My Math Forum Proof of Fourier Transform and Inverse Transform

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 December 16th, 2015, 02:03 PM #1 Senior Member     Joined: Oct 2014 From: EU Posts: 224 Thanks: 26 Math Focus: Calculus Proof of Fourier Transform and Inverse Transform Hi, Hi have some doubt in computing the proof of: $\displaystyle (1)\qquad \mathcal{F}[\mathrm e^{\mathrm i\omega_0t}] = 2\pi \delta(\omega -\omega_0)$ And also its inverse: $\displaystyle (2)\qquad\mathcal{F}[\mathrm e^{\mathrm i\omega_0}]^{-1} = \delta(t- t_0)$ Eq(2) is taken from the book by Oppenheim, Signals and Systems 2nd Ed. Simulating the results with an online solver for the equation (2), it results that equation (2) should be: $\displaystyle (2.1)\qquad\mathcal{F}[\mathrm e^{\mathrm i\omega_0}]^-1 = \sqrt{2 \pi} \delta(t- t_0)$ So who is right and who is wrong ? Also, it's not clear to me where does the $\displaystyle 2 \pi$ comes from in equation (1), the same for the $\displaystyle \sqrt{2 \pi}$ in equation (2.1). For example, I would solve equation (1) as follows: \displaystyle \begin{aligned} \mathcal F[\mathrm e^{\mathrm i\omega_0 t}] &= \int_{-\infty}^{\infty}\mathrm e^{\mathrm i\omega_0 t}\mathrm e^{-\mathrm i\omega t}\,\mathrm dt\\ & = \int_{0}^{2 \pi}\mathrm e^{\mathrm it(\omega_0 - \omega)}\,\mathrm dt\\ & = \left [ {\mathrm e^{\mathrm it(\omega_0 - \omega)} \over \mathrm i (\omega_0 - \omega)}\right ]_0^{2\pi}\\ & = {\mathrm e^{\mathrm i 2 \pi (\omega_0 - \omega)} - 1 \over \mathrm i (\omega_0 - \omega)} \end{aligned} Which is wrong or incomplete ... The same happens to me for the Inverse Fourier Transform: \displaystyle \begin{aligned} \mathcal F[\mathrm e^{\mathrm i\omega_0}]^{-1} &= {1 \over 2\pi}\int_{-\infty}^{\infty}\mathrm e^{\mathrm i\omega_0}\mathrm e^{\mathrm i\omega t}\,\mathrm d\omega\\ & = {\mathrm e^{\mathrm i\omega_0} \over 2\pi} \int_{0}^{2\pi}\mathrm e^{\mathrm i\omega t}\,\mathrm d\omega\\ & = {\mathrm e^{\mathrm i\omega_0} \over 2\pi}\left[ {\mathrm e^{\mathrm i\omega t} \over \mathrm i t}\right ]_0^{2\pi}\\ & = {\mathrm e^{\mathrm i\omega_0} \over 2\pi}\left ( {\mathrm e^{\mathrm i 2\pi t} - 1 \over \mathrm i t}\right ) \end{aligned} from which I am not able to determine the $\displaystyle \delta(t)$ neither the coefficient $\displaystyle \sqrt{2 \pi}$. I don't understand where I am wrong.. If I am so.. Thank you in advance for your help. szz

 Tags fourier, inverse, proof, transform

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