# Limit of a Complex Integral

## Homework Statement

Let U be open in C, f : U -> C continuous.
Prove that

$$\lim_{R\rightarrow 0} \int_0^{2\pi} f(Re^{it}) dt = 2\pi f(0)$$

## Homework Equations

$$\lim_{R\rightarrow 0} f(Re^{it}) dt = f(0)$$

Also

$$\int_0^{2\pi} \lim_{R\rightarrow 0} f(Re^{it}) dt = \int_0^{2\pi} f(0) = 2\pi f(0)$$

## The Attempt at a Solution

The question then just resolves to passing the limit inside the integral.
I would think then we'd use either uniform convergence or DCT, since the integral of a complex function is simply the sum of the integrals of its real and imaginary parts.

Would this be the correct way of doing it, or is there some other way from complex analysis instead?