Argument Principle

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Theorem

Let $f$ be a function meromorphic in the interior of some simply connected region $D$.

Let $f$ be holomorphic with no zeroes on the boundary of $D$.

Let $N$ denote the number of zeroes of $f$ in the interior of $D$, counted up to multiplicity.

Let $P$ denote the number of poles of $f$ in the interior of $D$, counted up to order.

Then:

$\displaystyle N - P = \frac 1 {2\pi i} \oint_D \frac { f'\left({z}\right) } { f\left({z}\right) } \, \mathrm d z$


Proof

Let $n_1, n_2, n_3 \ldots n_N$ denote the zeroes of $f$, and $p_1, p_2, p_3 \ldots p_P$ denote its poles. Then, there exists a function $g$ nonzero and holomorphic on the boundary and on the interior of $D$ such that:

$\displaystyle f\left({z}\right) = \frac { \prod_{k \mathop = 1}^N \left( { z - n_k } \right) } { \prod_{j \mathop = 1}^P \left( { z - p_j } \right) } g\left({z}\right)$

Taking the logarithmic derivative:

\(\displaystyle \mathcal L \left( { f\left( {z} \right) } \right)\) \(=\) \(\displaystyle \frac { f'\left({z}\right) } { f\left({z}\right) }\) Definition of Logarithmic Derivative of Meromorphic Function
\(\displaystyle \) \(=\) \(\displaystyle \mathcal L \left( { \prod_{k \mathop = 1}^N \left( { z - n_k } \right) } \right) - \mathcal L \left( { \prod_{j \mathop = 1}^P \left( { z - p_j } \right) } \right) + \mathcal L \left( { g\left({z}\right) } \right)\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = 1}^N \frac 1 {z - n_k} - \sum_{j \mathop = 1}^P \frac 1 {z - p_j} + \frac { g'\left({z}\right) } { g\left({z}\right) }\)

Then:

\(\displaystyle \oint_D \frac { f'\left({z}\right) } { f\left({z}\right) } \, \mathrm d z\) \(=\) \(\displaystyle \oint_D \left( \sum_{k \mathop = 1}^N \frac 1 {z - n_k} - \sum_{j \mathop = 1}^P \frac 1 {z - p_j} + \frac { g'\left({z}\right) } { g\left({z}\right) } \right) \, \mathrm d z\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = 1}^N \oint_D \frac 1 {z - n_k} \, \mathrm d z - \sum_{j \mathop =1}^P \oint_D \frac 1 {z - p_j} \, \mathrm d z + \oint_D \frac { g'\left({z}\right) } { g\left({z}\right) } \, \mathrm d z\) Linear Combination of Contour Integrals
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k=1}^N 2\pi i - \sum_{j \mathop = 1}^P 2\pi i + \oint_D \frac{ g'\left({z}\right) } { g\left({z}\right) } \, \mathrm d z\)
\(\displaystyle \) \(=\) \(\displaystyle 2\pi i \left( {N - P} \right) + \oint_D \frac{ g'\left({z}\right) } { g\left({z}\right) } \, \mathrm d z\)

By our construction of $g$, $\frac {g'} g$ is holomorphic on and inside $D$ so, by the Cauchy Integral Theorem, the integral is equal to zero. We therefore have:

$\displaystyle \oint_D \frac { f'\left({z}\right) } { f\left({z}\right) } \, \mathrm d z = 2\pi i \left( {N - P} \right)$

Hence:

$\displaystyle N - P = \frac 1 {2\pi i} \oint_D \frac { f'\left({z}\right) } { f\left({z}\right) } \, \mathrm d z$


$\blacksquare$