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Complex Analysis

Master analytic functions, contour integration, series representations, residue calculus, and conformal mappings for GATE Mathematics.

Analytic Functions Complex Integration Series Representations Residue Calculus Conformal Mappings

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Analytic Functions

The theory of complex differentiable functions and the remarkable consequences of the Cauchy-Riemann equations.

◆ Complex Differentiability

Definition — Analytic Function A function $f: \mathbb{C} \to \mathbb{C}$ 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. If $f$ is analytic at every point of a domain $D$, we say $f$ is analytic on $D$.

Write $f(z) = u(x,y) + iv(x,y)$ where $z = x + iy$. The complex derivative exists if and only if the Cauchy-Riemann equations hold:

$\dfrac{\partial u}{\partial x} = \dfrac{\partial v}{\partial y} \quad \text{and} \quad \dfrac{\partial u}{\partial y} = -\dfrac{\partial v}{\partial x}$

and the partial derivatives are continuous.

◆ Harmonic Functions

If $f = u + iv$ is analytic, then both $u$ and $v$ are harmonic (satisfy Laplace's equation $\nabla^2 u = 0$). Given a harmonic function $u$, the function $v$ satisfying the Cauchy-Riemann equations is called its harmonic conjugate.

★ Example

Verify that $f(z) = z^2$ is analytic and find its derivative.

Step 1: Write $f(z) = (x+iy)^2 = (x^2 - y^2) + i(2xy)$. So $u = x^2 - y^2$ and $v = 2xy$.

Step 2: Check Cauchy-Riemann: $u_x = 2x = v_y = 2x$ ✓ and $u_y = -2y = -v_x = -2y$ ✓.

Step 3: Since the C-R equations hold and the partials are continuous, $f$ is analytic everywhere (i.e., $f$ is entire).

Step 4: $f'(z) = u_x + iv_x = 2x + i(2y) = 2(x+iy) = 2z$.

★ Example

Show that $f(z) = \bar{z}$ (complex conjugate) is not analytic.

$f(z) = x - iy$, so $u = x$, $v = -y$. Then $u_x = 1$ but $v_y = -1$. Since $u_x \neq v_y$, the Cauchy-Riemann equations fail and $f$ is nowhere analytic.

★ Key Takeaways — Analytic Functions

Analytic functions are infinitely differentiable (unlike real differentiable functions)
The Cauchy-Riemann equations are the primary tool for checking analyticity
Common entire functions: polynomials, $e^z$, $\sin z$, $\cos z$

Complex Integration

Contour integration and the powerful theorems of Cauchy that form the core of complex analysis.

◆ Cauchy's Theorem and Integral Formula

Theorem — Cauchy's Integral Theorem If $f$ is analytic on and inside a simple closed contour $C$, then $\displaystyle\oint_C f(z)\,dz = 0$.

Theorem — Cauchy's Integral Formula If $f$ is analytic inside and on a simple closed contour $C$ and $z_0$ is inside $C$, then $$f(z_0) = \frac{1}{2\pi i}\oint_C \frac{f(z)}{z - z_0}\,dz$$ More generally: $f^{(n)}(z_0) = \frac{n!}{2\pi i}\oint_C \frac{f(z)}{(z - z_0)^{n+1}}\,dz$.

★ Example

Evaluate $\displaystyle\oint_{|z|=2} \frac{e^z}{z-1}\,dz$.

Step 1: The function $f(z) = e^z$ is entire, and $z_0 = 1$ lies inside $|z| = 2$.

Step 2: By Cauchy's integral formula: $\displaystyle\oint_{|z|=2} \frac{e^z}{z - 1}\,dz = 2\pi i \cdot f(1) = 2\pi i \cdot e$.

◆ Liouville's Theorem and Maximum Modulus Principle

Theorem — Liouville's Theorem Every bounded entire function is constant. (Consequence: the Fundamental Theorem of Algebra — every non-constant polynomial has a root in $\mathbb{C}$.)

