Complex Functions and Analyticity

A complex function is a function $f : D \subseteq \mathbb{C} \to \mathbb{C}$. We can write $f(z) = u(x, y) + iv(x, y)$ where $z = x + iy$ and $u, v$ are real-valued functions.

Example. $f(z) = z^2 = (x + iy)^2 = (x^2 - y^2) + i(2xy)$. Here $u = x^2 - y^2$ and $v = 2xy$.

Example. $f(z) = \bar{z} = x - iy$. Here $u = x$ and $v = -y$.

Example. $f(z) = |z|^2 = x^2 + y^2$. Here $u = x^2 + y^2$ and $v = 0$.

The limit $\lim_{z \to z_0} f(z) = L$ means: for every $\varepsilon \gt 0$There exists $\delta \gt 0$ Such that $0 \lt |z - z_0| \lt \delta$ implies $|f(z) - L| \lt \varepsilon$.

Unlike the real case, $z$ can approach $z_0$ from any direction in $\mathbb{C}$. This makes limits More restrictive.

Proposition 2.1. $\lim_{z \to z_0} f(z) = L$ if and only if $\lim_{(x,y) \to (x_0, y_0)} u(x, y) = a$ And $\lim_{(x,y) \to (x_0, y_0)} v(x, y) = b$ where $L = a + bi$.

Definition. $f$ is continuous at $z_0$ if $\lim_{z \to z_0} f(z) = f(z_0)$.

Definition. $f$ is differentiable at $z_0$ if

$f"(z_0) = \lim_{h \to 0} \frac{f(z_0 + h) - f(z_0)}{h}$

Exists (and is independent of how $h \to 0$ in $\mathbb{C}$).

Remark. The requirement that the limit be the same for all directions of approach of $h$ is what Makes complex differentiability far more restrictive than real differentiability.

Definition. A function $f$ is analytic (or holomorphic) on an open set $U \subseteq \mathbb{C}$ if $f$ is differentiable at every point of $U$. A function that is analytic On all of $\mathbb{C}$ is called entire.

Examples of entire functions: $z^n$, $e^z$, $\sin z$, $\cos z$Polynomials.

Example of a non-analytic function: $f(z) = \bar{z}$ is nowhere differentiable (except at $z = 0$ if we define it, but still not analytic there).

Many important functions in complex analysis are inherently multi-valued. To work with them as Single-valued functions, we must restrict the domain.

Definition. A branch of a multi-valued function $f$ is a single-valued analytic function $g$ Defined on a domain $D$ such that $g(z) \in f(z)$ for all $z \in D$.

The Complex Logarithm. We define $\log z = \ln|z| + i\arg(z)$Which is multi-valued because $\arg(z) = \mathrm{Arg}(z) + 2\pi k$ for $k \in \mathbb{Z}$. The principal branch is

$\mathrm{Log}\, z = \ln|z| + i\,\mathrm{Arg}(z)$

Defined on $\mathbb{C} \setminus (-\infty, 0]$. The negative real axis is called the branch cut.

Proposition 2.2. The principal branch $\mathrm{Log}\, z$ is analytic on $\mathbb{C} \setminus (-\infty, 0]$ and $\frac{d}{dz}\,\mathrm{Log}\, z = \frac{1}{z}$.

Complex Powers. For $z, \alpha \in \mathbb{C}$ with $z \neq 0$:

$z^\alpha = e^{\alpha \log z}$

This is multi-valued . When $\alpha$ is rational with reduced form $p/q$There are exactly $q$ distinct values.

Mistake 1: Assuming that differentiability in the real sense implies complex differentiability. A function $f(z) = u(x, y) + iv(x, y)$ being differentiable as a function of two real variables does not imply that $f$ is complex differentiable. The Cauchy-Riemann equations must also be satisfied. For example, $f(z) = \bar{z}$ is real-differentiable but not complex differentiable.

Mistake 2: Forgetting that analyticity requires differentiability in a neighborhood, not just at a point. A function is analytic at a point if it is differentiable in some open neighborhood of that point. A function can be differentiable at a point but not analytic there. For example, $f(z) = |z|^2$ is differentiable only at $z = 0$, so it is not analytic anywhere.

Mistake 3: Confusing the Cauchy-Riemann equations in Cartesian and polar form. In Cartesian coordinates, the Cauchy-Riemann equations are $u_x = v_y$ and $u_y = -v_x$. In polar coordinates, they are $u_r = (1/r)v_\theta$ and $v_r = -(1/r)u_\theta$. Do not mix up the two forms, as this leads to incorrect conclusions about analyticity.

Mistake 4: Assuming that a function with continuous partial derivatives satisfying the Cauchy-Riemann equations is automatically analytic. The Cauchy-Riemann equations are necessary but not sufficient for analyticity. Even if $u$ and $v$ have continuous partial derivatives and satisfy the Cauchy-Riemann equations, the function may not be analytic if the partial derivatives are not continuous in a neighborhood. However, if the partial derivatives are continuous in a neighborhood, then the function is analytic.

Mistake 5: Forgetting that analytic functions are infinitely differentiable. If $f$ is analytic on a domain $D$, then $f$ is infinitely differentiable on $D$. This is a powerful property that does not hold for real-differentiable functions. Do not assume that an analytic function has only finitely many derivatives.

flowchart TD
    A[2_Complex Functions And Analyticity] --> B[Key Concepts]
    A --> C[Core Principles]
    A --> D[Practical Applications]
    B --> E[Fundamental definitions]
    C --> F[Design patterns]
    D --> G[Real-world usage]

A complex function maps the plane to the plane, but requiring differentiability in the complex sense imposes a severe constraint. Unlike real functions that can wiggle independently in different directions, a complex differentiable function must satisfy the Cauchy-Riemann equations, linking its real and imaginary parts. This means the function preserves angles locally, making it conformal. Analytic functions are remarkably rigid: knowing the function on any small arc determines it everywhere. Branch cuts arise because multi-valued functions like logarithms require a choice of which sheet to use, creating artificial boundaries in an otherwise smooth landscape.

• Complex Numbers Review: Complex arithmetic and polar form provide the foundation for understanding complex functions.

• Cauchy-Riemann Equations: The Cauchy-Riemann equations are necessary and sufficient conditions for analyticity.

• Complex Integration: Integration of analytic functions leads to powerful results like Cauchy’s integral theorem.

• Classical Mechanics

• Electromagnetism