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Fourier Transform of Integrals Calculator

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Example
Created on 2024-06-20Asked by Henry Taylor (Solvelet student)
Find the Fourier transform of the integral ∫0xe−t2 dt \int_{0}^{x} e^{-t^2} \, dt .

Solution

To find the Fourier transform of the integral ∫0xe−t2 dt \int_{0}^{x} e^{-t^2} \, dt : F[∫0xe−t2 dt]=π erf(x), \mathcal{F}\left[\int_{0}^{x} e^{-t^2} \, dt\right] = \sqrt{\pi} \, \text{erf}(x), where erf(x) \text{erf}(x) is the error function. Solved on Solvelet with Basic AI Model
Some of the related questions asked by Mateo Green on Solvelet
1. Compute the Fourier transform of the integral of the function f(x)=e−x f(x) = e^{-x} from −∞ -\infty to ∞ \infty .2. Find the Fourier transform of the function g(x)=11+x2 g(x) = \frac{1}{1 + x^2} using the Fourier transform of its integral.
DefinitionIn mathematics, the Fourier transform of the difference of two piecewise continuous functions is a prperty of the Fourier transform which extends the concept of a Fourier series to functions which have two types of discontinuities. This is where this property comes into the picture where it states that the discrete frequencies component (harmonics) predict the presence of non-smooth portions of the function. Piecewise Continuous FunctionsThe usage of the Fourier transform of piecewise continuous functions to analyze signal in signal processing, to filter noise in image analysis, and to solve differential equations with discontinuous boundary conditions in mathematical physics. For instance, we can rephrase that the Fourier transform of a piecewise continuous function f(x) takes the following form F(ω)=∫f(x)e−2πiωxdx−∞∞, where F(ω) represents the frequency domain representation of the function
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