The Fourier transform is important in mathematics, engineering, and the physical sciences. It is a tool that breaks a waveform (a function or signal) into an alternate representation, characterized by sine and cosines. The Fourier Transform shows that any waveform can be re-written as the sum of sinusoidal functions. It is named after French mathematician Joseph Fourier.

Many linear boundary value and initial value problems in applied mathematics, mathematical physics, and engineering science can be effectively solved by the use of the Fourier transform, the Fourier cosine transform, or the Fourier sine transform.

- These transforms are very useful for solving differential or integral equations for the following reasons.

First, these equations are replaced by simple algebraic equations, which enable us to find the solution of the transform function. The solution of the given equation is then obtained in the original variables by inverting the transform solution.

Second, the Fourier transform of the elementary source term is used for determination of the fundamental solution that illustrates the basic ideas behind the construction and implementation of Green’s functions.

Third, the transform solution combined with the convolution theorem provides an elegant representation of the solution for the boundary value and initial value problems.

Fourier Transform of a function $f(x)$ is denoted by $\mathcal{F[f(x)]}$ and is defined as $$\mathcal{F[f(x)]}=F(u)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} f(t)e^{iut} dt$$ where $\mathcal{F}$ is called the Fourier transform operator of the Fourier transformation.

This is often called the complex Fourier transform.

The inverse Fourier transform of $F(u)$ is denoted by $\mathcal{F}^{-1}[F(u)]$ and is defined as $$\mathcal{F}^{-1}[F(u)]=f(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} F(u)e^{iux} du$$

**References:**

"Integral Transforms and Their Applications" by Dambaru Bhatta and Lokenath Debnath