Complex Variables

Complex Number

A number of the form $z = x + iy$, where $x$ and $y$ are real numbers and $i$ is the imaginary unit ($i² = -1$).

Complex Plane

A two-dimensional plane where the horizontal axis represents the real part and the vertical axis represents the imaginary part of a complex number.

Analytic Function

A complex function that is differentiable at every point in an open region of the complex plane.

Cauchy-Riemann Equations

A pair of first-order partial differential equations that a complex function must satisfy to be analytic.

Residue

The coefficient of the $(z-a)^-1$ term in the Laurent series expansion of a function $f(z)$ around an isolated singularity $a$.

Conformal Mapping

A transformation that preserves angles between curves in the complex plane.

Complex Number Arithmetic

Addition and subtraction

$(x₁ + iy₁) \± (x₂ + iy₂) = (x₁ \± x₂) + i(y₁ \± y₂)$

Complex Number Multiplication

$(x₁ + iy₁)(x₂ + iy₂) = (x_1x_2 - y_1y_2) + i(x_1y_2 + x_2y_1)$

Complex Conjugate

If $z = x + iy$, then $\bar{z} = x - iy$.

Modulus of a Complex Number

Distance from origin in complex plane

$|z| = \sqrt{z\bar{z}} = \sqrt{x² + y²}$

Euler's Formula

Relates exponential and trigonometric forms

$e^i\θ = \cos\θ + i\sin\θ$

Polar Form

where $r = |z|$ and $\θ = \arg(z)$

$z = r(\cos\θ + i\sin\θ) = re^i\θ$

Cauchy-Riemann Equations

For $f(z) = u(x, y) + iv(x, y)$, $f$ is analytic if $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$.

Cauchy's Integral Theorem

If $f(z)$ is analytic in a simply connected domain $D$, then for any simple closed contour $C$ in $D$, $\oint_C f(z) dz = 0$.

Cauchy's Integral Formula

If $f(z)$ is analytic in a simply connected domain $D$ and $C$ is a simple closed contour in $D$ with $a$ inside $C$, then $f(a) = \frac{1}{2\π i} \oint_C \frac{f(z)}{z-a} dz$.

Cauchy's Formula for Derivatives

If $f(z)$ is analytic in a simply connected domain $D$ and $C$ is a simple closed contour in $D$ with $a$ inside $C$, then $f⁽ⁿ⁾(a) = \frac{n!}{2\π i} \oint_C \frac{f(z)}{(z-a)ⁿ⁺¹} dz$.

Residue Theorem

If $f(z)$ is analytic inside and on a simple closed contour $C$, except for a finite number of isolated singular points $z₁, z₂, ..., z_n$ inside $C$, then $\oint_C f(z) dz = 2\π i \sum_k=1^n Res(f, z_k)$.