Complex Analysis/Exercises/Sheet 5

Exercise on Complex Analysis

Calculate the following integrals using the Cauchy integral formula:

$\int_{| z + 1 | = 1} \frac{d z}{(z + 1) (z - 1)^{3}}$
$\int_{| z | = 2} \frac{\sin z}{z + i} d z$
$\int_{| z | = r} \frac{d z}{(z - a)^{n} (z - b)^{m}}$ mit $| a | < r < | b |$ , $n, m \geq 1$ .

Let

S^{1} : = {z \in 𝐂 : | z | = 1}

. On

𝐂 ∖ S^{1}

, consider the function

f (z) : = \int_{S^{1}} \frac{d ζ}{ζ (ζ - z)}

Determine $f$ . At which points of $S^{1}$ does $f$ have a limit?

Use the Cauchy integral formula to determine

\int_{0}^{2 π} e^{k \sin t} \sin (k \sin t) d t

Hint: Consider

z \mapsto \frac{e^{k z}}{z}

on the unit circle.

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