Complex Analysis
Complex Analysis in GATE MA covers analytic functions, contour integration, series expansions, and residue calculus. Questions often combine a quick residue computation with a theorem such as Liouville or the maximum modulus principle.
Core rules
- f=u+iv is analytic on an open set iff u,v satisfy Cauchy–Riemann u_x=v_y, u_y=−v_x and have continuous partials.
- Cauchy's theorem: ∮_γ f dz=0 for f analytic on and inside a simple closed contour γ.
- Cauchy integral formula: f(z_0)= (1/2πi)∮ f(z)/(z−z_0) dz.
- Liouville: bounded entire functions are constant. Maximum modulus: |f| attains its maximum on the boundary, not in the interior unless f is constant.
- Laurent series and residues: Res(f,z_0)= coefficient a_{−1}; ∮_γ f dz = 2πi Σ Res inside γ.
- Morera's theorem: continuous f with zero integral over every closed triangle is analytic.
Worked example
Compute ∮{|z|=2} e^z/(z−1) dz. The integrand has a simple pole at z=1 inside the contour. Residue = lim{z→1}(z−1)e^z/(z−1)=e. By the residue theorem the integral is 2πi e.
Traps
- C-R equations alone need differentiability and continuity of partials; they are necessary but not sufficient without that.
- Do not apply Cauchy's theorem across a singularity; check which poles lie inside γ.
- Branch cuts matter for log and z^α; a residue at infinity has a sign convention.
- The argument principle counts zeros minus poles, not just poles.