The paper describes the development of the numerical Fourier transform with exponential sampling. This is called the exponential Fourier transform (e.f.t.). The e.f.t. gives quite a high accuracy and a very long observation time with a small number of samples in comparison with a conventional numerical Fourier transform with equally spaced sampling. Thus, the computational efficiency of the e.f.t. is much higher than the conventional Fourier transform. Various calculated examples using the e.f.t. are given and are compared with the results obtained by the conventional Fourier transform. From the results the superiority of the e.f.t. compared with the conventional Fourier transform becomes clear. The e.f.t. would be very useful to deal with electrical transients with a very wide range of frequency or time.