Integration of functions of Complex Variables

  1. We can show that
    [tex]
    \int_{0}^\infty e^{-kx}dx=\frac{1}{k}
    [/tex]
    for real $$k>0.$$

    Does this result hold for $$\Re k>0$$ belonging to complex numbers? The reason I have this question is because $$i\times\infty$$ is not $$\infty$$ and so u substitution would not work.
     
  2. jcsd
  3. Char. Limit

    Char. Limit 1,986
    Gold Member

    Let's try this. We'll extend k to a+ib, and see what happens to our integral:

    [tex]e^{-(a+ib)x} = e^{-a x} e^{-i b x} = e^{-a x} cos(b x) - i e^{-a x} sin(b x)[/tex]

    Integrating this, we get the following two integrals:

    [tex]\int_0^\infty e^{-a x} cos(b x) dx = \frac{a}{a^2 + b^2}[/tex]

    and

    [tex]\int_0^\infty e^{-a x} sin(b x) dx = \frac{b}{a^2 + b^2}[/tex]

    Summing these two, we get [itex]\frac{a - i b}{a^2 + b^2}[/itex], or 1/(a + i b). Note that to get this, we DID assume that Re(k)>0, and we got as our answer 1/k. So we can say that e^(-k x), integrated from 0 to infinity, will give 1/k, where k is any complex number with real part greater than zero.

    In other words, yes, that's correct.
     
  4. Thanks for responding

    I talked to my professor and he said that if you look at the Riemann sphere, you could just assume that i*infinity is equal to infinity. The reason I got confused was because there is a similar notation which appears in the formula for the inverse Laplace transform.
     
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