I am reading the article Mirela Vinerean:(adsbygoogle = window.adsbygoogle || []).push({});

http://www.math.kau.se/mirevine/mf2bess.pdf

On page 6, I have a question about

[tex]e^{\frac{x}{2}t} e^{-\frac{x}{2}\frac{1}{t}}=\sum^{\infty}_{n=-\infty}J_n(x)e^{jn\theta}=\sum_{n=0}^{\infty}J_n(x)[e^{jn\theta}+(-1)^ne^{-jn\theta}][/tex]

I think there is a mistake at the last term. If you look at n=0, the equation will be:

[tex]J_0(x)[e^0+(-1)^0e^{-0}]\;=\;j_0(x)[1+1]\;=\;2J_0(x)[/tex]

Which is not correct. The problem is n=0 is being repeated in both the n=+ve and n=-ve.

The equation should be:

[tex]e^{\frac{x}{2}t} e^{-\frac{x}{2}\frac{1}{t}}=\sum_{n=0}^{\infty}[J_n(x)e^{jn\theta}]\;+\;\sum_{n=1}^{\infty}[(-1)^ne^{-jn\theta}][/tex]

With this, the first term covers the original n=+ve from 0 to ∞. The second term covers the original from -∞ to -1. Now you only have one term contain n=0. Am I correct?

Thanks

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# Double check the derivation integral representation of Bessel Function

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