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Improper Integrals

  1. Mar 9, 2014 #1
    What are the real life applications of improper integrals? Why are they on the syllabus of every first course in calculus?
    I am looking for examples which have a real impact.
  2. jcsd
  3. Mar 9, 2014 #2


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    The definition of the Laplace transform is by the improper integral
    F(s) = \int_0^\infty e^{-st} f(t)\,dt.

    The definition of the Fourier transform is by the improper integral
    F(\omega) = \int_{-\infty}^{\infty} e^{i\omega t} f(t)\,dt.

    The cumulative distribution function of the standard Normal distribution is defined by the improper integral
    \Phi(z) = \int_{-\infty}^z \frac{1}{\sqrt{2\pi}} e^{-\frac12 z^2}\,dz

    The space of wavefunctions in quantum mechanics has as its inner product the improper integral
    \langle f,g \rangle = \int_{-\infty}^{\infty} f(x)g^{*}(x)\,dx

    Do you need further examples?
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