Here is an example of a Complex Fourier Transform.[br][br]The Fourier Transform is defined as [math]F\left(\omega\right)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f\left(x\right)e^{-i\omega x}dx[/math]. The Fourier Transform, transforms a function of [math]x[/math] into a new function of [math]\omega[/math]. This can be used to solve partial differential equations. [br][br]For the case [math]f\left(x\right)=e^{-a\left|x\right|}[/math] the Fourier Transform is [math]F\left(\omega\right)=\sqrt{\frac{2}{\pi}}\left(\frac{a}{a^2+\omega^2}\right)[/math]. Below is a graph where the value of the constant [math]a[/math] can be varied with a slider. The imaginary part of this transform is zero since the function is an even symmetric function ([math]f\left(-x\right)=f\left(x\right)[/math]).