If we replace the imaginary variable \(i\xi \) in the Fourier transform \(\displaystyle \hat {f}(\xi )=\int _{-\infty }^{\infty }f(x)e^{-i\xi x}dx \) by the complex variable \(s=\sigma +i\xi ,\) and set \(f(x)=0\) for all \(x<0,\) the function defined by the resulting integral, \(\displaystyle F(s)=\int _{0}^{\infty }f(x)e^{-sx}dx, \) is called the Laplace transform of \(f.\)