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Complex Analysis Cheat Sheet

Definitions

$${z = a + bi}$$

$${w = c + di}$$

$${|z| = \sqrt{a^2 + b^2} }$$

Complex conjugate:

$${\bar z = a - bi}$$

$${Re(z) = a}$$

$${Im(z) = b}$$

Formulas

Euler's Formula:

$${e^{ix} = \cos(x) + i \sin(x) }$$

$${Re(z) = \frac{1}{2} (z + \bar z) }$$

$${Im(z) = \frac{1}{2i} (z - \bar z) }$$

$${|z|^2 = a^2 + b^2 = z \bar z }$$

$${|z| = \sqrt{z \bar z} }$$

$${\overline{z + w} = \bar z + \bar w }$$

$${\overline{z w} = \bar z \bar w }$$

$${z \bar w = \overline{\bar z w} }$$

$${\bar z w = \overline{z \bar w} }$$

$${|z \bar w| = |\bar z w| = |z| |w| }$$

$${Re(z \bar w) \leq |Re(z \bar w)| \leq |z \bar w| = |z| |w| }$$

Triangle Inequality:

$${|z + w| \leq |z| + |w| }$$

Roots

$${z = a + bi}$$

$${r = |z| = \sqrt{a^2 + b^2} }$$

$${a = r \cos(t)}$$

$${b = r \sin(t)}$$

$${t = arg(z) = \tan^{-1} \left( \frac{b}{a} \right) }$$

$${z = r e^{i \left(t + 2 \pi m \right)}}$$

$${z^n = (a + bi)^n = r^n e^{i\left(nt + 2 \pi m \right)} }$$

$${w^n = z }$$

$${w = \sqrt[n]{z} = \sqrt[n]{r} e^{i \left( \frac{t}{n} + 2 \pi \frac{m}{n} \right)}, m,n \in \mathbb{Z}^+ }$$

$${0 \leq m \lt n }$$

$${1 + w + w^2 + \cdots + w^{n-1} = 0 }$$

De Moivre's Theorem

$${z^n = (a + bi)^n = \left( r e^{it} \right)^n = r^n e^{int} }$$

$${z^n = \left( r \cos t + r i \sin t \right)^n = r^n \left(\cos nt + i \sin nt \right) }$$

Logarithms

$${z = |z| e^{i arg(z)}}$$

$${\ln(z) = ln |z| + i \ arg(z) }$$

Nice Values

$${e^{i \pi} = -1 }$$

$${e^{i 2 \pi} = 1 }$$

$${e^{i \frac{\pi}{2} } = i }$$

$${e^{i \frac{3 \pi}{2} } = -i }$$

$${\sqrt{2} e^{i \frac{\pi}{4} } = 1 + i }$$

$${\sqrt{2} e^{-i \frac{\pi}{4} } = 1 - i }$$

$${e^{i \frac{\pi}{3} } = \frac{1}{2} + \frac{\sqrt{3}}{2} i }$$

$${i^2 = -1 }$$

$${i^3 = -i }$$

$${i^4 = 1 }$$