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.

Cauchy's results let engineers evaluate difficult real integrals, invert Laplace transforms, and analyse stability (Nyquist criterion) by walking around contours in the complex plane.

An analogy

Cauchy's theorem is like walking round a flat field and returning home: if there are no 'holes' (singularities) inside, the total climb is zero.

Parametrising contours

Circle: z = z₀ + re^(jt), dz = jre^(jt) dt. Line segments: z = z₁ + t(z₂ − z₁). Positive orientation is counter-clockwise.

Cauchy's theorem

For analytic f in a simply connected domain, integrals depend only on endpoints and vanish around closed loops — contours can be deformed freely without crossing singularities.

Integral formula

Values of an analytic function inside a contour are determined by its boundary values — and so are all its derivatives.

Consequences

Liouville: bounded entire functions are constant → fundamental theorem of algebra. Argument principle counts zeros minus poles inside a contour — the basis of the Nyquist stability criterion.

Formulas

• Contour integral: ∫_C f dz = ∫ₐᵇ f(z(t)) z′(t) dt
• Key integral: ∮_{|z − z₀| = r} (z − z₀)ⁿ dz = 2πj if n = −1, else 0
• Cauchy's theorem: f analytic inside and on C ⇒ ∮_C f dz = 0
• Cauchy integral formula: f(z₀) = (1/2πj) ∮ f(z)/(z − z₀) dz; f⁽ⁿ⁾(z₀) = (n!/2πj) ∮ f/(z − z₀)ⁿ⁺¹ dz

Worked example: Cauchy integral formula

Given: ∮_{|z|=2} eᶻ/(z − 1) dz

Find: Value.

• f = eᶻ analytic, z₀ = 1 inside
• = 2πj f(1) = 2πj e

Answer: 2πe j ≈ 17.08j

Worked example: Basic contour integral

Given: ∮_{|z|=1} dz/z

Find: Value.

• z = e^(jt), dz = je^(jt) dt
• ∫₀^{2π} j dt

Answer: 2πj

Common mistakes

• Wrong: Applying Cauchy's theorem with a singularity inside. — Right: Check analyticity everywhere inside.
• Wrong: Clockwise contours without sign change. — Right: Clockwise gives the negative.

Applications

• Stability analysis: Nyquist criterion via the argument principle.
• Inverse transforms: Bromwich contour for Laplace inversion.
• Real integrals: Evaluated via residues (next lessons).

Key takeaways

• Parametrise, substitute, integrate.
• Analytic inside ⇒ ∮ = 0.
• f(z₀) = (1/2πj)∮ f/(z − z₀) dz.
• Basis of residues and Nyquist.

Going further

• Morera's theorem: Converse of Cauchy's theorem.
• Argument principle: Zeros − poles = winding number.
• Nyquist plots: Encirclements of −1 (M13).