Residue at Simple Pole
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Theorem
Let $f: \C \to \C$ be a function meromorphic on some region, $D$, containing $a$.
Let $f$ have a simple pole at $a$.
Then the residue of $f$ at $a$ is given by:
- $\ds \Res f a = \lim_{z \mathop \to a} \paren {z - a} \map f z$
Proof
By Existence of Laurent Series, there exists a Laurent series:
- $\ds \map f z = \sum_{n \mathop = -\infty}^\infty c_n \paren {z - a}^n$
which is convergent in $D \setminus \set a$, where $\sequence {c_n}$ is a doubly infinite sequence of complex coefficients.
We are given that $f$ has only a simple pole at $a$.
Thus $c_n = 0$ for $n < -1$.
So we can write:
- $\ds \map f z = \sum_{n \mathop = 0}^\infty c_n \paren {z - a}^n + \frac {c_{-1} } {z - a}$
Then:
\(\ds \lim_{z \mathop \to a} \paren {z - a} \map f z\) | \(=\) | \(\ds \lim_{z \mathop \to a} \paren {z - a} \paren {\sum_{n \mathop = 0}^\infty c_n \paren {z - a}^n + \frac {c_{-1} } {z - a} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{z \mathop \to a} \paren {\sum_{n \mathop = 0}^\infty c_n \paren {z - a}^{n + 1} + c_{-1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty c_n \paren {a - a}^{n + 1} + c_{-1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0 \sum_{n \mathop = 0}^\infty c_n + c_{-1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds c_{-1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \Res f a\) | Definition of Residue |
$\blacksquare$