There is an intimate connection between power series and Fourier series. If we expand a function f in a power series inside a circle of radius R, this means that f ( z ) = ∑ ∑ n = 0 ∞ ∞ c n z n , with suitably defined coefficients whose real and imaginary parts are given by c n = a n + i b n . Therefore f ( z ) = ∑ ∑ n = 0 ∞ ∞ [ a n r n cos ⁡ ⁡ n θ θ − − b n r n sin ⁡ ⁡ n θ θ ] + i ∑ ∑ n = 1 ∞ ∞ [ a n r n sin ⁡ ⁡ n θ θ + b n r n cos ⁡ ⁡ n θ θ ] , which is a Fourier series for f. These trigonometric functions can themselves be expanded, using multiple angle formulae.