Integrating $\int_{-\pi}^{\pi} \exp(\kappa \cos(\theta-\theta_p) + j 2\pi f_m\tau\cos\theta)\, d\theta$, where $j=\sqrt{-1}$

Question

The formula is: $$\int_{-\pi}^{\pi} \exp(\kappa \cos(\theta-\theta_p) + j 2\pi f_m\tau\cos\theta)\, d\theta$$ and if ignore insignificant parameters, it can be expressed as: $$ \int_{-\pi}^{\pi} \exp(\cos(\theta-\theta_p) + j\cos\theta)\, d\theta$$ Here, $j=\sqrt{-1}$, and $\theta_p$ is a parameter.

I already know $$\int_{0}^{\pi} \exp( z\cos \theta)\,d\theta = \pi I_0(z)$$ but I don't know how to due with the parameter $\theta_p $ in cosine function. I also tried to use some trigonometric transformation for $\cos(\theta-\theta_p)$, but it didn't work.

And the answer is:

$$2\pi I_0(\sqrt{\kappa^2-4\pi^2f_m^2\tau^2+j4\pi\kappa cos\theta_p f_m\tau}) $$

It seems that some numerical square tricks are used.

Any suggestions?

Answer 1

I think I know it.

First, to get the even part of following function, and the method is here.

$$\exp(\kappa \cos(\theta-\theta_p))$$

Then, if I get the odd part A and even part B. Then, the original equation will look something like this:

$$\int_{-\pi}^{\pi} (A+B) \exp(j 2\pi f_m\tau\cos\theta)\, d\theta $$

Since odd function multiply even function will be odd function, and the odd function over a symmetric interval with respect to $0$ is zero. Then, I get:

$$ \int_{-\pi}^{\pi} B\exp(j 2\pi f_m\tau\cos\theta)\, d\theta $$

Thirdly, using the identity of the difference of the sum of angles for function B's exponent. Here, the formula that should be used is:

$$ cos(\theta-\theta_p) = cos(\theta)cos(\theta_p) + sin(\theta)sin(\theta_p) $$

Finally, using an integral equation $3.338$ from the book: Table of Integrals, Series, and Products, Fifth Edition. The equation is:

$$ \int_{-\pi}^{\pi} \frac {exp[\frac{a+b sinx+c cosx}{1+psinx+qcosx}]}{1+p\sin x+q\cos x} dx= \frac {2\pi }{\sqrt {1-p^ {2}-q^ {2}}} e^ {-\alpha } I_ {0} ( \beta ) $$

where $ \alpha $ = $ \frac {bp+cq-a}{1-p^ {2}-q^ {2}} $ ; $ \beta $ = $ \sqrt {\alpha ^ {2}-\frac {a^ {2}-b^ {2}-c^ {2}}{1-p^ {2}-q^ {2}}} $. And $ p^ {2}+q^ {2} < 1$, $I_0$ is zero-order Bessel functions of an imaginary argument.

I'm not giving the exact steps, but if you follow the instructions above, you can get the final answer. And you will find the final equation can translate exactly what we got in the previous part into the answer.