Complex Analysis

UPSC Mathematics Optional · Paper II · Analytic Functions to Residues

1. Complex Numbers & Functions 2. Analytic Functions & CR Equations 3. Harmonic Functions 4. Elementary Functions 5. Complex Integration 6. Cauchy's Theorems 7. Taylor & Laurent Series 8. Singularities & Residues 9. Contour Integration 10. Bilinear Transformations & Conformal Mapping

§1 · Complex Numbers & Functions

Complex Number Identities PYQ★★

\(z=x+iy=re^{i\theta}\). \(|z|=r=\sqrt{x^2+y^2}\). \(\arg z=\theta=\arctan(y/x)\).

\(\bar z=x-iy\). \(|z|^2=z\bar z\). \(\text{Re}(z)=\frac{z+\bar z}{2}\). \(\text{Im}(z)=\frac{z-\bar z}{2i}\).

de Moivre: \((\cos\theta+i\sin\theta)^n=\cos n\theta+i\sin n\theta\).

Euler: \(e^{i\theta}=\cos\theta+i\sin\theta\). Hence \(e^{i\pi}+1=0\).

Triangle inequality: \(|z_1+z_2|\le|z_1|+|z_2|\). \(\big||z_1|-|z_2|\big|\le|z_1-z_2|\).

Roots of Unity & nth Roots PYQ★★

nth roots of \(re^{i\theta}\): \(r^{1/n}e^{i(\theta+2k\pi)/n}\), \(k=0,1,\ldots,n-1\).

nth roots of unity: \(e^{2\pi ik/n}\), \(k=0,\ldots,n-1\). Sum = 0.

§2 · Analytic Functions & CR Equations

Analytic / Holomorphic Function

\(f(z)\) is analytic at \(z_0\) if it is differentiable in some neighbourhood of \(z_0\).

\(f'(z_0)=\lim_{\Delta z\to 0}\dfrac{f(z_0+\Delta z)-f(z_0)}{\Delta z}\) (limit must be same from all directions).

Cauchy-Riemann Equations PYQ★★★

If \(f(z)=u(x,y)+iv(x,y)\) is differentiable at \(z=x+iy\), then:

\[\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y},\qquad\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}\]

Sufficient condition: If \(u,v\) have continuous partial derivatives satisfying CR equations, then \(f\) is analytic.

\(f'(z)=u_x+iv_x=u_x-iu_y=v_y+iv_x\).

Polar form: \(u_r=\dfrac{1}{r}v_\theta\), \(v_r=-\dfrac{1}{r}u_\theta\).

§3 · Harmonic Functions

Harmonic Conjugate PYQ★★★

If \(f=u+iv\) is analytic, then \(u\) and \(v\) are both harmonic: \(\nabla^2 u=0\), \(\nabla^2 v=0\).

Level curves of \(u\) and \(v\) intersect orthogonally.

Finding \(v\) given \(u\): Use \(v_x=-u_y\), \(v_y=u_x\); integrate.

Milne-Thomson method: \(f(z)=\phi(z,0)-i\int\phi_1(z,0)dz\) where \(\phi_1=\partial u/\partial x\Big|_{y=0}\).

§4 · Elementary Functions

Complex Exponential, Trig, Log PYQ★★

\(e^z=e^x(\cos y+i\sin y)\). Periodic with period \(2\pi i\).
\(\sin z=\dfrac{e^{iz}-e^{-iz}}{2i}\), \(\cos z=\dfrac{e^{iz}+e^{-iz}}{2}\).
\(\sinh z=\dfrac{e^z-e^{-z}}{2}=-i\sin(iz)\), \(\cosh z=\cos(iz)\).
\(\log z=\ln|z|+i\arg z\) (multi-valued; principal value uses \(\arg z\in(-\pi,\pi]\)).
\(z^w=e^{w\log z}\) (multi-valued in general).

§5 · Complex Integration

Contour Integral

\(\displaystyle\int_C f(z)\,dz=\int_a^b f(z(t))z'(t)\,dt\)

where \(z=z(t)\), \(t\in[a,b]\) parametrizes \(C\).

Bound: \(\left|\int_C f\,dz\right|\le ML\) where \(M=\max|f|\) on \(C\), \(L=\) length of \(C\) (ML-inequality).

§6 · Cauchy's Theorems

Cauchy's Integral Theorem PYQ★★★

If \(f\) is analytic in a simply connected domain \(D\), then \(\oint_C f(z)\,dz=0\) for every closed contour \(C\) in \(D\).

Independence of path: The integral depends only on endpoints.

Cauchy's Integral Formula PYQ★★★

\[f(z_0)=\frac{1}{2\pi i}\oint_C\frac{f(z)}{z-z_0}\,dz\]

Generalized: \[f^{(n)}(z_0)=\frac{n!}{2\pi i}\oint_C\frac{f(z)}{(z-z_0)^{n+1}}\,dz\]

Consequence: Analytic functions are infinitely differentiable (holomorphic ⟺ analytic as power series).

Liouville's Theorem & FTA PYQ★★

Liouville: A bounded entire function is constant.

Fundamental Theorem of Algebra: Every non-constant polynomial has a root in \(\mathbb{C}\).

