Functions of a complex variable

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• ConceptComplex plane

  The complex plane is the plane $\mathbb{R}^2$ in which the point $(a, b)$ represents the complex number $a + b i$. The horizontal axis is the real axis and the vertical axis is the imaginary axis. In this picture, addition of complex numbers is addition of vectors. The modulus $|z|$ is the distance from $z$ to $0$, and the argument of $z$ is the angle from the positive real axis to $z$.

• ConceptRiemann sphere

  The Riemann sphere is the complex plane with one added point $\infty$, see the definition. Stereographic projection identifies it with the unit sphere in $\mathbb{R}^3$. It is the natural domain and target of Möbius transformations and of meromorphic functions.

• ConceptStereographic projection

  Stereographic projection maps the unit sphere minus one point, its north pole, bijectively onto a plane, see the definition. Sending the north pole to a point at infinity makes the sphere a model of the Riemann sphere $\C \cup \set{\infty}$. The map preserves angles and sends circles on the sphere to circles or lines in the plane. It is the bridge between the geometry of the sphere and complex analysis on the extended plane.

• ConceptHolomorphic function

  A holomorphic function is a complex function that is complex differentiable at each point of an open subset of $\mathbb{C}$, see the definition. Holomorphic functions are the main objects of complex analysis. They are infinitely differentiable and they are equal to their Taylor series near each point.

• TheoremCauchy–Riemann equations

  Let $U \subseteq \mathbb{C}$ be an open set and let $f : U \to \mathbb{C}$ be a function. Identify $\mathbb{C}$ with $\mathbb{R}^2$ by $x + i y \leftrightarrow (x, y)$. Let $u = \operatorname{Re} f$ and $v = \operatorname{Im} f$ be the real and imaginary parts of $f$, as real functions of $(x, y)$ on $U$. Let $z_0 = x_0 + i y_0 \in U$. Then $f$ is complex differentiable at $z_0$ if and only if the following two conditions hold. First, $u$ and $v$ are differentiable at $(x_0, y_0)$ as functions of two real variables. Second, they satisfy $$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \qquad \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}$$ at $(x_0, y_0)$. In this case $f'(z_0) = \frac{\partial u}{\partial x}(x_0, y_0) + i \frac{\partial v}{\partial x}(x_0, y_0)$.

• ConceptWirtinger derivatives

  The Wirtinger derivatives $\partial / \partial z$ and $\partial / \partial \bar z$ are two first-order differential operators on functions of $z = x + iy$, see the definition. They let one treat $z$ and $\bar z$ as if they were independent variables. A continuously differentiable function is holomorphic exactly when its $\partial / \partial \bar z$ derivative is zero, which is a compact form of the Cauchy–Riemann equations.