REFERENCE SHEET

Contour integration & Cauchy formulas

Contour integrals connect local values of a holomorphic function to its behavior around a boundary.

Before this Holomorphic functions & power series, Integration & accumulation

f(a) = (1/(2πi)) ∮γ f(z)/(z−a) dz

Definitions

Contour integral

For a piecewise C¹ parametrization γ:[a,b]→ℂ, ∫γ f(z)dz=∫ₐᵇ f(γ(t))γ′(t)dt.

Primitive

A holomorphic F with F′=f. Then contour integrals depend only on endpoints.

Winding number

For a closed contour avoiding a, Ind(γ,a)=(1/(2πi))∮γ dz/(z−a), an integer counting oriented turns.

Key theorems

Cauchy integral theorem

A holomorphic function on a simply connected open subset of ℂ has integral zero around every closed piecewise C¹ contour in that domain.

Cauchy integral formula

If f is holomorphic on a neighborhood of a closed disk, γ is its positively oriented boundary, and a is inside, the displayed formula holds.

Derivative formula

Under the same disk hypotheses, f⁽ⁿ⁾(a)=n!/(2πi) ∮γ f(z)/(z−a)ⁿ⁺¹ dz.

Notable applications

Exact evaluation

Extract function values and derivatives from integrals without finding a real-variable antiderivative.

Bounds

Cauchy estimates control derivatives using bounds on a surrounding circle.

Work it out

Try before revealing

01 / Compute

Evaluate ∮|z|=2 eᶻ/z dz counterclockwise.

Hint

Use the formula at a=0.

Solution

The integral is 2πi e⁰=2πi.

02 / Compute

Evaluate ∮|z|=1 dz/z directly with z=eⁱᵗ.

Hint

Use dz=i eⁱᵗdt, 0≤t≤2π.