Difference between revisions of "Derivative of Bessel Y with respect to its order"
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==Theorem== | ==Theorem== | ||
| − | The following formula holds: | + | The following formula holds for $\nu \neq 0, \pm 1, \pm 2, \ldots$: |
| − | $$Y_{\nu}(z)= | + | $$\dfrac{\partial}{\partial \nu} Y_{\nu}(z)=\cot(\nu \pi) \left[ \dfrac{\partial}{\partial \nu} J_{\nu}(z)-\pi Y_{\nu}(z) \right] - \csc(\nu \pi) \dfrac{\partial}{\partial \nu} J_{-\nu}(z)-\pi J_{\nu}(z),$$ |
| + | where $Y_{\nu}$ denotes the [[Bessel Y|Bessel function of the second kind]], $\cot$ denotes the [[cotangent]], $J_{\nu}$ denotes the [[Bessel J|Bessel function of the first kind]], $\pi$ denotes [[pi]], and $\csc$ denotes the [[cosecant]]. | ||
==Proof== | ==Proof== | ||
==References== | ==References== | ||
* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Derivative of Bessel J with respect to its order|next=findme}}: 9.1.65 | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Derivative of Bessel J with respect to its order|next=findme}}: 9.1.65 | ||
Revision as of 22:44, 19 June 2016
Theorem
The following formula holds for $\nu \neq 0, \pm 1, \pm 2, \ldots$: $$\dfrac{\partial}{\partial \nu} Y_{\nu}(z)=\cot(\nu \pi) \left[ \dfrac{\partial}{\partial \nu} J_{\nu}(z)-\pi Y_{\nu}(z) \right] - \csc(\nu \pi) \dfrac{\partial}{\partial \nu} J_{-\nu}(z)-\pi J_{\nu}(z),$$ where $Y_{\nu}$ denotes the Bessel function of the second kind, $\cot$ denotes the cotangent, $J_{\nu}$ denotes the Bessel function of the first kind, $\pi$ denotes pi, and $\csc$ denotes the cosecant.
Proof
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 9.1.65
