Difference between revisions of "Integral of Bessel J for nu=1"
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(Created page with "==Theorem== The following formula holds: $$\displaystyle\int_0^z J_1(t) \mathrm{d}t = 1-J_0(z),$$ where $J_1$ denotes the Bessel function of the first kind. ==Pr...") |
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==Proof== | ==Proof== | ||
| − | + | Recall, from definition, that | |
| + | $$J_1(t) = \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^kt^{2k+1}}{k! \Gamma(k+2)2^{2k+1}}.$$ | ||
==References== | ==References== | ||
* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Integral of Bessel J for nu=n+1|next=findme}}: $11.1.6$ | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Integral of Bessel J for nu=n+1|next=findme}}: $11.1.6$ | ||
Revision as of 23:10, 20 February 2018
Theorem
The following formula holds: $$\displaystyle\int_0^z J_1(t) \mathrm{d}t = 1-J_0(z),$$ where $J_1$ denotes the Bessel function of the first kind.
Proof
Recall, from definition, that $$J_1(t) = \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^kt^{2k+1}}{k! \Gamma(k+2)2^{2k+1}}.$$
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $11.1.6$
