Difference between revisions of "Weber function"

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{{:Relationship between Weber function 0 and Struve function 0}}
 
{{:Relationship between Weber function 0 and Struve function 0}}
 
{{:Relationship between Weber function 1 and Struve function 1}}
 
{{:Relationship between Weber function 1 and Struve function 1}}
 +
{{:Relationship between Weber function 2 and Struve function 2}}
  
 
=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 18:21, 28 June 2015

The Weber function is defined by $$\mathbf{E}_{\nu}(z)=\dfrac{1}{\pi} \displaystyle\int_0^{\pi} \sin(\nu \theta - z \sin(\theta))d\theta.$$

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

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

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.

Proof

References

Theorem

The following formula holds: $$\mathbf{E}_2(z)=\dfrac{2z}{3\pi} - \mathbf{H}_2(z),$$ where $\mathbf{E}_2$ denotes a Weber function and $\mathbf{H}_2$ denotes a Struve function.

Proof

References

References

Abramowitz and Stegun