Integrate sin(z sin(t)) with respect to what variable?
hi all, i need some help to solve this problem..
i knew we have to make the equation with jacobi-anger expansion,
so that we get the series form of that equation, and then integrate it
but i cant find the exact jacobi-anger expansion in any web for sin(z sin t) sin(z cos t) cos(z sin t) sin(z sin t)
i tried wiki but i think it doesnt correct
so, please help me find the RIGHT expansion, or any of you guys have a different method to solve it?
thank you very much
If you look up the Jacobi - Anger expansion, you get this page Jacobi - Anger expansion. At the bottom is the expansion of [LaTeX ERROR: Convert failed] :
[LaTeX ERROR: Convert failed]
Do you know how to go from the left hand side to the right hand side? If you do, then do that. Then you need to integrate the series with respect to t, which shouldn't be too difficult.
well, i haven't tried anything but to make the left side into the right side in here Jacobi - Anger expansion
and so far i couldn't get anything..
i would appreciate whoever can make the equation in the left side into the right side..
As a start let's show the correctness of some of the formulas on that page. Let's start with the following identity:
[LaTeX ERROR: Convert failed]
It is correct. Now we want to make the left hand side look like the one provided in the article, so we want to make:
[LaTeX ERROR: Convert failed]
Note that[LaTeX ERROR: Convert failed] which looks a lot like the complex definition for cosine, except that we need t to be [LaTeX ERROR: Convert failed] and instead of a minus in the middle, we need a plus. After a few tries, just plug in [LaTeX ERROR: Convert failed] , because we need a factor of [LaTeX ERROR: Convert failed] , in order to obtain the desired exponent.
[LaTeX ERROR: Convert failed]
Here we've used that [LaTeX ERROR: Convert failed] and hence the plus sign between the exponential functions. Now, let's plug in [LaTeX ERROR: Convert failed] in the doubly infinite sum:
[LaTeX ERROR: Convert failed]
Thus we have the first identity [LaTeX ERROR: Convert failed]
For reasons I will explain later, we need to find an analogous identity for [LaTeX ERROR: Convert failed] (It has to do with the complex definitions of sine and cosine). Let's use the initial identity. This time, however, we want to make [LaTeX ERROR: Convert failed] , so just use [LaTeX ERROR: Convert failed] to get:
[LaTeX ERROR: Convert failed]
For the next step, we need to exploit the symmetry of the Bessel function [LaTeX ERROR: Convert failed] for any integer n. Take the doubly infinite series and split it into 2 parts:
[LaTeX ERROR: Convert failed]
Lets work on the first sum. We need to flip the limits to make them look like the ones on the right sum. We do this by flipping the sign on every n in the summand:
[LaTeX ERROR: Convert failed]
Similarly to before, [LaTeX ERROR: Convert failed] . Then, using the Bessel function identity, we get:
[LaTeX ERROR: Convert failed]
Now we are ready to combine the two sums:
[LaTeX ERROR: Convert failed]
[LaTeX ERROR: Convert failed]
In a similar fashion, we can derive the other identity and I suggest you do it as practice. It is:
[LaTeX ERROR: Convert failed] (I'm not 100% sure my derivation and expression are correct)
Now, finally, let's derive the expression for [LaTeX ERROR: Convert failed] . We need to use the complex number definition for sin(t). Then we plug in [LaTeX ERROR: Convert failed] :
[LaTeX ERROR: Convert failed]
Now, you see that we need to use the second series to expand this.
[LaTeX ERROR: Convert failed]
[LaTeX ERROR: Convert failed]
[LaTeX ERROR: Convert failed]
Note that [LaTeX ERROR: Convert failed] and [LaTeX ERROR: Convert failed] .
[LaTeX ERROR: Convert failed]
Let's not forget the factor 1/(2i). After the division, we get
[LaTeX ERROR: Convert failed]