Oxford MATH203 - Complex Analysis Flashcards

Question 1

Euler's Formula & Polar Representation

Accepted Answer

Any complex number $z = x + iy$ can be written in polar form as $z = r e^{i	heta} = r(\cos	heta + i\sin	heta)$, where $r = |z| = \sqrt{x^2 + y^2}$ is the modulus and $	heta = \arg(z) = \arctan(y/x)$ is the argument.

Question 2

De Moivre's Theorem

Accepted Answer

For any real number $n$ and angle $	heta$:
$$[r(\cos	heta + i\sin	heta)]^n = r^n (\cos(n	heta) + i\sin(n	heta)) = r^n e^{in	heta}$$

Question 3

Analytic (Holomorphic) Function

Accepted Answer

A complex function $f(z)$ is **analytic** (or **holomorphic**) at a point $z_0$ if it is complex-differentiable at every point in some open neighborhood around $z_0$.

Question 4

Cauchy-Riemann Equations (Cartesian)

Accepted Answer

For $f(z) = u(x,y) + i v(x,y)$ to be analytic at $z = x + iy$, $u$ and $v$ must have continuous partial derivatives satisfying:
$$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad 	ext{and} \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$$

Question 5

Cauchy-Riemann Equations (Polar Form)

Accepted Answer

For $f(z) = u(r,	heta) + i v(r,	heta)$, the Cauchy-Riemann equations in polar coordinates are:
$$\frac{\partial u}{\partial r} = \frac{1}{r} \frac{\partial v}{\partial 	heta} \quad 	ext{and} \quad \frac{\partial v}{\partial r} = -\frac{1}{r} \frac{\partial u}{\partial 	heta}$$

Question 6

Harmonic Functions & Conjugates

Accepted Answer

If $f(z) = u + iv$ is analytic, both real component $u$ and imaginary component $v$ satisfy Laplace's equation $
abla^2 u = u_{xx} + u_{yy} = 0$. Here, $v$ is called the **harmonic conjugate** of $u$.

Question 7

Complex Contour Integration

Accepted Answer

The integral of $f(z)$ along a smooth parameterized curve $\gamma(t)$ for $a \le t \le b$ is given by:
$$\int_\gamma f(z) dz = \int_a^b f(\gamma(t)) \gamma'(t) dt$$

Question 8

Cauchy's Integral Theorem

Accepted Answer

If $f(z)$ is analytic inside and on a simply connected region bounded by a simple closed contour $C$, then:
$$\oint_C f(z) dz = 0$$

Question 9

Cauchy's Integral Formula

Accepted Answer

If $f(z)$ is analytic inside and on a positively oriented simple closed contour $C$, and $z_0$ lies in the interior of $C$:
$$f(z_0) = \frac{1}{2\pi i} \oint_C \frac{f(z)}{z - z_0} dz$$

Question 10

Cauchy's Integral Formula for Derivatives

Accepted Answer

Extending Cauchy's formula, the $n$-th derivative of an analytic function $f(z)$ at $z_0$ is:
$$f^{(n)}(z_0) = \frac{n!}{2\pi i} \oint_C \frac{f(z)}{(z - z_0)^{n+1}} dz$$

Question 11

Liouville's Theorem

Accepted Answer

Every **entire** function (analytic on the entire complex plane $\mathbb{C}$) that is **bounded** across $\mathbb{C}$ must be constant.

Question 12

Fundamental Theorem of Algebra

Accepted Answer

Every non-constant polynomial $P(z) = a_n z^n + a_{n-1} z^{n-1} + \dots + a_0$ with complex coefficients has at least one complex root in $\mathbb{C}$, and exactly $n$ roots counted with multiplicity.

Question 13

Complex Taylor Series

Accepted Answer

If $f(z)$ is analytic in a disk $|z - z_0| < R$, it can be represented uniquely as a power series:
$$f(z) = \sum_{n=0}^{\infty} a_n (z - z_0)^n \quad 	ext{where} \quad a_n = \frac{f^{(n)}(z_0)}{n!}$$

Question 14

Laurent Series

Accepted Answer

For $f(z)$ analytic in an annulus $r < |z - z_0| < R$, $f(z)$ expands to:
$$f(z) = \sum_{n=-\infty}^{\infty} a_n (z - z_0)^n = \sum_{n=0}^{\infty} a_n (z - z_0)^n + \sum_{n=1}^{\infty} \frac{b_n}{(z - z_0)^n}$$

Question 15

Classification of Isolated Singularities

Accepted Answer

Based on the principal part $\sum_{n=1}^\infty \frac{b_n}{(z-z_0)^n}$ of the Laurent series:
1. **Removable Singularity**: $b_n = 0$ for all $n \ge 1$.
2. **Pole of Order $m$**: $b_m 
eq 0$ and $b_n = 0$ for all $n > m$.
3. **Essential Singularity**: Infinitely many non-zero $b_n$ terms.

Question 16

Residue Definition

Accepted Answer

The **residue** of $f(z)$ at an isolated singularity $z_0$ is the coefficient $a_{-1}$ (or $b_1$) of the $\frac{1}{z - z_0}$ term in its Laurent expansion, written $	ext{Res}(f, z_0)$.

Question 17

Residue Formula for Poles

Accepted Answer

For a **simple pole** ($m=1$):
$$	ext{Res}(f, z_0) = \lim_{z 	o z_0} (z - z_0)f(z)$$
For a **pole of order $m$**:
$$	ext{Res}(f, z_0) = \frac{1}{(m-1)!} \lim_{z 	o z_0} \frac{d^{m-1}}{dz^{m-1}} \left[ (z - z_0)^m f(z) ight]$$

Question 18

Cauchy's Residue Theorem

Accepted Answer

If $f(z)$ is analytic inside and on a positively oriented simple closed contour $C$, except for a finite set of isolated singularities $z_1, z_2, \dots, z_k$ inside $C$:
$$\oint_C f(z) dz = 2\pi i \sum_{j=1}^k 	ext{Res}(f, z_j)$$

Question 19

Jordan's Lemma

Accepted Answer

If $f(z) 	o 0$ uniformly as $|z| 	o \infty$ on a semicircular arc $C_R = \{R e^{i	heta} : 0 \le 	heta \le \pi\}$, then for $a > 0$:
$$\lim_{R 	o \infty} \int_{C_R} f(z) e^{iaz} dz = 0$$

Question 20

Real Integral Evaluation via Residues: $\int_{-\infty}^\infty \frac{dx}{x^2+1}$

Accepted Answer

Consider $f(z) = \frac{1}{z^2+1}$ over a upper-half plane semicircular contour. $f(z)$ has a simple pole at $z = i$ with residue $	ext{Res}(f, i) = \lim_{z 	o i} \frac{1}{z+i} = \frac{1}{2i}$. Applying Residue Theorem:
$$\int_{-\infty}^\infty \frac{dx}{x^2+1} = 2\pi i \left(\frac{1}{2i}ight) = \pi$$

Question 21

Rouché's Theorem

Accepted Answer

If $f(z)$ and $g(z)$ are analytic inside and on a simple closed contour $C$, and $|g(z)| < |f(z)|$ at every point on $C$, then $f(z)$ and $f(z) + g(z)$ have the exact same number of zeros inside $C$.

Oxford MATH203 - Complex Analysis

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