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Fourier Transform--Cosine


$\displaystyle {\mathcal F}[\cos(2\pi k_0x)]$ $\textstyle =$ $\displaystyle \int_{-\infty}^\infty e^{-2\pi ikx}\left({e^{2\pi ik_0x}+e^{-2\pi ik_0 x}\over 2}\right)\,dx$  
  $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\int_{-\infty}^\infty [e^{-2\pi i(k-k_0)x}+e^{-2\pi i(k+k_0)x}]\,dx$  
  $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}[\delta(k-k_0)+\delta(k+k_0)],$  

where $\delta(x)$ is the Delta Function.

See also Cosine, Fourier Transform--Sine




© 1996-9 Eric W. Weisstein
1999-05-26