Engineering Mathematics IV: Complex Analysis & PDEs

The complex plane and fields in space and time. Complex numbers and phasors, analytic functions and Cauchy–Riemann equations, contour integration and Cauchy's theorems, Laurent series and residues, partial differential equations (heat, wave, Laplace), and separation of variables.

• Complex Numbers — Complex numbers make oscillations and AC circuits algebraic. Learn Cartesian, polar and exponential forms, modulus, argument and conjugate, the Argand plane, Euler's formula e^(jθ) = cos θ + j sin θ, De Moivre's theorem, n-th roots, and phasor/impedance applications.
• Analytic Functions & Cauchy–Riemann — Complex functions map the plane to itself. Learn limits, continuity and differentiability in ℂ, analytic (holomorphic) functions, the Cauchy–Riemann equations, harmonic functions and conjugates, elementary functions (eᶻ, sin z, log z), and conformal mapping in potential flow and electrostatics.
• Complex Integration & Cauchy's Theorems — Integrals along paths in the complex plane have remarkable properties. Learn contours and parametrisation, contour integrals, Cauchy's integral theorem (zero around analytic regions), path independence, the Cauchy integral formula and its derivative form, and consequences such as Liouville's theorem.
• Taylor & Laurent Series, Singularities — Laurent series extend Taylor series with negative powers, describing functions near singularities. Learn complex Taylor series and convergence discs, Laurent series in annuli, principal part, classification of singularities (removable, poles of order m, essential), and residues as the 1/(z − z₀) coefficient.
• Residue Theorem & Applications — The residue theorem evaluates contour integrals by summing residues. Learn residue formulas for simple and higher-order poles, the residue theorem, evaluating real integrals (rational functions, trigonometric integrals over [0, 2π], Fourier-type integrals with Jordan's lemma), and inverse Laplace transforms via residues.
• Partial Differential Equations — PDEs describe fields varying in space and time. Learn order, linearity and classification (elliptic, parabolic, hyperbolic), the heat, wave and Laplace equations with their physical meaning, initial and boundary conditions (Dirichlet, Neumann, Robin), well-posedness, and solution strategies.
• Separation of Variables — Separation of variables turns a PDE into ODEs. Learn the procedure (product solution, separation constant), eigenvalue problems from boundary conditions, superposition and Fourier series for initial conditions, and full solutions of the heat equation, wave equation (standing waves) and Laplace's equation on a rectangle — with a Python evaluation.