The complex number field $\mathbb{C}$ extends the real domain $\mathbb{R}$ by adjoining the imaginary unit $i$, defined fundamentally by $i^2 = -1$. Any complex variable $z \in \mathbb{C}$ is expressed in Cartesian form as $z = x + iy$, where $x = \text{Re}(z)$ and $y = \text{Im}(z)$.
Via Taylor series expansion of $e^x$, $\cos x$, and $\sin x$, Euler's formula establishes the profound identity connecting exponential and trigonometric functions:
$$e^{i\theta} = \cos\theta + i\sin\theta$$
For $\theta = \pi$, this simplifies to Euler's identity, $e^{i\pi} + 1 = 0$. In polar coordinates, $z = r e^{i\theta}$ where $r = \vert{}z\vert{} = \sqrt{x^2 + y^2}$ and $\theta = \arg(z) = \arctan(y/x)$.
A complex function $f(z) = u(x,y) + i v(x,y)$ is complex-differentiable (holomorphic) at a point $z_0$ if and only if its real and imaginary parts satisfy the Cauchy-Riemann equations:
$$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad \text{and} \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$$
If $f(z)$ is analytic inside and on a simple closed contour $C$, the value of $f(a)$ at any interior point $a$ is completely determined by its boundary values:
$$f(a) = \frac{1}{2\pi i} \oint_C \frac{f(z)}{z - a} \, dz$$
When evaluating contour integrals with isolated singularities $z_k$, the Residue Theorem reduces the integral to a discrete sum:
$$\oint_C f(z)\,dz = 2\pi i \sum_{k=1}^n \text{Res}(f, z_k)$$