Argument Principle
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links.
To discuss this in more detail, feel free to use the talk page.
Once you have done so, you can remove this instance of {{MissingLinks}} from the code.
Theorem
Let $\gamma$ be a closed contour.
Let $D$ be the region enclosed by $\gamma$.
Let $f$ be a function meromorphic on $D$.
Let $f$ be holomorphic with no zeroes on $\gamma$.
Let $N$ denote the number of zeroes of $f$ in $D$, counted up to multiplicity.
Let $P$ denote the number of poles of $f$ in $D$, counted up to order.
order
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this point in more detail, feel free to use the talk page.
If you are able to explain it, then when you have done so you can remove this instance of {{Explain}} from the code.
Then:
- $\ds N - P = \frac 1 {2\pi i} \oint_\gamma \frac {\map {f'} z} {\map f z} \rd z$
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 non-zero holomorphic function $g$ such that:
- $\ds \map f x = \frac {\prod_{k \mathop = 1}^N \paren {z - n_k} } {\prod_{j \mathop = 1}^P \paren {z - p_j} } \map g z$
Taking the logarithmic derivative:
| \(\ds \map \LL {\map f z}\) | \(=\) | \(\ds \frac {\map {f'} z} {\map f z}\) | Definition of Logarithmic Derivative of Meromorphic Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \LL {\prod_{k \mathop = 1}^N \paren {z - n_k} } - \map \LL {\prod_{j \mathop = 1}^P \paren {z - p_j} } + \map \LL {\map g z}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 1}^N \frac 1 {z - n_k} - \sum_{j \mathop = 1}^P \frac 1 {z - p_j} + \frac {\map {g'} z} {\map g z}\) |
Then:
| \(\ds \oint_\gamma \frac {\map {f'} z} {\map f z} \rd z\) | \(=\) | \(\ds \oint_\gamma \paren {\sum_{k \mathop = 1}^N \frac 1 {z - n_k} - \sum_{j \mathop = 1}^P \frac 1 {z - p_j} + \frac {\map {g'} z} {\map g z} } \rd z\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 1}^N \oint_\gamma \frac 1 {z - n_k} \rd z - \sum_{j \mathop = 1}^P \oint_\gamma \frac 1 {z - p_j} \rd z + \oint_\gamma \frac {\map {g'} z} {\map g z} \rd z\) | Linear Combination of Contour Integrals | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 1}^N 2 \pi i - \sum_{j \mathop = 1}^P 2 \pi i + \oint_\gamma \frac {\map {g'} z} {\map g z} \rd z\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 2\pi i \paren {N - P} + \oint_\gamma \frac {\map {g'} z} {\map g z} \rd z\) |
By our construction of $g$, $\frac {g'} g$ is holomorphic on $D$ so, by the Cauchy Integral Theorem, the integral is equal to zero.
We therefore have:
- $\ds \oint_\gamma \frac {\map {f'} z} {\map f z} \rd z = 2\pi i \paren {N - P}$
Hence:
- $\ds N - P = \frac 1 {2\pi i} \oint_\gamma \frac {\map {f'} z} {\map f z} \rd z$
$\blacksquare$