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Evaluate the line integral $ \int\limits_C {\left( {z - z_0 } \right)^m dz} $ , where $C$ is the circle centered at $z_0$ with radius r>0, and:

$i)$ $m$ is an integer $m \geqslant 0 $

$ii)$ m is an integer $m<0$

I want to see how can I solve this , to see some examples of integrals )=.

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up vote 2 down vote accepted

Parameterize the circle by $C(t) = z_0 + r e^{it} $ with $t\in (-\pi,\pi].$ Using the definition of a path integral, $$ \int_{\gamma} f(z) dz = \int^b_a f(\gamma(t)) \gamma'(t) dt$$

we have $$ \int_{C} (z-z_0)^m dz = \int^{\pi}_{-\pi} r^m e^{imt} ire^{it} dt = ir^{m+1} \int^{\pi}_{-\pi} e^{i (m+1)t} dt.$$

Now if $m+1\neq 0$ then the last integral is simply zero, seen either by direct evaluation or noting that the exponential goes through exactly $|m+1|$ periods. If $m+1=0$ then the integrand is $1$ and the value of the integral is thus $2\pi i.$

To summarize, $$\int_C (z-z_0)^m \, dz = \left\{ \begin{array}{lr} 2\pi i \text{ if }m=-1 \\ 0 \text{ if } m\in \mathbb{Z}\setminus \{-1\} \end{array} \right.$$

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