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 December 16th, 2015, 01:03 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 Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post jsmith613 Calculus 1 October 1st, 2015 09:02 AM fannushofi Calculus 1 December 15th, 2014 03:37 PM Dmath Elementary Math 7 April 15th, 2014 08:54 PM progrocklover Real Analysis 1 March 24th, 2011 08:29 PM carnagr Complex Analysis 0 April 2nd, 2009 04:41 PM

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