Fourier Series and Transforms Calculator

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Example

Created on 2024-06-20Asked by Madison Jackson (Solvelet student)

Find the Fourier series and Fourier transform of the function

f(x) = e^{-x^2}

Solution

To find the Fourier series and Fourier transform of the function

f(x) = e^{-x^2}

\text{Fourier Series:}

The Fourier series of

f(x)

is not straightforward to compute due to the function's non-periodic nature. It can be expressed in terms of a power series or as a Gaussian integral.

\text{Fourier Transform:}

The Fourier transform of

f(x) = e^{-x^2}

is given by:

F(\omega) = \mathcal{F}[e^{-x^2}] = \sqrt{\pi} e^{-\frac{\omega^2}{4}}.

Solved on Solvelet with Basic AI Model

Some of the related questions asked by Eleanor Thomas on Solvelet

1. Compute the Fourier series of the function

f(x) = x^3

on the interval

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.2. z

DefinitionFourier series and transforms are mathematical techniques used to represent and analyze functions in terms of their frequency components. Fourier series decompose a periodic function into a sum of sine and cosine cathetus, while Fourier transforms generalize the concepts to non-periodic functions representing them as integrals of complex exponentials over the entire real line. In addition, Fourier series and transforms provide information on the frequency content, amplitude, and phase of functions. Fourier series and transforms are used in areas that use varying frequencies such as signal processing, image analysis, quantum mechanics, and communication systems to view the signal, filter noise, and secure information. For example, F(k) = ∫- ∞ ∞ f(x) e-2πikx dx is the Fourier transform of the function f(x), with F(k) representing the frequency domain of the function.

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