# Fourier sine and cosine transform formula

The Fourier Transform is a mathematical technique that transforms a function of time, x(t), to a function of frequency, X(ω). It is closely related to the Fourier Series . If you are familiar with the Fourier Series , the following derivation may be helpful. The Fourier Transform is a mathematical technique that transforms a function of time, x(t), to a function of frequency, X(ω). It is closely related to the Fourier Series . If you are familiar with the Fourier Series , the following derivation may be helpful. The Fourier Transform 1.1 Fourier transforms as integrals There are several ways to de ne the Fourier transform of a function f: R ! C. In this section, we de ne it using an integral representation and state some basic uniqueness and inversion properties, without proof. Thereafter, For an odd function, the Fourier transform is purely imaginary. For a general real function, the Fourier transform will have both real and imaginary parts. We can write f˜(k)=f˜c(k)+if˜ s(k) (18) where f˜ s(k) is the Fourier sine transform and f˜c(k) the Fourier cosine transform. One hardly ever uses Fourier sine and cosine transforms.