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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.