12.1 Complex Numbers and Analytic Functions

12.1 Complex Numbers and Analytic Functions

Complex analysis on the GRE Mathematics Subject Test balances geometric intuition in the Argand plane with differential analysis. Complex differentiability imposes extreme rigidity: a complex differentiable function is infinitely differentiable, analytic, and locally conformal.

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Complex Numbers, Polar Form, and Roots of Unity

A complex number $z \in \mathbb{C}$ is expressed as $z = x + iy$ with $x = \operatorname{Re}(z), y = \operatorname{Im}(z) \in \mathbb{R}$. In polar form: $z = r(\cos\theta + i\sin\theta) = r e^{i\theta}$ where $r = |z| = \sqrt{x^2 + y^2}$ is the modulus and $\theta = \operatorname{Arg}(z) \in (-\pi, \pi]$ is the principal argument. De Moivre's formula yields $(e^{i\theta})^n = e^{in\theta} = \cos(n\theta) + i\sin(n\theta)$.

Roots of Unity

The equation $z^n = 1$ has $n$ distinct roots of unity in $\mathbb{C}$: $\omega_k = e^{2\pi i k / n}, \quad k = 0, 1, \dots, n-1$ These form the vertices of a regular $n$-gon inscribed in $|z| = 1$. The sum of all $n$-th roots vanishes for $n \ge 2$: $\sum_{k=0}^{n-1} \omega_k = 0$. For any $w = R e^{i\phi} \neq 0$, the $n$ roots of $z^n = w$ are $z_k = R^{1/n} e^{i(\phi + 2\pi k)/n}$.

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Complex Differentiability and Cauchy-Riemann Equations

Let $U \subseteq \mathbb{C}$ be open. The complex derivative of $f$ at $z_0 \in U$ is: $f'(z_0) = \lim_{\Delta z \to 0} \frac{f(z_0 + \Delta z) - f(z_0)}{\Delta z}$ Because $\Delta z \to 0$ along arbitrary paths in $\mathbb{C}$, complex differentiability requires consistency across all directions.

Cartesian Cauchy-Riemann Equations

Writing $f(z) = u(x, y) + i v(x, y)$:

• Necessary & Sufficient: If $u, v$ have continuous first partial derivatives satisfying: $u_x = v_y \quad \text{and} \quad u_y = -v_x$ then $f(z)$ is analytic. The derivative is computed via: $f'(z) = u_x + i v_x = v_y - i u_y = u_x - i u_y$

Polar Cauchy-Riemann Equations

In polar coordinates $z = r e^{i\theta}$ with $f(z) = u(r, \theta) + i v(r, \theta)$: $u_r = \frac{1}{r} v_\theta \quad \text{and} \quad v_r = -\frac{1}{r} u_\theta$ with derivative $f'(z) = e^{-i\theta}(u_r + i v_r) = \frac{e^{-i\theta}}{r}(v_\theta - i u_\theta)$.

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Harmonic Functions and Conjugates

A function $u(x, y)$ is harmonic on $D$ if it satisfies Laplace's equation: $\nabla^2 u = u_{xx} + u_{yy} = 0$

• Analyticity Implies Harmonicity: If $f = u + iv$ is analytic, Clairaut's theorem gives $u_{xx} + u_{yy} = (v_y)_x + (-v_x)_y = 0$.
• Finding Conjugates: Given harmonic $u$, integrate $v_y = u_x$ with respect to $y$, then substitute into $v_x = -u_y$ to solve for the remaining function of $x$.

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Entire Functions and Rigidity

A function analytic on all of $\mathbb{C}$ is entire. Analytic functions exhibit strong rigidity on connected domains:

• If $\operatorname{Re}(f)$, $\operatorname{Im}(f)$, or $|f(z)|$ is constant, then $f(z)$ is constant.
• If $\overline{f(z)}$ is analytic, then $f(z)$ is constant.
• If $f'(z) = 0$ on a connected domain, $f(z)$ is constant.

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Conformal Mapping Property

A mapping $w = f(z)$ is conformal at $z_0$ if it preserves both the magnitude and orientation of angles between intersecting smooth curves.

• Condition: $f$ is conformal at $z_0$ if and only if $f$ is analytic and $f'(z_0) \neq 0$.
• Jacobian: $\det J_f = u_x v_y - u_y v_x = u_x^2 + u_y^2 = |f'(z)|^2 > 0$.
• Critical Points: Where $f'(z) = 0$, conformality fails and angles are multiplied by the local branching order $m$.

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Comparison: Real vs. Complex Differentiability

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Step-by-Step Worked Problems

Problem 1: Reconstructing an Analytic Function

Find the harmonic conjugate $v(x, y)$ of $u(x, y) = 2x - 2xy$ satisfying $v(0, 0) = 3$.

Solution:

1. Check harmonicity: $u_{xx} + u_{yy} = 0 + 0 = 0$.
2. Integrate $v_y = u_x = 2 - 2y$: $v(x, y) = 2y - y^2 + g(x)$.
3. Differentiate and equate to $-u_y$: $v_x = g'(x) = -(-2x) = 2x \implies g(x) = x^2 + C$.
4. Use $v(0, 0) = 3$ to find $C = 3$: $v(x, y) = x^2 - y^2 + 2y + 3$.
5. Reconstruct $f(z) = u + iv = i z^2 + 2z + 3i$.

Problem 2: Polar Cauchy-Riemann

Verify analyticity of $f(z) = z^2 = r^2 e^{2i\theta}$ using polar coordinates.

Solution:

1. $u = r^2 \cos(2\theta)$, $v = r^2 \sin(2\theta)$.
2. $u_r = 2r \cos(2\theta)$ and $\frac{1}{r} v_\theta = 2r \cos(2\theta)$, so $u_r = \frac{1}{r} v_\theta$.
3. $v_r = 2r \sin(2\theta)$ and $-\frac{1}{r} u_\theta = 2r \sin(2\theta)$, so $v_r = -\frac{1}{r} u_\theta$.
4. C-R holds with continuous partials; $f'(z) = e^{-i\theta}(u_r + i v_r) = 2r e^{i\theta} = 2z$.

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GRE Exam Traps & Pitfalls

Trap 1: The Cauchy-Riemann Sign Error Remember $u_y = -v_x$. The negative sign belongs to $u_y = -v_x$, while $u_x = v_y$ is positive.

Trap 2: Harmonic Conjugate Direction If $v$ is conjugate to $u$, then $u$ is not conjugate to $v$. The conjugate of $v$ is $-u$, since $v - iu = -i(u + iv)$.

Trap 3: Conformality Breakdown Conformality fails at critical points where $f'(z) = 0$. For $f(z) = z^2$, angles at the origin are doubled.