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Inverse Laplace Transform Step by Step

  1. Jul 5, 2005 #1
    Hi - I really need someone to show me step by step how to do an Inverse Laplace transform using a contour integral. The one I would like to understand is the frequency function 1/sqrt(s)

    Thank you if you can help me out.

    Steve
     
  2. jcsd
  3. Jul 6, 2005 #2

    lurflurf

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    Homework Helper

    This is the contour integral that gives the inverse.
    [tex] f(t)= \frac{1}{2{\pi}i} \int_{a-i\infty}^{a+i\infty} g(s)e^{st} ds [/tex]
    or for the specific function.
    [tex] f(t)=\frac{1}{2{\pi}i}\int_{a-i\infty}^{a+i\infty} \frac{e^{st}}{\sqrt{s}} ds [/tex]
    We need to take "a" far enough to the right that we avoid problems.
    Here we may take a=0, as even though the function has problems at zero, they are not major. You can consider a small right half circle and see the integral is small.
    [tex] f(t)=\frac{1}{2{\pi}i}\int_{-i\infty}^{i\infty} \frac{e^{st}}{\sqrt{s}} ds [/tex]
    we can clean the integral up with a substitution i u=s t
    [tex] f(t)=\frac{1}{2{\pi}\sqrt{it}}\int_{-\infty}^{\infty} \frac{e^{iu}}{\sqrt{u}} du [/tex]
    This integral can be written in terms of "know" real integrals.
    [tex]\int_0^\infty \frac{sin(x)}{\sqrt{x}} dx=\int_0^\infty \frac{cos(x)}{\sqrt{x}} dx=\sqrt{\frac{\pi}{2}}[/tex]
    thus the answer
    [tex]f(t)=\frac{2+2i}{2{\pi}\sqrt{it}}\sqrt{\frac{\pi}{2}}[/tex]

    [tex]f(t)=\frac{1}{\sqrt{{\pi}t}}[/tex]
    You can also do a real inversion.
    [tex]f(t)=\lim_{k\rightarrow\infty}\frac{(-1)^k}{k!}g^{(k)}(\frac{k}{t})(\frac{k}{t})^{k+1}[/tex]
     
    Last edited: Jul 6, 2005
  4. Jul 6, 2005 #3
    Thank you.

    Thanks for the reply I really appreciate it!

    Steve
     
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