Engineering Mathematics II

Complex Variables

PreviousNext

Module III: Complex Variables. This chapter introduces analytic functions, Cauchy Riemann equations, complex line integrals, Cauchy's theorems, series representations and the residue theorem. These ideas form the foundation of complex analysis and are widely used in signal processing, electrical engineering and mathematical modelling.

1. Complex Functions

A complex function maps complex numbers to complex numbers:f(z) = u(x, y) + i v(x, y), where z = x + i y. Here u and v are real-valued functions of two variables.

  • Limit and continuity: defined analogously to real functions, but must be independent of the path of approach.
  • Differentiability: f is differentiable at z₀ iflim (Δz→0) [f(z₀ + Δz) - f(z₀)] / Δzexists and has the same value for all directions Δz.
  • Analytic function: differentiable in a neighborhood. Also called holomorphic.
  • Entire function: analytic everywhere (examples:e^z, sin z, cos z).

Diagram: complex-plane-mapping.png

2. Cauchy Riemann (CR) Equations

CR equations are necessary conditions for differentiability. For f(z) = u(x, y) + i v(x, y):

u_x = v_yandu_y = -v_x

If u and v have continuous partial derivatives and satisfy CR equations in a neighborhood, then f is analytic there.

Example

Let f(z) = z² = (x² − y²) + i(2xy). u = x² − y², v = 2xy. Then u_x = 2x, v_y = 2x, and u_y = -2y, v_x = 2y.

CR equations satisfied hence f(z) is analytic everywhere.

Diagram: cr-equations-visual.png

3. Line Integrals in the Complex Plane

A line integral of f(z) along a curve C is defined as:

∫_C f(z) dz = ∫ (u dx - v dy) + i ∫ (v dx + u dy)
  • Computed by parametrizing the curve z(t).
  • Value depends on the path unless f is analytic everywhere inside C.

Diagram: complex-path-integral.png

4. Cauchy's Integral Theorem

A landmark result: If f is analytic and C is a closed curve in a simply connected region, then the line integral is zero.

∮_C f(z) dz = 0
  • Independence of path.
  • Exists a primitive F such that F'(z) = f(z).
Example

∮ z³ dz around |z| = 1 is 0 since z³ is analytic everywhere.

Diagram: cauchy-theorem.png

5. Cauchy's Integral Formula

Provides the value of an analytic function inside a curve using boundary values:

f(a) = 1/(2πi) ∮ (f(z)/(z-a)) dz
  • Higher derivatives:f^(n)(a) = n!/(2πi) ∮ f(z)/(z-a)^(n+1) dz

Diagram: cif-geometry.png

6. Taylor and Laurent Series

Analytic functions behave like power series. Taylor series is valid inside a disk where the function is analytic.

  • Taylor series:f(z) = Σ f^(n)(a)/n! * (z-a)^n
  • Laurent series: includes negative powers; valid in annular regions.
Example

For f(z) = 1/(z (z-1)), a Laurent expansion around z = 0 gives:f(z) = -1/z + 1/(z-1).

Diagram: taylor-laurent-region.png

7. Zeros, Singularities and Residues

A point z = a is a zero of f if f(a) = 0. A point where f fails to be analytic is a singularity.

  • Removable singularity: function can be redefined to become analytic.
  • Pole: behaves like (z - a)^(-n).
  • Essential singularity: extremely unstable behavior (Casorati theorem).

Residue

Residue is the coefficient of 1/(z - a) in the Laurent series of f.

Example

Residue of f(z) = 1/(z² + 1) at z = i is1/(2i).

8. Residue Theorem

One of the most powerful tools in complex analysis:

∮ f(z) dz = 2πi * Σ residues inside the contour

Used extensively to evaluate real integrals that are difficult using real-variable calculus.

Example

Evaluate∮ dz/(z² + 1)around a contour containing z = i and z = -i. Total residue = 1/(2i) + (-1/(2i)) = 0. Integral = 0.

Diagram: residue-theorem-contour.png

Numerical AnalysisIntegral Calculus