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Laplace transform of complex exponential

  1. Aug 24, 2005 #1
    I just want to be sure I understand this correctly, usually L[f(t)] = 1/(s-a), where [tex]f(t) = e^{at}[/tex], but if it is a complex number would still be 1/(s - complex_number)?
    techinically, i think it should be, since every number can be reprsented as complex number. Just want to be sure about this with Laplace transform.
    thanks much.
  2. jcsd
  3. Aug 25, 2005 #2


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    Definitely! You can demonstrate it by direct integration.
  4. Aug 25, 2005 #3


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    Yes and it is a nice way to find the laplace transforms of sin and cos.
    L[cos(a x)+i sin(a x)]=L[exp(i x)]=1/(s-a i)=(s+a i)/(s^2+a^2)
    hence (equating real and imaginary parts)
    L[cos(a x)]=s/(s^2+a^2)
    L[sin(a x)]=a/(s^2+a^2)
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