Circle integral

Definition

Let $c \in \mathbb{C}$ and let $R > 0$ be a real number. Let $f$ be a complex function that is continuous on the circle $\{z : |z - c| = R\}$. The circle integral of $f$ over this circle is the contour integral along the path $\gamma(\theta) = c + R e^{i\theta}$, $\theta \in [0, 2\pi]$: $$\oint_{|z - c| = R} f(z) \, dz = \int_0^{2\pi} f(c + R e^{i\theta}) \, i R e^{i\theta} \, d\theta.$$

Remarks

The path goes once around the circle, in the positive (counterclockwise) direction. The length of the path is $2 \pi R$. The basic example is the integral of 1/z: $\oint_{|z| = 1} \frac{dz}{z} = 2 \pi i$.

Explanations

L1 human

A circle integral is a contour integral where the path is a circle, traveled once counterclockwise. Circles are the paths that appear most in complex analysis, for example in Cauchy's integral formula.