramowitz and Stegun] | [http://dualaud.net/specialfunctionswiki/abramowitz_and_stegun-1.03/page_498.htm Abramowitz and Stegun] | ||
Revision as of 18:19, 28 June 2015
The Weber function is defined by $$\mathbf{E}_{\nu}(z)=\dfrac{1}{\pi} \displaystyle\int_0^{\pi} \sin(\nu \theta - z \sin(\theta))d\theta.$$
Contents
Properties
Theorem
The following formula holds: $$\sin(\nu \pi)\mathbf{E}_{\nu}(z)=\mathbf{J}_{-\nu}(z)-\cos(\nu \pi)\mathbf{J}_{\nu}(z),$$ where $\mathbf{E}_{\nu}$ denotes a Weber function and $\mathbf{J}_{\nu}$ denotes an Anger function.
Proof
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 12.3.5
Theorem
The following formula holds: $$\sin(\nu\pi)\mathbf{J}_{\nu}(z)=\cos(\nu \pi)\mathbf{E}_{\nu}(z)-\mathbf{E}_{-\nu}(z),$$ where $\mathbf{J}_{\nu}$ denotes an Anger function and $\mathbf{E}_{\nu}$ denotes a Weber function.
Proof
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 12.3.4
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.
