Bessel function transformation and also cos variation

[baby_1]

Homework Statement

In a article I have found this transformation (exp to bessel function) . I have two questions.

Homework Equations

The Attempt at a Solution

a)where did the Cos go after setting n=1 and n=-1 ? in the third equations ( it is equal to -wmt-pi/2)? why?)
b)how did the writer transform the exp to bessel function?

Thanks[/B]

 

[DrDu]

See, https://en.wikipedia.org/wiki/Jacobi–Anger_expansion
I think equation 2 is not entirely correct, namely, the cos should no longer be in the exponent. Then you can express the cos in the usual way as a sum of two exponential functions to arrive at 3.