Fourier Transforms

f̂k a−a1 e ikx dxa 1ike ikx −a2sinkakNote that the function has a discontinuity at x a but its FT is still defined.Example 3Suppose that fx is a real-valued even function, then determine its FT and show that it alsoeven.By definitionf̂k − −fx e ikx dxfx coskx isinkx dx −fx coskx dx i −fx sinkx dx 2−fx coskx dx 2f̂ckwhere the properties of coskx and sinxk being even and odd respectively has beenemployed together with the knowledge that fx is even. It is clear that f̂k is even, sincecoskx cos−kx and hence f̂−k f̂k.3.2.2 Derivatives and other useful identitiesFT of Derivative. The notation f̂k employed in the definition of the complex FTessentially defines a function of k. It is also possible to consider the definition as anoperator on fx, sothatFfx f k −fx e ikx dx .and F indicates the FT operator. This enables us to determine the FT of the derivatives offx,Ff ′ x −f ′ x e ikx dx fx e ikx −− ik −by definitionfx e ikx dxintegrating by parts Ff ′ x − ik Ffx − ik f̂k . (3.2.3)In this derivation , the property that fx → 0as|x| → if fx is absolutely integrable hasbeen employed.6