09 Complex Integrals
Line Integral¶
For \(\int f(z) \ dz\), put \(z = r \cdot e^{i \theta}\)
ML Inequality¶
maximum value / upper bound of integral
\[ \left| \int_C f(z) \ dz \right| \le M \times L \]
where
- \(M =\) max value of \(f(z)\)
- \(L =\) length of contour \(C\)
Theorems¶
| Theorem | Cauchy-Goursat | Cauchy-Integral | Cauchy-Integral for derivatives | Cauchy Residue |
|---|---|---|---|---|
| Condition | \(f(z)\) is analytic inside/on \(C\) | - \(f(z)\) is analytic inside/on \(C\) - \(z_0\) is a point inside \(C\) | - \(f(z)\) is analytic inside/on \(C\) - \(z_0\) is a point inside \(C\) | |
| Identity | \(\int_C f(z) \ dz = 0\) | \(\int_C \frac{f(z)}{z-z_0} dz = 2 \pi i \cdot f(z_0)\) | \(\int_C \frac{f(z)}{(z-z_0)^{n+1}} dz = \frac{2 \pi i}{n!} \times f^{(n)}(z_0)\) | $\int_C f(z) dz = 2 \pi i \times \ |
| [\text{Sum of residues at poles lying inside/on } C]$ | ||||
| add for multiple points | ❌ | ✅ | ✅ |
| Theorem | Condition | Identity | add for multiple points |
|---|---|---|---|
| Cauchy-Goursat | \(f(z)\) is analytic inside/on \(C\) | \(\int_C f(z) \ dz = 0\) | ❌ |
| Cauchy-Integral | - \(f(z)\) is analytic inside/on \(C\) - \(z_0\) is a point inside \(C\) | \(\int_C \frac{f(z)}{z-z_0} dz = 2 \pi i \cdot f(z_0)\) | âś… |
| Cauchy-Integral for derivatives | - \(f(z)\) is analytic inside/on \(C\) - \(z_0\) is a point inside \(C\) | \(\int_C \frac{f(z)}{(z-z_0)^{n+1}} dz = \frac{2 \pi i}{n!} \times f^{(n)}(z_0)\) | âś… |
| Cauchy Residue | \(\int_C f(z) \ dz = 2 \pi i \times (\sum R)\) | ❌ |
\(\sum R =\) Sum of residues at poles lying inside/on \(C\)
Residue¶
| Type | \(R\) |
|---|---|
| Simple Pole | \(\lim_{z \to z_0} (z-z_0) f(z)\) |
| Pole of order \(m\) | \(\dfrac{1}{m-1} \times \dfrac{d^{m-1}}{dz^{m-1}} [(z-z_0)^m f(z)]_{z = z_0}\) |
| \(\dfrac{P(z_0)}{\textcolor{orange}{Q}(z_0)}, P(z_0) \ne 0, Q(z_0) = 0\) | \(\dfrac{P(z_0)}{\textcolor{orange}{Q'}(z_0)}\) |
Laurent’s Series¶
\[ \begin{aligned} f(z) &= \sum_0^\infty a_n (z-z_0)^n + \underbrace{ \sum_1^\infty \frac{b_n}{(z - z_0)^n} }_\text{Principal Part} \\ a_n &= \frac{1}{2 \pi i} \times \int \frac{f(z)}{(z-z_0)^{ \textcolor{orange}{n}+1 }} \\ b_n &= \frac{1}{2 \pi i} \times \int \frac{f(z)}{(z-z_0)^{ \textcolor{orange}{-n}+1 }} \end{aligned} \]
The following equation is only valid if \(0 < |z| < 1\)
\[ \begin{aligned} (1+z)^{-1} &= 1 - z + z^2 - z^3 + \dots \\ (1-z)^{-1} &= 1 + z + z^2 + z^3 + \dots \\(1+z)^{-2} &= 1 - 2z + 3z^2 - 4z^3 + \dots \\(1-z)^{-2} &= 1 + 2z + 3z^2 + 4z^3 + \dots \end{aligned} \]
Singular Points¶
Take all \(n\) points \((\pm n\pi, \pm 2n\pi, \dots)\)
Isolated Point¶
No other singular point in close neighborhood
Poles¶
isolated points are poles too
poles of order \(m=1\) are simple poles