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5.E: Fourier Transform (Exercises)

  • Page ID
    2155
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    These are homework exercises to accompany Miersemann's "Partial Differential Equations" Textmap. This is a textbook targeted for a one semester first course on differential equations, aimed at engineering students. Partial differential equations are differential equations that contains unknown multivariable functions and their partial derivatives. Prerequisite for the course is the basic calculus sequence.

    Q5.1

    Show

    \[
    \int_{{\mathbb{R}}^n}{\rm e}^{-|y|^2/2}\ dy=(2\pi)^{n/2}.
    \]

    Q5.2

    Show that \(u\in \mathcal{S}(\mathbb{R}n^)\) implies \(\hat{u},\ \widetilde{u}\in\mathcal{S}(\mathbb{R}^n)\).

    Q5.3

    Give examples for functions \(p(x,\xi)\) which satisfy \(p(x,\xi)\in S^m\).

    Q5.4

    Find a formal solution of Cauchy's initial value problem for the wave equation by using Fourier's transform.

    Contributors:


    This page titled 5.E: Fourier Transform (Exercises) is shared under a not declared license and was authored, remixed, and/or curated by Erich Miersemann.

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