Complex Analysis
Analytic functions, contour integration, residue theory, conformal mappings, and the deep connections between complex differentiability and geometry at the CSIR NET level.
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01
Analytic Functions & Cauchy-Riemann Equations
Complex differentiability, the Cauchy-Riemann equations, harmonic functions, and elementary analytic functions.
Complex Differentiability
Definition
A function $f: \Omega \to \mathbb{C}$ is analytic (holomorphic) at $z_0$ if $f'(z_0) = \lim_{z \to z_0} \frac{f(z) - f(z_0)}{z - z_0}$ exists. Analyticity on a domain means analyticity at every point.
Cauchy-Riemann Equations
Writing $f(x + iy) = u(x,y) + iv(x,y)$, $f$ is analytic at $z_0$ if and only if $u, v$ have continuous partial derivatives satisfying $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$ at $z_0$.
A consequence: if $f$ is analytic, then $u$ and $v$ are harmonic: $\nabla^2 u = \nabla^2 v = 0$. Conversely, every harmonic function on a simply connected domain is the real part of an analytic function.
📝 Example
Show that $f(z) = \bar{z}$ is not analytic anywhere.
$f(x+iy) = x - iy$, so $u = x$, $v = -y$. Then $u_x = 1$ but $v_y = -1$. Since $u_x \neq v_y$, the Cauchy-Riemann equations fail everywhere. Hence $f(z) = \bar{z}$ is nowhere analytic.
02
Power Series & Radius of Convergence
Taylor series of analytic functions, radius of convergence, and identity theorem.
Power Series Representation
Taylor's Theorem for Analytic Functions
If $f$ is analytic on a disk $D(z_0, R)$, then $f(z) = \sum_{n=0}^{\infty} a_n (z - z_0)^n$ with $a_n = \frac{f^{(n)}(z_0)}{n!} = \frac{1}{2\pi i} \oint_\gamma \frac{f(\zeta)}{(\zeta - z_0)^{n+1}} d\zeta$. The series converges uniformly on compact subsets.
Radius of Convergence
The radius of convergence is $R = \frac{1}{\limsup_{n\to\infty} |a_n|^{1/n}}$ (Cauchy-Hadamard formula). $R$ equals the distance from $z_0$ to the nearest singularity of $f$.
Identity Theorem
If $f$ and $g$ are analytic on a connected domain $\Omega$ and $f = g$ on a set with a limit point in $\Omega$, then $f \equiv g$ on $\Omega$.
📝 Example
Find the radius of convergence of $\sum_{n=0}^{\infty} \frac{z^n}{n!}$.
$|a_n|^{1/n} = (1/n!)^{1/n} \to 0$ (by Stirling's approximation). So $R = 1/0 = \infty$. This is the Taylor series of $e^z$, which is entire (analytic on all of $\mathbb{C}$).
03
Cauchy's Theorem & Integral Formula
Contour integration, Cauchy-Goursat theorem, Cauchy integral formula, and consequences including Liouville's theorem.
Cauchy's Integral Theorem & Formula
Cauchy-Goursat Theorem
If $f$ is analytic on and inside a simple closed contour $\gamma$, then $\oint_\gamma f(z) \, dz = 0$.
Cauchy Integral Formula
If $f$ is analytic inside and on a simple closed contour $\gamma$ and $z_0$ is inside $\gamma$, then $f(z_0) = \frac{1}{2\pi i} \oint_\gamma \frac{f(z)}{z - z_0} \, dz$. More generally, $f^{(n)}(z_0) = \frac{n!}{2\pi i} \oint_\gamma \frac{f(z)}{(z - z_0)^{n+1}} \, dz$.
Liouville's Theorem
Every bounded entire function is constant. Proof: by Cauchy's estimate, $|f'(z_0)| \leq \frac{M}{R}$ for any $R > 0$. Letting $R \to \infty$ gives $f' \equiv 0$.
Maximum Modulus Principle
If $f$ is analytic and non-constant on a domain $\Omega$, then $|f|$ has no local maximum in $\Omega$. In particular, $\max_{\bar{\Omega}} |f|$ is attained on $\partial \Omega$.
