Bessel's Integrals with Cosine or Sine?

[FQVBSina_Jesse]

TL;DR

Bessel's integral form: is it e to the power of a cosine or sine?

Hello all,

This is knowledge needed to solve my take-home final exam but I just want to ask about the definition of Bessel's integrals. This is not a problem on the exam. Wikipedia says the integral is defined as:

$$J_n(x) = \frac {1} {2\pi} \int_{-\pi}^{\pi} e^{i(xsin(\theta) - n\theta)} \, d\theta$$

My professor wrote it as:

$$J_m(Z) = \frac {1} {2\pi} \int_{-\pi}^{\pi} e^{i(Zcos(\theta))} e^{(- im\theta)} \, d\theta$$

Ignoring notation differences and I understand that cosine and sine form an orthogonal basis and are essentially the same as they can be easily expressed in terms of each other, but how do I justify that these two expressions are EXACTLY the same without any modifications with negative signs and such?

Thanks!

Jesse

 

[marcusl]

I don't think they have the same constants. To see this, try defining a new variable equal to theta+pi/2, and evaluating the integral.