Complex Analysis

A complex number is $z = a + bi$ where $a, b \in \mathbb{R}$ and $i^2 = -1$. We call $a = \mathrm{Re}(z)$ the real part and $b = \mathrm{Im}(z)$ the imaginary part.

Arithmetic: $(a + bi) + (c + di) = (a + c) + (b + d)i$ and $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$.

Proposition 1.1 (Properties of Complex Arithmetic). For all $z, w \in \mathbb{C}$:

1. $z + w = w + z$ and $zw = wz$ (commutativity)
2. $(z + w) + u = z + (w + u)$ and $(zw)u = z(wu)$ (associativity)
3. $z(w + u) = zw + zu$ (distributivity)
4. There exist additive identity $0$ and multiplicative identity $1$.
5. Every $z \neq 0$ has a multiplicative inverse $\frac{1}{z} = \frac{\bar{z}}{|z|^2}$.

Remark. The complex field $\mathbb{C}$ cannot be ordered: there is no total ordering on $\mathbb{C}$ Compatible with the field operations. In particular, $i^2 = -1$ precludes any such ordering.

Definition. The complex conjugate of $z = a + bi$ is $\bar{z} = a - bi$.

Proposition 1.2. For all $z, w \in \mathbb{C}$:

1. $\overline{z + w} = \bar{z} + \bar{w}$ and $\overline{zw} = \bar{z}\bar{w}$
2. $z\bar{z} = |z|^2$
3. $z + \bar{z} = 2\,\mathrm{Re}(z)$ and $z - \bar{z} = 2i\,\mathrm{Im}(z)$
4. $\bar{\bar{z}} = z$

Definition. The modulus (or absolute value) of $z = a + bi$ is $|z| = \sqrt{a^2 + b^2}$.

Proposition 1.3 (Modulus Properties). For all $z, w \in \mathbb{C}$:

1. $|z| \geq 0$ with equality iff $z = 0$
2. $|zw| = |z||w|$
3. $|z + w| \leq |z| + |w|$ (triangle inequality)
4. $\bigl||z| - |w|\bigr| \leq |z - w|$ (reverse triangle inequality)

Proof of (3). $|z + w|^2 = (z + w)(\bar{z} + \bar{w}) = |z|^2 + z\bar{w} + \bar{z}w + |w|^2 = |z|^2 + 2\,\mathrm{Re}(z\bar{w}) + |w|^2 \leq |z|^2 + 2|z||w| + |w|^2 = (|z| + |w|)^2$. The inequality follows from $\mathrm{Re}(z\bar{w}) \leq |z\bar{w}| = |z||w|$. $\blacksquare$

Every non-zero complex number can be written in polar form:

$z = r(\cos\theta + i\sin\theta) = re^{i\theta}$

Where $r = |z| = \sqrt{a^2 + b^2}$ is the modulus and $\theta = \arg(z)$ is the argument.

Definition. The principal argument $\mathrm{Arg}(z)$ is the unique $\theta \in (-\pi, \pi]$ Such that $z = |z|e^{i\theta}$. The argument $\arg(z)$ is multi-valued: $\arg(z) = \mathrm{Arg}(z) + 2\pi k$ for $k \in \mathbb{Z}$.

Proposition 1.4. If $z_1 = r_1 e^{i\theta_1}$ and $z_2 = r_2 e^{i\theta_2}$Then $z_1 z_2 = r_1 r_2 e^{i(\theta_1 + \theta_2)}$ and $z_1/z_2 = (r_1/r_2)\, e^{i(\theta_1 - \theta_2)}$.

Euler’s formula: $e^{i\theta} = \cos\theta + i\sin\theta$.

De Moivre’s theorem: $(e^{i\theta})^n = e^{in\theta}$So

$(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$

Proposition 1.5. De Moivre’s theorem holds for all integers $n$Including negative values.

Proof. For $n \geq 0$It follows by induction from the multiplication law $e^{i\alpha}e^{i\beta} = e^{i(\alpha + \beta)}$. For $n \lt 0$Write $n = -m$ with $m \gt 0$: $(\cos\theta + i\sin\theta)^n = \frac{1}{(\cos\theta + i\sin\theta)^m} = \frac{1}{\cos(m\theta) + i\sin(m\theta)} = \cos(-m\theta) + i\sin(-m\theta) = \cos(n\theta) + i\sin(n\theta)$. $\blacksquare$

Example. Compute $(1 + i)^{20}$.

$1 + i = \sqrt{2}\,e^{i\pi/4}$So $(1 + i)^{20} = (\sqrt{2})^{20}\, e^{20\pi i/4} = 2^{10}\, e^{5\pi i} = 1024\,e^{\pi i} = -1024$.

Definition. An $n$-th root of $w \in \mathbb{C}$ is a complex number $z$ such that $z^n = w$.

Proposition 1.6. Every non-zero $w \in \mathbb{C}$ has exactly $n$ distinct $n$-th roots. If $w = \rho\, e^{i\phi}$Then

$z_k = \rho^{1/n}\, e^{i(\phi + 2\pi k)/n}, \quad k = 0, 1, \ldots, n - 1$

Where $\rho^{1/n} \gt 0$ is the positive real $n$-th root of $\rho$.

Proof. If $z^n = w$Write $z = r\,e^{i\theta}$. Then $r^n e^{in\theta} = \rho\, e^{i\phi}$ So $r = \rho^{1/n}$ and $n\theta = \phi + 2\pi k$. For $k = 0, 1, \ldots, n-1$ these give distinct Values of $\theta$; for $k \geq n$ they repeat. $\blacksquare$

Remark. The $n$-th roots of $w$ lie equally spaced on a circle of radius $\rho^{1/n}$Forming a Regular $n$-gon.