Maximum Modulus Principle: If \(f\) analytic and non-constant in domain \(D\), then \(|f|\) attains no maximum in the interior of \(D\).

§7 · Taylor & Laurent Series

Taylor Series PYQ★★★

If \(f\) analytic in \(|z-z_0|

Laurent Series PYQ★★★

If \(f\) analytic in annulus \(r<|z-z_0|

\[f(z)=\sum_{n=-\infty}^{\infty}a_n(z-z_0)^n, \quad a_n=\frac{1}{2\pi i}\oint_C\frac{f(z)}{(z-z_0)^{n+1}}\,dz\]

Principal part: terms with negative powers. Analytic part: non-negative powers.

§8 · Singularities & Residues

Types of Singularities PYQ★★★

Removable singularity: \(\lim_{z\to z_0}f(z)\) exists and is finite. Laurent series has no principal part.
Pole of order \(m\): principal part has finitely many terms, highest being \(a_{-m}/(z-z_0)^m\), \(a_{-m}\ne 0\).
Essential singularity: principal part has infinitely many terms. Picard: \(f\) takes every complex value (with at most one exception) near an essential singularity.

Residue Calculation PYQ★★★

Residue of \(f\) at \(z_0\) = coefficient \(a_{-1}\) in Laurent expansion.

Simple pole: \(\text{Res}(f,z_0)=\lim_{z\to z_0}(z-z_0)f(z)\).
Pole of order \(m\): \(\text{Res}(f,z_0)=\dfrac{1}{(m-1)!}\lim_{z\to z_0}\dfrac{d^{m-1}}{dz^{m-1}}\left[(z-z_0)^m f(z)\right]\).
Quotient \(p(z)/q(z)\) at simple zero of \(q\): \(\text{Res}=\dfrac{p(z_0)}{q'(z_0)}\).

Cauchy's Residue Theorem PYQ★★★

\[\oint_C f(z)\,dz=2\pi i\sum_k\text{Res}(f,z_k)\]

where the sum is over all singularities \(z_k\) inside \(C\). \(C\) positively oriented.

§9 · Contour Integration

Real Integrals via Residues PYQ★★★

Type 1: \(\displaystyle\int_0^{2\pi} R(\cos\theta,\sin\theta)\,d\theta\). Put \(z=e^{i\theta}\): \(\cos\theta=\dfrac{z+z^{-1}}{2}\), \(\sin\theta=\dfrac{z-z^{-1}}{2i}\), \(d\theta=\dfrac{dz}{iz}\).

Type 2: \(\displaystyle\int_{-\infty}^\infty \frac{P(x)}{Q(x)}\,dx=2\pi i\sum\text{Res in upper half-plane}\).

Condition: \(\deg Q\ge\deg P+2\), \(Q\) has no real zeros.

Type 3: \(\displaystyle\int_{-\infty}^\infty \frac{P(x)}{Q(x)}e^{imx}\,dx\): close in UHP if \(m>0\), LHP if \(m<0\) (Jordan's lemma).

Jordan's Lemma: \(\int_{C_R}f(z)e^{iaz}\,dz\to 0\) as \(R\to\infty\) if \(a>0\) and \(|f(z)|\to 0\) on upper semicircle.

Indentation around real poles

If \(f\) has a simple pole at \(x_0\in\mathbb{R}\): indent with small semicircle of radius \(\varepsilon\). Contribution = \(\pm\pi i\cdot\text{Res}(f,x_0)\) (sign depends on direction: \(-\pi i\) going above, \(+\pi i\) going below).

§10 · Bilinear Transformations & Conformal Mapping

Bilinear (Möbius) Transformation PYQ★★★

\(w=\dfrac{az+b}{cz+d}\), \(ad-bc\ne 0\).

Maps circles/lines to circles/lines.
Preserves cross-ratio: \((z_1,z_2;z_3,z_4)=\dfrac{(z_1-z_3)(z_2-z_4)}{(z_1-z_4)(z_2-z_3)}\) is invariant.
Unique bilinear map sending 3 given points to 3 other given points.
Fixed points: solve \(z=\dfrac{az+b}{cz+d}\) → quadratic with at most 2 roots.

Conformal Mapping PYQ★★★

An analytic function \(f\) is conformal at \(z_0\) if \(f'(z_0)\ne 0\): preserves angles between curves.

Key mappings:

\(w=z^2\): maps upper half-plane to full plane minus negative real axis.
\(w=e^z\): maps horizontal strip \(0<\text{Im}\,z<\pi\) to upper half-plane.
\(w=\dfrac{1}{z}\): inversion — maps unit disk (excl 0) to outside unit disk.
\(w=\dfrac{z-i}{z+i}\): maps upper half-plane to unit disk \(|w|<1\).
Joukowski: \(w=z+1/z\): maps outside unit circle to \(\mathbb{C}\) minus \([-2,2]\).
\(w=\sin z\): maps \(-\pi/2<\text{Re}\,z<\pi/2\) strip to \(w\)-plane minus slits.

Riemann Mapping Theorem

Any simply connected domain \(\Omega\ne\mathbb{C}\) is conformally equivalent to the unit disk. (Existence only — not constructive.)