Interesting definite integral involving exp and trig

I'm trying to evaluate the following integrals:

$$\int_0^{2\pi} e^{\kappa \cos(\phi - \mu)} \cos(\phi) d\phi$$ $$\int_0^{2\pi} e^{\kappa \cos(\phi - \mu)} \sin(\phi) d\phi$$

for which I want to find an easily computable function. This may be either a closed form, or something in terms of special functions that are available in most scientific computing libraries (e.g. scipy.special).

I found the following result on wikipedia, which may be useful. $$\int_0^{2\pi} e^{x \cos(\theta)} d\theta = 2\pi I_0(x),$$ where $I_0(x)$ is the modified Bessel function of the first kind, of order 0.

I want to apologize in advance if this is a basic question; I have not had formal training in advanced calculus (beyond highschool), except for a few online lectures and wikipedia reading.

Any help is greatly appreciated.

• Wenn you just write out function names like that, $\TeX$ interprets that as a juxtaposition of variable names and formats it accordingly. To get the appropriate font and spacing, you can use predefined commands like \cos, or, if you need an operator name for which there isn't a predefined command, you can use \operatorname{name}. – joriki May 7 '13 at 11:46
• You are trying to solve for what: $\kappa$ or $\mu$ ?? Or maybe you are trying to evaluate the integrals? – GEdgar May 7 '13 at 12:12
• Fixed the terminology: solve -> evaluate. – John von N. May 7 '13 at 12:31

Consider $$f(x):=\int_0^{2\pi} e^{x \cos(\phi)}\;d\phi=2\pi I_0(x)$$
since $I'_0(x)=I_1(x)$ and using derivation under the integral sign this becomes : $$f'(x)=\int_0^{2\pi} e^{x \cos(\phi)}\;\cos(\phi) \;d\phi=2\pi I'_0(x)=2\pi I_1(x)$$

Let's rewrite your first integral (using the substitution $\theta:=\phi - \mu$) : \begin{align} I_c:&=\int_{-\mu}^{2\pi-\mu} e^{\kappa \cos(\theta)} \cos(\theta+\mu)\;d\theta\\ &=\int_0^{2\pi} e^{\kappa \cos(\theta)} \cos(\theta+\mu)\;d\theta\\ &=\cos(\mu)\int_0^{2\pi} e^{\kappa \cos(\theta)} \cos(\theta)\;d\theta-\sin(\mu)\int_0^{2\pi} e^{\kappa \cos(\theta)} \sin(\theta)\;d\theta\\ &=2\pi\cos(\mu)I_1(\kappa)+\sin(\mu)\int_0^{2\pi} e^{\kappa \cos(\theta)} \;d\cos(\theta)\\ &=2\pi\cos(\mu)I_1(\kappa)+\sin(\mu)\frac{e^{\kappa\cos(2\pi)}-e^{\kappa\cos(0)}}{\kappa}\\ I_c&=2\pi\cos(\mu)\;I_1(\kappa) \end{align}

The same way I got the second integral as :

$\qquad\qquad\qquad I_s=2\pi\sin(\mu)\;I_1(\kappa)$

For interesting properties and rewriting of the modified Bessel function $I_1$ see the excellent references available :

• +1 for the answer. That said, although I make much use of the DLMF, I find it sorely lacking compared to A & S with respect to computational use (i.e., approximations, integrals, sums, products, etc.) – Ron Gordon May 7 '13 at 10:23
• Awesome, thanks! – John von N. May 7 '13 at 10:27
• You are right @Ron A&S is missing here (I updated my answer). I don't know why I forgot it here since I am fond of this book ! – Raymond Manzoni May 7 '13 at 11:32
• You are welcome @JohnvonN. (really? :-)). Should something be unclear let me know... – Raymond Manzoni May 7 '13 at 11:34