## Related questions with answers

Suppose $f$ is differentiable at points on a closed path $\gamma$ and at all points in the region G enclosed by $\gamma,$ except possibly at a finite number of poles of $f$ in G. Let Z be the number of zeros of $f$ in G, and P the number of poles of $f$ in G, with each zero and pole counted as many times as its multiplicity. Show that $\frac{1}{2 \pi i} \oint_{\gamma} \frac{f^{\prime}(z)}{f(z)} d z=Z-P.$ This formula is known as the argument principle.

Solution

VerifiedLet $f(z)$ have zeros $z_1,z_2,\ldots,z_n$ of orders $r_1,r_2,\ldots,r_n$ respectively and have poles $\xi_1,\xi_2,\ldots,\xi_m$ of orders $s_1,s_2,\ldots, s_m$ respectively in $G$ so that

${\color{#4257b2}Z=r_1+r_2+\cdots+r_n}~~\text{and}~~{\color{#4257b2}P=s_1+s_2+\cdots+s_m}$

Clearly, the only singularities of $\frac{f'(z)}{f(z)}$ are $z_1,z_2,\ldots,z_n;\xi_1,\xi_2,\ldots,\xi_m$.

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