Theorem — Maximum Modulus Principle If $f$ is analytic and non-constant on a domain $D$, then $|f|$ has no local maximum in $D$. On a closed bounded region, $|f|$ attains its maximum on the boundary.

★ Key Takeaways — Complex Integration

Cauchy's integral formula is the single most powerful tool in complex analysis
Analytic functions are determined by their boundary values (via the integral formula)
Liouville's theorem is the key to proving the Fundamental Theorem of Algebra

Series Representations

Taylor and Laurent series provide local representations of analytic functions, and the nature of singularities determines the behaviour of the function.

◆ Taylor and Laurent Series

If $f$ is analytic in a disk $|z - z_0| < R$, it has a Taylor series: $f(z) = \sum_{n=0}^{\infty} a_n (z - z_0)^n$ where $a_n = \frac{f^{(n)}(z_0)}{n!}$.

If $f$ is analytic in an annulus $r < |z - z_0| < R$, it has a Laurent series: $f(z) = \sum_{n=-\infty}^{\infty} a_n (z - z_0)^n$. The terms with $n < 0$ form the principal part.

Definition — Classification of Singularities An isolated singularity $z_0$ of $f$ is:

Removable if the principal part is empty (all $a_n = 0$ for $n < 0$)
A pole of order $m$ if $a_{-m} \neq 0$ and $a_n = 0$ for $n < -m$
Essential if infinitely many $a_n \neq 0$ for $n < 0$

◆ Computing Laurent Series

★ Example

Find the Laurent series of $f(z) = \dfrac{1}{z(z-1)}$ in the annulus $0 < |z| < 1$.

Step 1: Use partial fractions: $\dfrac{1}{z(z-1)} = \dfrac{-1}{z} + \dfrac{1}{z-1}$.

Step 2: For $|z| < 1$, expand $\dfrac{1}{z-1} = \dfrac{-1}{1-z} = -\sum_{n=0}^{\infty} z^n$.

Step 3: Therefore $f(z) = -\dfrac{1}{z} - \sum_{n=0}^{\infty} z^n = -\dfrac{1}{z} - 1 - z - z^2 - \cdots$

The principal part is $-1/z$, so $z = 0$ is a simple pole with residue $-1$.

★ Example

Classify the singularity of $f(z) = e^{1/z}$ at $z = 0$.

$e^{1/z} = \sum_{n=0}^{\infty} \frac{1}{n! \, z^n} = 1 + \frac{1}{z} + \frac{1}{2z^2} + \frac{1}{6z^3} + \cdots$

The principal part has infinitely many nonzero terms, so $z = 0$ is an essential singularity.

★ Key Takeaways — Series Representations

The Laurent series is unique for a given annulus, but different annuli give different series
The residue of $f$ at $z_0$ is the coefficient $a_{-1}$ in the Laurent expansion
Essential singularities exhibit wild behaviour: by Picard's theorem, $f$ takes every value (with at most one exception) infinitely often near an essential singularity

Residue Calculus

The residue theorem transforms complex integrals into algebraic computations and provides powerful methods for evaluating real integrals.

◆ The Residue Theorem

Theorem — Residue Theorem If $f$ is analytic inside and on a simple closed contour $C$ except for isolated singularities $z_1, z_2, \ldots, z_n$ inside $C$, then $$\oint_C f(z)\,dz = 2\pi i \sum_{k=1}^{n} \text{Res}(f, z_k)$$

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 rule: If $f = g/h$ where $g(z_0) \neq 0$ and $h$ has a simple zero at $z_0$, then $\text{Res}(f, z_0) = g(z_0)/h'(z_0)$

★ Example

Evaluate $\displaystyle\oint_{|z|=2} \frac{z}{(z-1)(z+1)}\,dz$.

Step 1: Singularities inside $|z|=2$: simple poles at $z = 1$ and $z = -1$.

Step 2: $\text{Res}(f, 1) = \lim_{z \to 1} (z-1) \cdot \frac{z}{(z-1)(z+1)} = \frac{1}{2}$.

