Fourier Transform—Cosine

Author

Eric W. Weisstein
May 6, 2003

This notebook downloaded from http://mathworld.wolfram.com/notebooks/IntegralTransforms/FourierTransformCosine.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/FourierTransformCosine.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Initialization

SetOptions[FourierTransform,FourierParameters->{0,-2Pi}];
DiracSimplify[expr_]:=
expr/.DiracDelta[x_]:>DiracDelta[Factor[x]]/.DiracDelta[c_?NumericQ r_]:>DiracDelta[r]/c

Cosine

cos(2 π (Subscript[k, 0]) x)

FourierTransform[Cos[2Pi (Subscript[k, 0]) x],x,k]

Factor[DiracSimplify[%]]//TraditionalForm

cos(π x)

FourierTransform[Cos[Pi x],x,k]

DiracSimplify[%]

(DiracDelta[HoldForm[-1/2+ k]]+DiracDelta[HoldForm[1/2+ k]])/2//TraditionalForm