📝 Example
Evaluate $\oint_{|z|=2} \frac{e^z}{z(z-1)} \, dz$.
Both $z = 0$ and $z = 1$ are inside $|z| = 2$. By partial fractions: $\frac{e^z}{z(z-1)} = \frac{-e^z}{z} + \frac{e^z}{z-1}$. By Cauchy's formula: $\oint \frac{-e^z}{z} dz = -2\pi i \cdot e^0 = -2\pi i$ and $\oint \frac{e^z}{z-1} dz = 2\pi i \cdot e^1 = 2\pi i e$. Total: $2\pi i(e - 1)$.
04
Laurent Series & Singularities
Laurent expansions, classification of isolated singularities, and behavior near singular points.
Laurent Series
Laurent's Theorem
If $f$ is analytic in the annulus $r < |z - z_0| < R$, then $f(z) = \sum_{n=-\infty}^{\infty} a_n (z - z_0)^n$, where $a_n = \frac{1}{2\pi i} \oint_\gamma \frac{f(\zeta)}{(\zeta - z_0)^{n+1}} d\zeta$. The sum $\sum_{n=-1}^{-\infty} a_n(z-z_0)^n$ is the principal part.
Classification of Singularities
An isolated singularity $z_0$ of $f$ is:
- Removable if the principal part is zero ($a_n = 0$ for all $n < 0$)
- A pole of order $m$ if $a_{-m} \neq 0$ and $a_n = 0$ for $n < -m$
- An essential singularity if infinitely many $a_n \neq 0$ for $n < 0$
Casorati-Weierstrass Theorem
If $z_0$ is an essential singularity of $f$, then for every $w \in \mathbb{C}$ and every $\varepsilon, \delta > 0$, there exists $z$ with $0 < |z - z_0| < \delta$ and $|f(z) - w| < \varepsilon$. (The stronger Picard theorem: $f$ takes every value in $\mathbb{C}$ with at most one exception.)
📝 Example
Classify the singularity of $f(z) = e^{1/z}$ at $z = 0$.
$e^{1/z} = \sum_{n=0}^{\infty} \frac{1}{n! z^n} = 1 + \frac{1}{z} + \frac{1}{2z^2} + \cdots$. The principal part has infinitely many nonzero terms, so $z = 0$ is an essential singularity.
05
Residue Calculus & Applications
The residue theorem, computation of real integrals, argument principle, and Rouche's theorem.
Residue Theorem
Residue Theorem
If $f$ is analytic inside a simple closed contour $\gamma$ except at isolated singularities $z_1, \ldots, z_k$ inside $\gamma$, then $\oint_\gamma f(z) \, dz = 2\pi i \sum_{j=1}^{k} \text{Res}(f, z_j)$.
Computing Residues
- Simple pole: $\text{Res}(f, z_0) = \lim_{z \to z_0} (z - z_0) f(z)$
- Pole of order $m$: $\text{Res}(f, z_0) = \frac{1}{(m-1)!} \lim_{z \to z_0} \frac{d^{m-1}}{dz^{m-1}} [(z-z_0)^m f(z)]$
- Quotient form: If $f = g/h$ with $g(z_0) \neq 0$, $h(z_0) = 0$, $h'(z_0) \neq 0$: $\text{Res}(f, z_0) = \frac{g(z_0)}{h'(z_0)}$
Argument Principle & Rouche's Theorem
Argument Principle
If $f$ is meromorphic inside a simple closed contour $\gamma$ with $Z$ zeros and $P$ poles (counted with multiplicity), and $f \neq 0$ on $\gamma$, then $\frac{1}{2\pi i} \oint_\gamma \frac{f'(z)}{f(z)} \, dz = Z - P$.
Rouche's Theorem
If $f$ and $g$ are analytic inside and on a simple closed contour $\gamma$, and $|g(z)| < |f(z)|$ on $\gamma$, then $f$ and $f + g$ have the same number of zeros inside $\gamma$.
📝 Example
How many roots does $p(z) = z^5 + 3z + 1$ have inside $|z| < 1$?
