FT2

From Example Problems
Jump to: navigation, search

Find the Fourier transform of f(t)={\begin{cases}1&|t|<1\\0&|t|>1\end{cases}}\,

F(\omega )=\int _{{-1}}^{1}e^{{-i\omega t}}\,dt={\frac  {e^{{-i\omega t}}}{-i\omega }}{\Bigg |}_{{-1}}^{1}={\frac  {2}{\omega }}\cdot {\frac  {e^{{i\omega }}-e^{{-i\omega }}}{2i}}={\frac  {2\sin(\omega )}{\omega }},\,\,\,\omega \neq 0\,

When \omega =0\,, e^{{-i\omega t}}=1\,

In this case,

F(0)=\lim _{{\omega \rightarrow 0}}F(\omega )=2\,.

Main Page : Partial Differential Equations : Fourier Transforms