Cauchy Integral Formula

The Cauchy Integral Formula is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk.

For an interior point and derivative order m, I = (2πi/m!) f^(m)(a). The circle is traversed counterclockwise; reversing orientation changes the sign. This formula uses holomorphic numerator data and a kernel point strictly inside the contour.

Worked Example

Consider the integral of f(z)/(z-a)^m+1 dz around a counterclockwise circle |z| = R.

Let f(z) = z², with the kernel point a = 1, contour radius R = 2, and derivative order m = 0.

We know that since |a| = 1 < 2, the point lies inside the contour. By the Cauchy Integral Formula:

f(a) = 1 / (2πi) ∫ f(z) / (z-a) dz

Evaluate the function at a: f(1) = 1² = 1
Calculate the normalized integral: I / (2πi) = f(1) / 0! = 1
The complex integral is purely imaginary: I = 2πi(1) = 2πi ≈ 6.28318530718i

Experiment

Use the interactive visualization to explore different kernel points and contours:

Set Analytic polynomial f to z^3.
Set Kernel point a to 1.
Set Contour radius R to 2.
Set Derivative order m to 1.
Observe that the status is Inside and the normalized_integral is 3.

See the interactive module at https://riano.app/module/cauchy-integral-formula.

Limitations

The visualization uses specific bounded parameters. It only handles polynomial analytic boundaries of degree up to 3 and derivatives up to order 2. A kernel point is not always a pole: for f(z)=z^2, a=0, m=0, the quotient simplifies to z and the kernel point is removable. In contrast m=2 leaves 1/z and gives integral 2πi. Outside the circle the integral is zero. At |a|=R all listed nonzero real points are nonremovable, so the ordinary contour integral is undefined and numeric integral outputs are hidden.

References

The theoretical background and formulas are derived from MIT 18.04 Complex Variables: Topic 4 Notes.

Cauchy Integral Formula