# Properties of the Fourier transform

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## Symmetry example 4 of the FT: Purely imaginary even function

$F.4(x)=F.4(-x)=i⋅2⋅cos3x=i⋅F.1(x)⇔C.4(k)=i⋅C.1(k)=0+i⋅reC.1(k)$

Multiplication by $i$ shifts the real part in example 1 into the imaginary part of the spectrum without reversing the sign. It applies

$in theC.4(-3)=in theC.4(3)andreC.4(k)=0$

This finding applies to everyone $k$ and because of the linearity for any linear combination of $i⋅coskx$Functions.

theorem
For any purely imaginary even function, the spectrum is purely imaginary and straight (symmetrical).