Bessel Functions and Shifted Integral Limits: How Are They Related?

[dfrenette]

A nth order bessel function of the first kind is defined as:

Jn(B)=(1/2pi)*integral(exp(jBsin(x)-jnx))dx

where the integral limits are -pi to pi

I have an expression that is the exact same as above, but the limits are shifted by 90 degrees; from -pi/2 to 3pi/2

My question is how does this new expression relate to Bessel functions? My first thought was that the two function are equal since the integral limits are over one period. But I am not sure.

 

[edgepflow]

I think they are different. I can not offer a mathematical proof, but using MathCAD function for Jn and its numerical integration, I obtained different numbers.

 

[JJacquelin]

My question is how does this new expression relate to Bessel functions? My first thought was that the two function are equal since the integral limits are over one period. But I am not sure.

You are right. The two functions are equal.
A more tedious proof is given in the joint page :

 

[dfrenette]

Super,
Thanks for the proof!

I did set up a little mathcad file (attached) that shows the two implementations, proving to myself that indeed they are equal, but this proof will come in handy.

Where did you get it from so I can cite it?

Thanks again,

Darren

 

[edgepflow]

You are right. The two functions are equal.
A more tedious proof is given in the joint page :

Nice proof. I stand corrected.

The Bessel function I tried in MathCAD is: Jn(m, x) which returns Jm(x). Is this the same thing?

 

[JJacquelin]

Where did you get it from so I can cite it?

It's written by myself. You can copy it. No need to cite an author : the calculus is rather classical.

 

[dfrenette]

Perfect. Thanks again.