Difference between revisions of "Integral of Bessel J for nu=1"

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==Proof==
 
==Proof==
 
Recall, from definition, that  
 
Recall, from definition, that  
−
$$J_1(t) = \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^kt^{2k+1}}{k! \Gamma(k+2)2^{2k+1}}.$$
+
$$J_1(t) = \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^kt^{2k+1}}{k! (k+1)! 2^{2k+1}}.$$
 +
Integrating from $0$ to $z$ yields
 +
$$\begin{array}{ll}
 +
\displaystyle\int_0^z J_1(t) \mathrm{d}t &= \displaystyle\int_0^z \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^kt^{2k+1}}{k! (k+1)! 2^{2k+1}} \mathrm{d}t \\
 +
&=\displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k}{k! (k+1)! 2^{2k+1}} \displaystyle\int_0^z t^{2k+1} \mathrm{d}t \\
 +
&=\displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k t^{2k+2}}{(k+1)!^2 2^{2k+2}}  \\
 +
&=\displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k t^{2(k+1)}}{(k+1)!^2 2^{2(k+1)}}  \\
 +
\end{array}$$
 
==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 23:14, 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 $$J_1(t) = \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^kt^{2k+1}}{k! (k+1)! 2^{2k+1}}.$$ Integrating from $0$ to $z$ yields $$\begin{array}{ll} \displaystyle\int_0^z J_1(t) \mathrm{d}t &= \displaystyle\int_0^z \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^kt^{2k+1}}{k! (k+1)! 2^{2k+1}} \mathrm{d}t \\ &=\displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k}{k! (k+1)! 2^{2k+1}} \displaystyle\int_0^z t^{2k+1} \mathrm{d}t \\ &=\displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k t^{2k+2}}{(k+1)!^2 2^{2k+2}} \\ &=\displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k t^{2(k+1)}}{(k+1)!^2 2^{2(k+1)}} \\ \end{array}$$

References

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