My mathematical physics test

Calculus Level 4

Let \( k \) be a real number. Evaluate the integral

\[ \frac{1}{\pi} \int_{0}^{\pi} e^{k \cos(\theta)} \cos(k \sin(\theta)) d\theta. \]

  • Hint: you may want to solve this one, first: Suppose C is the circumference \( |z| = 1 \). Let \( k \) and \( a < 1 \) be real numbers and \( n \) a natural number. Evaluate the integral

\[ I_{n}(z, a) = \oint_{C} \frac{e^{kz}}{(z-a)^{n}} dz. \]

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