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