On $|z| = 1$: $|3z| = 3$ and $|z^5 + 1| \leq |z|^5 + 1 = 2 < 3$. By Rouche's theorem with $f(z) = 3z$ and $g(z) = z^5 + 1$: $p(z) = f(z) + g(z)$ has the same number of zeros as $f(z) = 3z$ inside $|z| < 1$, which is 1.
Evaluation of Real Integrals
📝 Example
Evaluate $\int_0^{\infty} \frac{1}{x^4 + 1} \, dx$.
Consider $\oint_\gamma \frac{dz}{z^4+1}$ over a semicircular contour in the upper half-plane. The poles in the upper half-plane are $z_1 = e^{i\pi/4}$ and $z_2 = e^{3i\pi/4}$. Computing residues: $\text{Res}(f, z_k) = \frac{1}{4z_k^3} = \frac{z_k}{4z_k^4} = \frac{z_k}{-4}$. Summing and using $2\pi i$ gives $\int_{-\infty}^{\infty} \frac{dx}{x^4+1} = \frac{\pi}{\sqrt{2}}$, so $\int_0^{\infty} = \frac{\pi}{2\sqrt{2}}$.
06
Conformal Mappings & Riemann Mapping Theorem
Conformal maps, Mobius transformations, and the fundamental existence theorem for biholomorphic maps.
Conformal Mappings
Definition
A conformal mapping is an analytic function with nonzero derivative. It preserves angles and orientation. Equivalently, a holomorphic bijection (biholomorphism) between domains.
Mobius Transformations
$T(z) = \frac{az + b}{cz + d}$ with $ad - bc \neq 0$ maps the extended plane $\hat{\mathbb{C}}$ to itself biholomorphically. They map circles/lines to circles/lines, and are determined by the images of three points.
Riemann Mapping Theorem
Every simply connected domain $\Omega \subsetneq \mathbb{C}$ is biholomorphic to the open unit disk $\mathbb{D}$. The biholomorphism is unique up to the choice of a point in $\Omega$ mapping to $0$ and a rotation.
Schwarz Lemma
If $f: \mathbb{D} \to \mathbb{D}$ is analytic with $f(0) = 0$, then $|f(z)| \leq |z|$ and $|f'(0)| \leq 1$. Equality holds iff $f(z) = e^{i\theta} z$ for some $\theta$.
📝 Example
Find a conformal map from the upper half-plane $\mathbb{H} = \{z : \text{Im}(z) > 0\}$ to the unit disk $\mathbb{D}$.
The Cayley transform $w = \frac{z - i}{z + i}$ maps $\mathbb{H}$ to $\mathbb{D}$. Verification: for $z \in \mathbb{R}$, $|w| = \frac{|z - i|}{|z + i|} = 1$ (boundary maps to boundary). For $z = i$: $w = 0$. Since it is a Mobius transformation, it is conformal.
★ Key Takeaways
- Complex differentiability (Cauchy-Riemann) is far stronger than real differentiability: analytic functions are infinitely differentiable
- Cauchy's integral formula is the engine of complex analysis: it gives Taylor coefficients, Liouville's theorem, and the maximum principle
- Laurent series classify singularities; the residue (coefficient $a_{-1}$) is the key to contour integration
- Rouche's theorem is the main tool for counting zeros: compare with a simpler function on the boundary
- Every simply connected proper subdomain of $\mathbb{C}$ is biholomorphic to the disk (Riemann mapping theorem)
- Schwarz lemma and its generalizations (Schwarz-Pick) are central to hyperbolic geometry and automorphism groups
✍ Practice Problems
Problem 1
If $u(x,y) = x^3 - 3xy^2 + 2x$, find a harmonic conjugate $v$ and the corresponding analytic function $f(z)$.
Show Solution ▼
$u_x = 3x^2 - 3y^2 + 2 = v_y$, so $v = 3x^2 y - y^3 + 2y + \phi(x)$. $u_y = -6xy = -v_x = -(6xy + \phi'(x))$, giving $\phi'(x) = 0$, so $\phi = C$. Take $v = 3x^2 y - y^3 + 2y$. Then $f(z) = u + iv = z^3 + 2z$ (verify: $(x+iy)^3 + 2(x+iy) = x^3 - 3xy^2 + 2x + i(3x^2 y - y^3 + 2y)$).
