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

From specialfunctionswiki
Jump to: navigation, search
(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...")
 
 
(5 intermediate revisions by the same user not shown)
Line 5: Line 5:
  
 
==Proof==
 
==Proof==
 +
Recall, from definition, that
 +
$$J_0(t) = \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k t^{2k}}{(k!)^2 2^{2k}},$$
 +
and
 +
$$J_1(t) = \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^kt^{2k+1}}{k! (k+1)! 2^{2k+1}}.$$
 +
Integrating from $0$ to $z$ yields
 +
$$\begin{array}{ll}
 +
\displaystyle\int_0^z J_1(t) \mathrm{d}t &= \displaystyle\int_0^z \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^kt^{2k+1}}{k! (k+1)! 2^{2k+1}} \mathrm{d}t \\
 +
&=\displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k}{k! (k+1)! 2^{2k+1}} \displaystyle\int_0^z t^{2k+1} \mathrm{d}t \\
 +
&=\displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k t^{2k+2}}{(k+1)!^2 2^{2k+2}}  \\
 +
&=\displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k t^{2(k+1)}}{(k+1)!^2 2^{2(k+1)}}  \\
 +
&\stackrel{\mathrm{reindex}}{=} \displaystyle\sum_{k=1}^{\infty} \dfrac{(-1)^{k-1} t^{2k}}{(k!)^2 2^{2k}} \\
 +
&=1-\displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^{k} t^{2k}}{(k!)^2 2^{2k}} \\
 +
&=1-J_0(z),
 +
\end{array}$$
 +
completing the proof.
  
 
==References==
 
==References==
Line 10: Line 25:
  
 
[[Category:Theorem]]
 
[[Category:Theorem]]
−
[[Category:Unproven]]
+
[[Category:Proved]]
 +
[[Category:Justify]]

Latest revision as of 23:18, 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 $$J_0(t) = \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k t^{2k}}{(k!)^2 2^{2k}},$$ and $$J_1(t) = \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^kt^{2k+1}}{k! (k+1)! 2^{2k+1}}.$$ Integrating from $0$ to $z$ yields $$\begin{array}{ll} \displaystyle\int_0^z J_1(t) \mathrm{d}t &= \displaystyle\int_0^z \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^kt^{2k+1}}{k! (k+1)! 2^{2k+1}} \mathrm{d}t \\ &=\displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k}{k! (k+1)! 2^{2k+1}} \displaystyle\int_0^z t^{2k+1} \mathrm{d}t \\ &=\displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k t^{2k+2}}{(k+1)!^2 2^{2k+2}} \\ &=\displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k t^{2(k+1)}}{(k+1)!^2 2^{2(k+1)}} \\ &\stackrel{\mathrm{reindex}}{=} \displaystyle\sum_{k=1}^{\infty} \dfrac{(-1)^{k-1} t^{2k}}{(k!)^2 2^{2k}} \\ &=1-\displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^{k} t^{2k}}{(k!)^2 2^{2k}} \\ &=1-J_0(z), \end{array}$$ completing the proof.

References

Retrieved from "http://specialfunctionswiki.org/index.php?title=Integral_of_Bessel_J_for_nu%3D1&oldid=9063"