Expressing trigonometric function in terms of integral of Bessel function

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Observe that $2(n+1)(2\,n+1)!=(2\,n+2)!$. Then $$ \sum_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{2(n+1)(2n+1)!}=\frac1x\sum_{n=0}^\infty \frac{(-1)^nx^{2n+2}}{(2\,n+2)!}. $$

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Shibli
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Shibli

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Updated on May 18, 2020

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  • Shibli
    Shibli over 3 years

    I am trying to show that,

    \begin{align*} \frac{1-\cos x}{x} = \int_{0}^{\pi/2}J_1(x\cos\theta)\,\mathrm{d}\theta \end{align*}

    I did the following but cannot figure out how to continue.

    \begin{align*} \int_{0}^{\pi/2}J_1(x\cos\theta)\,\mathrm{d}\theta &=\sum_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{2(n+1)(n!)^24^n} \int_{0}^{\pi/2}\cos^{2n+1}\theta\,d\theta\cr &=\sum_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{2(n+1)(n!)^24^n}\cdot \frac{(n!)^24^n}{(2n+1)!}\cr &=\sum_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{2(n+1)(2n+1)!} \end{align*}

  • Shibli
    Shibli over 7 years
    So you mean that the summation of series is $1-\cos x$. I googled but could not find a reference about the expansion of $1-\cos x$. Could you refer me to a document?
  • Julián Aguirre
    Julián Aguirre over 7 years
    You know the series for the Bessel function but you do not know the one for $\cos x$?