PU-205 · Mathematical Methods II — Complex Analysis & Asymptotics
The unit builds complex analysis from the ground up — analyticity, Cauchy's theorem, series, and residues — showing how a single rigidity condition (differentiability in the complex sense) forces the entire global structure of a function. It then turns that machinery outward into asymptotics, where contour deformation and saddle points extract the large-parameter behaviour of the integrals that pervade quantum mechanics, statistical physics, and wave propagation.
Lectures
| L01 | The Geometry of the Complex Plane — |
| L02 | Complex Functions, Limits & Differentiability |
| L03 | The Cauchy-Riemann Equations & Harmonic Functions |
| L04 | Elementary Functions: exp, log & Powers |
| L05 | Branch Points, Cuts & Riemann Surfaces |
| L06 | Contour Integrals & the Cauchy-Goursat Theorem |
| L07 | The Cauchy Integral Formula |
| L08 | Liouville, Maximum Modulus & the FTA |
| L09 | Taylor Series & Radius of Convergence |
| L10 | Laurent Series & Classifying Singularities |
| L11 | The Residue Theorem |
| L12 | Real Integrals by Residues & Jordan's Lemma |
| L13 | Keyhole & Branch-Cut Contours |
| L14 | The Argument Principle & Rouché's Theorem |
| L15 | Analytic Continuation & the Identity Theorem |
| L16 | The Gamma Function & Reflection Formula |
| L17 | Conformal Mapping: Angles & Analyticity |
| L18 | Conformal Maps in Electrostatics & Ideal Flow |
| L19 | Asymptotic Series & Optimal Truncation |
| L20 | Watson's Lemma & Laplace-Type Integrals |
| L21 | Laplace's Method |
| L22 | Saddle Points in the Complex Plane |
| L23 | The Method of Steepest Descent |
| L24 | Stirling's Formula via Saddle Point |
| L25 | The Method of Stationary Phase |
| L26 | Wave Propagation, Dispersion & Diffraction |
| L27 | Divergent Series, Borel Summation & Stokes Phenomenon |
| L28 | Synthesis: Special Functions & Their Asymptotics |
Derivations homed in this unit
Cauchy-Riemann Equations & Harmonic Conjugates
Derives that complex differentiability forces the Cauchy-Riemann equations and that the real and imaginary parts of an analytic function are conjugate harmonic functions.
Cauchy-Goursat Theorem
Proves that the contour integral of an analytic function over any closed curve in a simply connected domain vanishes, without assuming continuity of the derivative.
Cauchy Integral Formula & Derivative Formula
Derives that an analytic function and all its derivatives at an interior point are determined by its boundary values through a contour integral, giving Liouville's theorem and the fundamental theorem of algebra.
Taylor Series of Analytic Functions
Shows that an analytic function equals its Taylor series inside any disc within its domain, with radius of convergence set by the nearest singularity.
Laurent Series & Classification of Singularities
Derives the two-sided Laurent expansion of a function analytic in an annulus and classifies isolated singularities as removable, pole, or essential.
The Residue Theorem
Proves that a closed contour integral equals 2πi times the sum of enclosed residues, identified as Laurent coefficients of the 1/(z-z0) term.
Real Integrals by Residues & Jordan's Lemma
Establishes Jordan's lemma and the semicircle/keyhole techniques that reduce definite real and oscillatory integrals to residue sums.
Argument Principle & Rouché's Theorem
Derives that a contour integral of f'/f counts enclosed zeros minus poles, and deduces Rouché's theorem for locating roots.
Branch Points, Cuts & Riemann Surfaces
Analyses multivalued functions such as log and fractional powers, defining branch points, branch cuts, and the Riemann surface that single-values them.
Analytic Continuation & the Identity Theorem
Proves that an analytic function is uniquely determined by its values on any set with a limit point, licensing continuation beyond the original domain.
Gamma Function: Continuation & Reflection Formula
Extends the factorial to the complex plane via analytic continuation and derives the reflection formula Γ(z)Γ(1-z)=π/sin(πz) by residues.
Conformal Mapping & Plane Potential Theory
Shows analytic maps preserve angles and Laplace's equation, letting boundary-value problems in electrostatics and ideal flow be solved by mapping to simple domains.
Asymptotic Series & Optimal Truncation
Defines Poincaré asymptotic expansions, shows why divergent series still approximate, and derives the least-term truncation error bound.
Watson's Lemma
Derives the full asymptotic expansion of a Laplace-type integral from the small-argument expansion of its integrand, term by term via the Gamma function.
Laplace's Method for Exponential Integrals
Derives the leading large-parameter asymptotics of ∫e^{λφ(t)} integrals by localising the contribution at the maximum of φ and Gaussian expansion.
Method of Steepest Descent
Extends Laplace's method to the complex plane by deforming the contour through a saddle point along the path of steepest descent.
Stirling's Formula via Saddle Point
Derives the Stirling asymptotic expansion of n! and Γ(z) by applying the method of steepest descent to the integral representation of the Gamma function.
Method of Stationary Phase
Derives the large-parameter asymptotics of oscillatory integrals ∫e^{iλφ(t)} from stationary points of the phase, with the Fresnel-integral phase factor.