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I am trying to calculate the Fourier transform of $f(t)=Ae^{-i\omega_0 t}$

I'm getting an infinity which is giving me problems. Here are my steps:

$$F(\omega)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty f(t)e^{i\omega t}dt$$ $$=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty Ae^{-i\omega_0 t}e^{i\omega t}dt$$ $$=\frac{A}{\sqrt{2\pi}}\int_{-\infty}^\infty e^{t(i\omega - i\omega_0)}dt$$ $$=\frac{A}{\sqrt{2\pi}(i\omega - i\omega_0)} |e^{t(i\omega - i\omega_0)}|_{-\infty}^{\infty}$$ $$=\frac{A}{\sqrt{2\pi}(i\omega - i\omega_0)}(\infty - 0)$$

Where am I going wrong? Thanks.

  • 1
    You are going wrong nowhere. The Fourier transform (in the simple-minded setting) is undefined.2012-10-16

1 Answers 1