Questions
~2 questions per paper
Difficulty
Medium-Hard
Importance
High yield for PSU technical papers
Overview
Complex analysis deals with functions of a complex variable and is a cornerstone of engineering mathematics in PSU exams. Mastery of analytic properties and residue integration allows you to solve complex integrals and series expansions that are otherwise unsolvable via real calculus. Focusing on the Cauchy-Riemann conditions and Residue Theorem is essential for solving high-weightage numerical problems.
Analytic Functions & Cauchy–Riemann
A function is analytic at a point if it is differentiable in some neighborhood of that point. The Cauchy-Riemann equations are the necessary and sufficient conditions for a function to be analytic, serving as the foundation for identifying potential candidates for integration.
- f(z) = u(x,y) + iv(x,y)
- Cauchy-Riemann: du/dx = dv/dy and du/dy = -dv/dx
- Harmonic condition: del^2(u) = 0 and del^2(v) = 0
- Necessary condition: u and v must have continuous first-order partial derivatives
Cauchy's Integral Theorem & Formula
These principles allow for the evaluation of line integrals of analytic functions over closed contours. Cauchy's Theorem states that the integral of an analytic function around a closed loop is zero, while the Integral Formula computes values within a domain based on boundary data.
- Cauchy Integral Theorem: Integral f(z) dz = 0 if analytic inside C
- Cauchy Integral Formula: f(a) = (1/2*pi*i) * Integral f(z)/(z-a) dz
- Generalized Formula: f^n(a) = (n! / 2*pi*i) * Integral f(z)/(z-a)^(n+1) dz
Taylor & Laurent Series
Power series expansions are used to represent complex functions near specific points or within an annulus. While Taylor series require analyticity at the point of expansion, Laurent series allow for the inclusion of negative powers to account for singularities.
- Taylor series: f(z) = Sum from n=0 to infinity of [f^n(a)/n!] * (z-a)^n
- Laurent series: f(z) = Sum from n=-infinity to infinity of a_n * (z-a)^n
- Singularities: Removable, Poles, and Essential
- Radius of convergence determines the disk where the series holds
Residue Theorem
The Residue Theorem is the most powerful tool for evaluating complex definite integrals by summing residues at isolated singularities enclosed within a contour. This method is the primary source of numerical problems in competitive engineering exams.
- Integral f(z) dz = 2*pi*i * Sum of Residues
- Residue at simple pole (z=a): Lim z->a [(z-a) * f(z)]
- Residue at pole of order m: (1/(m-1)!) * Lim z->a [d^(m-1)/dz^(m-1) * ((z-a)^m * f(z))]
- Residue at infinity: -Residue at 0
Formula Sheet
f(z) = u + iv
du/dx = dv/dy, du/dy = -dv/dx
f(a) = (1/2*pi*i) * integral f(z)/(z-a) dz
Integral f(z) dz = 2*pi*i * sum(Residues)
Res(f,a) = lim(z->a) (z-a)f(z)
Res(f,a) = (1/(n-1)!) * lim(z->a) d^(n-1)/dz^(n-1) [(z-a)^n * f(z)]
Exam Tip
Always verify if a function has poles inside the given contour before calculating residues; if the function is analytic everywhere inside the contour, the integral is zero.
Common Mistakes
- Forgetting to check if the pole actually lies inside the specified contour before calculating the residue.
- Incorrectly identifying the order of a pole or failing to use the derivative formula for higher-order poles.
- Confusing the Cauchy-Riemann equations signs, specifically missing the negative sign for du/dy.
More Revision Notes
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