# Questions tagged [fourier-transform]

The Fourier transform is important in mathematics, engineering, and the physical sciences. It is a tool that breaks a waveform (a function or signal) into an alternate representation, characterized by sine and cosines. The Fourier Transform shows that any waveform can be re-written as the sum of sinusoidal functions. It is named after French mathematician Joseph Fourier.

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### Evaluate the Fourier series equation below up to the 3rd harmonic and sketch the line spectrum for n=3

Fourier enter image description here
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### Correlation between FT of function constisting of a product and FT of its factors

Given is as function $C(x)=A(x)B(x)$ and I want to find the connection between its fourier transforms. So I want to find a function $$\tilde{C}(k)=\tilde{C}(\tilde{A}(k),\tilde{B}(k))$$ ($\tilde{}$...
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### A curious derivation of the Gaussian integral from Plancherel's Theorem

I was playing around with the Plancherel's theorem and stumbled across a derivation of the Gaussian integral which I have never seen before. Could folks check to see if derivation is correct? If so, ...
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### Finite-duration function in the time domain, means what in the frequency domain?

I'm trying to understand exactly what restrictions a finite duration time domain signal has on its Fourier transform. I found a helpful table in The Fourier Transform and its Applications by Ronald ...
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### Complicated Fourier transform

Let $t \in \mathbb{R}, k \in \mathbb{N}$. Then $$t^k \widehat{e^{-at} \mathbf{1}_{\mathbb{R}^+}}(t)(\xi) = \frac{k!}{(a+i\xi)^{k+1}}$$ (image) How to show this result ? I tried a IPP in the integral,...
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### Fourier transform on two variables

I have got a project to compute numericaly Poisson's and Laplace's equation with spectral method. I have implemented it and now I want to check it's error with analitical approach and here is a ...
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### Show that the solution to the Neumann problem is $u(x,y)=c+\frac{1}{2\pi}\int_{-\infty}^{\infty}{f(\xi)\ln[y^2+(x-\xi)^2]} d\xi$

Show that the solution to the following Neumann problem \begin{cases} u_{xx} + u_{yy} =0 \hspace{0.5cm} -\infty <x<\infty, y>0\\ u_{y}(x,0) = f(x), \hspace{0.5cm} -\infty <x<\infty\\ u(...
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### Calculate $\int_{\mathbb{R}}e^{ix\omega}(\frac{\sin(\omega)}{\omega})^2 d\omega$ using convolution theorem.

I have study Fourier transformation and found this exercise: Calculate $\int_{\mathbb{R}}e^{ix\omega}(\frac{\sin(\omega)}{\omega})^2 d\omega$ using convolution theorem. I have notice that integral ...
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### Fourier transform of a generalized complex exponential function

I have the following function $g(k)$, which I would like to perform the inverse Fourier transform upon, i.e., $g(k)\to \tilde{g}(x)$: \begin{align} g(k) \equiv \prod_{n=0}^{\infty} e^{i^n \alpha_n k^n}...
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### The Fourier Coefficients of $\cos(x)$

To begin I simply went with the definition: $$f(t)=\sum_{n=-\infty}^{\infty} f_k\cdot e^{ikw_ot} \implies f_k=\frac{1}{T}\int_{0}^{T} f(t)\cdot e^{-ikw_ot} \,dt$$ And before plugging $\cos(x)$ I first ...