FT1

From Example Problems
Jump to: navigation, search

Find the Fourier transform of f(t)=e^{{-|t|}}\,

The Fourier transform is given by

{\hat  {f}}(\omega )=\int _{R}f(t)e^{{-i\omega t}}\,dt\,

In this problem,

{\hat  {f}}(\omega )=\int _{R}e^{{-|t|}}e^{{-i\omega t}}\,dt\,

=\int _{{-\infty }}^{0}e^{{(1-i\omega )t}}\,dt+\int _{0}^{\infty }e^{{-(1+i\omega )t}}\,dt\,

={\frac  {1}{1+i\omega }}+{\frac  {1}{1-i\omega }}={\frac  {2}{1+\omega ^{2}}}\,

Main Page : Partial Differential Equations : Fourier Transforms