Problem 2
Evaluate $\oint_{|z|=3} \frac{z^2}{(z-1)(z-2)^2} \, dz$ using residues.
Show Solution ▼
Singularities inside $|z|=3$: simple pole at $z=1$, double pole at $z=2$. $\text{Res}(f,1) = \lim_{z\to 1} (z-1)\frac{z^2}{(z-1)(z-2)^2} = \frac{1}{1} = 1$. $\text{Res}(f,2) = \lim_{z\to 2} \frac{d}{dz}\left[\frac{z^2}{z-1}\right] = \lim_{z\to 2} \frac{2z(z-1)-z^2}{(z-1)^2} = \frac{4-4}{1} = 0$. Wait: $\frac{d}{dz}\left[\frac{z^2}{z-1}\right] = \frac{z^2-2z}{(z-1)^2}$. At $z=2$: $\frac{4-4}{1} = 0$. So $\oint = 2\pi i(1 + 0) = 2\pi i$.
Problem 3
Find the number of zeros of $f(z) = z^7 - 5z^3 + 12$ in $|z| < 1$.
Show Solution ▼
On $|z| = 1$: $|12| = 12$ and $|z^7 - 5z^3| \leq 1 + 5 = 6 < 12$. By Rouche's theorem with $f_1(z) = 12$ and $g(z) = z^7 - 5z^3$: $f(z) = f_1(z) + g(z)$ has the same number of zeros as $f_1(z) = 12$ inside $|z| < 1$, which is 0.
Problem 4
Let $f$ be entire with $|f(z)| \leq 3|z|^2 + 7$ for all $z$. Prove $f$ is a polynomial of degree at most $2$.
Show Solution ▼
By Cauchy's inequality: $|f^{(n)}(0)| \leq \frac{n!}{R^n} \max_{|z|=R} |f(z)| \leq \frac{n!}{R^n}(3R^2 + 7)$. For $n \geq 3$: $|f^{(n)}(0)| \leq n! \cdot \frac{3R^2 + 7}{R^n} \to 0$ as $R \to \infty$. So $f^{(n)}(0) = 0$ for all $n \geq 3$, meaning $f(z) = a_0 + a_1 z + a_2 z^2$.
Problem 5
Evaluate $\int_0^{2\pi} \frac{d\theta}{5 + 4\cos\theta}$.
Show Solution ▼
Let $z = e^{i\theta}$, $d\theta = dz/(iz)$, $\cos\theta = (z + z^{-1})/2$. The integral becomes $\oint_{|z|=1} \frac{1}{5 + 2(z + z^{-1})} \frac{dz}{iz} = \oint \frac{dz}{i(2z^2 + 5z + 2)} = \oint \frac{dz}{i(2z+1)(z+2)}$. The pole inside $|z|=1$ is $z = -1/2$. $\text{Res} = \frac{1}{i \cdot 2 \cdot (-1/2 + 2)} = \frac{1}{3i}$. Integral $= 2\pi i \cdot \frac{1}{3i} = \frac{2\pi}{3}$.
🎯 Test Your Understanding
1. If $f$ is entire and $\text{Re}(f) \leq M$ for some constant $M$, then $f$ is:
A Constant
B A polynomial of degree $\leq 1$
C Bounded
D Cannot be determined
2. The function $f(z) = \frac{\sin z}{z}$ has at $z = 0$:
A A removable singularity
B A simple pole
C An essential singularity
D Not a singularity
3. The residue of $\frac{e^z}{z^3}$ at $z = 0$ is:
A $1$
B $1/2$
C $1/6$
D $0$
4. The number of zeros of $z^4 - 6z + 3$ in $1 < |z| < 2$ is:
A 0
B 2
C 3
D 4
5. Which of the following is a conformal map from the unit disk to itself?
A $f(z) = z^2$
B $f(z) = e^z$
C $f(z) = \frac{z - a}{1 - \bar{a}z}$ for $|a| < 1$
D $f(z) = 2z$