Step 3: $\text{Res}(f, -1) = \lim_{z \to -1} (z+1) \cdot \frac{z}{(z-1)(z+1)} = \frac{-1}{-2} = \frac{1}{2}$.

Step 4: By the residue theorem: $\oint_{|z|=2} f(z)\,dz = 2\pi i \left(\frac{1}{2} + \frac{1}{2}\right) = 2\pi i$.

◆ Applications and Related Theorems

Evaluating real integrals: The residue theorem is used to compute integrals like $\int_0^{2\pi} R(\cos\theta, \sin\theta)\,d\theta$ (substitute $z = e^{i\theta}$) and $\int_{-\infty}^{\infty} f(x)\,dx$ (close the contour in the upper or lower half-plane).

Theorem — Argument Principle If $f$ is meromorphic inside a simple closed contour $C$ with $Z$ zeros and $P$ poles (counted with multiplicity), then $\frac{1}{2\pi i}\oint_C \frac{f'(z)}{f(z)}\,dz = Z - P$.

Theorem — Rouche's Theorem If $f$ and $g$ are analytic inside and on $C$, and $|g(z)| < |f(z)|$ on $C$, then $f$ and $f + g$ have the same number of zeros inside $C$.

★ Key Takeaways — Residue Calculus

The residue theorem reduces contour integration to computing residues at singularities
For GATE, master the three residue computation formulas (simple pole, higher order, quotient)
Rouche's theorem is the standard tool for counting zeros of polynomials in a region

Conformal Mappings

Angle-preserving maps and bilinear transformations that allow us to transform domains while preserving analytic structure.

◆ Bilinear (Mobius) Transformations

Definition — Mobius Transformation A Mobius transformation (bilinear or fractional linear transformation) is a map of the form $w = \frac{az + b}{cz + d}$ where $a, b, c, d \in \mathbb{C}$ and $ad - bc \neq 0$.

Mobius transformations map circles and lines to circles and lines
They are conformal (angle-preserving) everywhere except at the pole $z = -d/c$
Given three distinct points $z_1, z_2, z_3$ and their images $w_1, w_2, w_3$, the Mobius transformation is uniquely determined
The cross-ratio $(z, z_1; z_2, z_3) = \frac{(z - z_1)(z_2 - z_3)}{(z - z_3)(z_2 - z_1)}$ is preserved under Mobius transformations

◆ Standard Conformal Maps

★ Example

Find the Mobius transformation mapping $z_1 = 0, z_2 = 1, z_3 = \infty$ to $w_1 = -1, w_2 = -i, w_3 = 1$.

Step 1: Use cross-ratios. Set $(w, w_1; w_2, w_3) = (z, z_1; z_2, z_3)$.

Step 2: $(z, 0; 1, \infty) = \frac{(z-0)(1-\infty)}{(z-\infty)(1-0)} = z$ (in the limit).

Step 3: $(w, -1; -i, 1) = \frac{(w+1)(-i-1)}{(w-1)(-i+1)}$.

Step 4: Set equal: $\frac{(w+1)(-i-1)}{(w-1)(-i+1)} = z$. Solve for $w$:

$(w+1)(-1-i) = z(w-1)(1-i)$, so $w(-1-i) + (-1-i) = zw(1-i) - z(1-i)$.

$w[(-1-i) - z(1-i)] = -z(1-i) + (1+i)$, giving $w = \frac{(1+i) - z(1-i)}{(-1-i) - z(1-i)}$.

Important standard maps:

$w = e^z$ maps horizontal strips to sectors/wedges
$w = z^2$ maps the first quadrant to the upper half-plane
$w = \frac{z-a}{1-\bar{a}z}$ (with $|a| < 1$) is an automorphism of the unit disk
$w = \frac{z-i}{z+i}$ maps the upper half-plane to the unit disk (Cayley transform)

★ Key Takeaways — Conformal Mappings

An analytic function is conformal at every point where its derivative is nonzero
Mobius transformations are uniquely determined by three point-image pairs
The cross-ratio is invariant under Mobius transformations — use it to find the map

✎ Practice Problems

Problem 1

Determine whether $f(z) = |z|^2$ is analytic anywhere.