The $n$-th roots of unity are the solutions of $z^n = 1$:

$z_k = e^{2\pi i k / n}, \quad k = 0, 1, \ldots, n - 1$

They form a regular $n$-gon on the unit circle in the complex plane.

Proposition 1.7. If $\omega = e^{2\pi i/n}$ is a primitive $n$-th root of unity, then $\sum_{k=0}^{n-1} \omega^k = 0$ and $\sum_{k=0}^{n-1} \omega^{jk} = 0$ for any $j$ not divisible by $n$.

Proof. The sum $\sum_{k=0}^{n-1} \omega^k = \frac{1 - \omega^n}{1 - \omega} = \frac{1 - 1}{1 - \omega} = 0$ Provided $\omega \neq 1$. For $j$ not divisible by $n$, $\omega^j$ is a non-trivial root of unity, So the same argument applies. $\blacksquare$

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.

Theorem 3.1 (Cauchy-Riemann Equations). If $f(z) = u(x, y) + iv(x, y)$ is differentiable at $z = x + iy$Then

$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$

Proof. Compute the limit along the real axis ($h \in \mathbb{R}$, $h \to 0$):

$f'(z) = \lim_{h \to 0} \frac{u(x+h, y) - u(x, y)}{h} + i\lim_{h \to 0} \frac{v(x+h, y) - v(x, y)}{h} = \frac{\partial u}{\partial x} + i\frac{\partial v}{\partial x}$

Compute along the imaginary axis ($h = ik$, $k \in \mathbb{R}$, $k \to 0$):

$f'(z) = \lim_{k \to 0} \frac{u(x, y+k) - u(x, y)}{ik} + i\lim_{k \to 0} \frac{v(x, y+k) - v(x, y)}{ik} = -i\frac{\partial u}{\partial y} + \frac{\partial v}{\partial y}$

Equating real and imaginary parts: $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ And $\frac{\partial v}{\partial x} = -\frac{\partial u}{\partial y}$. $\blacksquare$

Theorem 3.2. If $u$ and $v$ have continuous first partial derivatives on an open set $U$ and Satisfy the Cauchy-Riemann equations on $U$Then $f = u + iv$ is analytic on $U$.

Proof. Since $u_x, u_y, v_x, v_y$ are continuous on $U$, $u$ and $v$ are (real) differentiable. Let $\Delta z = \Delta x + i\Delta y$. By real differentiability:

$u(x + \Delta x, y + \Delta y) - u(x, y) = u_x\,\Delta x + u_y\,\Delta y + \varepsilon_1$ $v(x + \Delta x, y + \Delta y) - v(x, y) = v_x\,\Delta x + v_y\,\Delta y + \varepsilon_2$

Where $\varepsilon_1, \varepsilon_2 = o(|\Delta z|)$. Therefore

$\frac{f(z + \Delta z) - f(z)}{\Delta z} = \frac{(u_x + iv_x)\Delta x + (u_y + iv_y)\Delta y + \varepsilon_1 + i\varepsilon_2}{\Delta x + i\Delta y}$

By CR: $u_y + iv_y = -v_x + iu_x = i(u_x + iv_x)$. Substituting:

$= (u_x + iv_x)\frac{\Delta x + i\Delta y}{\Delta x + i\Delta y} + \frac{o(|\Delta z|)}{\Delta z} \to u_x + iv_x$

As $\Delta z \to 0$. $\blacksquare$

When the Cauchy-Riemann equations hold:

$f'(z) = \frac{\partial u}{\partial x} + i\frac{\partial v}{\partial x} = \frac{\partial v}{\partial y} - i\frac{\partial u}{\partial y}$

Definition. A real-valued function $\phi(x, y)$ is harmonic if $\phi_{xx} + \phi_{yy} = 0$ (Laplace’s equation).

Proposition 3.3. If $f = u + iv$ is analytic, then $u$ and $v$ are harmonic.

Proof. From the Cauchy-Riemann equations: $u_x = v_y$ and $u_y = -v_x$. Differentiating: $u_{xx} = v_{yx}$ and $u_{yy} = -v_{xy}$. By equality of mixed partials, $u_{xx} + u_{yy} = v_{yx} - v_{xy} = 0$. Similarly for $v$. $\blacksquare$

Definition. If $u$ and $v$ are harmonic on $U$ and satisfy the Cauchy-Riemann equations, then $v$ is the harmonic conjugate of $u$.

Proposition 3.4. If $U$ is a connected domain and $u$ is harmonic on $U$Then $u$ has A harmonic conjugate on $U$Unique up to an additive constant.

Proof. Define $v(x, y) = \int_{(x_0, y_0)}^{(x, y)} (-u_y\, dx + u_x\, dy)$. The integrand is closed (since $(-u_y)_y = -u_{yy} = u_{xx} = (u_x)_x$) and since $U$ is Connected, $v$ is well-defined (path-independent) by Green’s theorem. Then $v_x = -u_y$ and $v_y = u_x$Which are the CR equations. $\blacksquare$

A contour (or piecewise smooth path) in $\mathbb{C}$ is a continuous function $\gamma : [a, b] \to \mathbb{C}$ that is differentiable except at finitely many points, with a Continuous derivative everywhere it exists.

A simple closed contour is a contour with $\gamma(a) = \gamma(b)$ and no other Self-intersections.

Definition. For a contour $\gamma$ and a continuous function $f$ on $\gamma$:

$\int_{\gamma} f(z)\, dz = \int_a^b f(\gamma(t))\gamma'(t)\, dt$