Show Solution ▼

$f(z) = x^2 + y^2$, so $u = x^2 + y^2$ and $v = 0$. The C-R equations require $u_x = v_y$ and $u_y = -v_x$: $2x = 0$ and $2y = 0$. These hold only at the origin $z = 0$, but analyticity requires the equations to hold in a neighbourhood. Therefore $f$ is nowhere analytic (though it is differentiable at $z=0$ in the real sense).

Problem 2

Evaluate $\displaystyle\oint_{|z|=3} \frac{\sin z}{z^2}\,dz$.

Show Solution ▼

The singularity $z = 0$ is inside $|z|=3$. By Cauchy's integral formula for derivatives: $\oint \frac{f(z)}{(z-0)^2}\,dz = 2\pi i f'(0)$ where $f(z) = \sin z$. Since $f'(z) = \cos z$, we get $f'(0) = 1$. Therefore the integral equals $2\pi i$.

Problem 3

Find the residue of $f(z) = \dfrac{z^2}{(z^2+1)^2}$ at $z = i$.

Show Solution ▼

$z = i$ is a pole of order 2. Write $(z^2+1)^2 = (z-i)^2(z+i)^2$. Let $g(z) = z^2/(z+i)^2$. Then $\text{Res}(f, i) = g'(i) = \frac{d}{dz}\left[\frac{z^2}{(z+i)^2}\right]_{z=i} = \frac{2z(z+i)^2 - z^2 \cdot 2(z+i)}{(z+i)^4}\Big|_{z=i} = \frac{2i(2i)^2 - i^2 \cdot 2(2i)}{(2i)^4} = \frac{-8i + 4i}{16} = \frac{-4i}{16} = \frac{-i}{4}$.

Problem 4

Using Rouche's theorem, find the number of zeros of $p(z) = z^5 + 3z + 1$ inside $|z| = 1$.

Show Solution ▼

On $|z| = 1$: let $f(z) = 3z$ and $g(z) = z^5 + 1$. Then $|f(z)| = 3$ and $|g(z)| \leq |z|^5 + 1 = 2 < 3 = |f(z)|$. By Rouche's theorem, $f(z) + g(z) = z^5 + 3z + 1$ has the same number of zeros inside $|z| = 1$ as $f(z) = 3z$, which has exactly 1 zero (at $z=0$). So $p(z)$ has 1 zero inside $|z| = 1$.

Problem 5

Find the image of the upper half-plane $\text{Im}(z) > 0$ under the map $w = e^{i\pi z}$.

Show Solution ▼

Let $z = x + iy$ with $y > 0$. Then $w = e^{i\pi(x+iy)} = e^{-\pi y} \cdot e^{i\pi x}$. Since $y > 0$, $|w| = e^{-\pi y} < 1$. Since $x$ ranges over all of $\mathbb{R}$, $\arg(w) = \pi x$ takes all values. Therefore the image is the punctured open unit disk $\{w : 0 < |w| < 1\}$.

⚙ Interactive Quiz

1. If $f(z) = u + iv$ is analytic and $u = e^x \cos y$, then $v$ equals:

A $e^x \cos y$

B $e^x \sin y + C$

C $-e^x \sin y$

D $e^y \sin x$

2. The value of $\displaystyle\oint_{|z|=1} \frac{1}{z}\,dz$ is:

A $0$

B $\pi i$

C $2\pi i$

D $1$

3. The function $f(z) = \dfrac{\sin z}{z}$ has what type of singularity at $z = 0$?

A Removable singularity

B Simple pole

C Essential singularity

D Pole of order 2

4. How many zeros does $z^7 - 5z^3 + 12$ have inside $|z| = 1$?

A 0

B 1

C 3

D 7

5. A Mobius transformation maps circles and lines to:

A Only circles

B Only lines

C Circles or lines

D Ellipses