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How to integrate COMPLEX-valued functions?

  1. Nov 11, 2009 #1
    1. The problem statement, all variables and given/known data
    How do we evaluate integrals of COMPLEX-valued functions (like the following)?

    2
    ∫ exp[-(2+3i)x] dx
    0

    This integral is appearing in my PDE course in the topic of Fourier transform, but I have no background of how to integrate complex-valued functions. Looking back at my calculus textbook, all functions are assumed to be REAL-valued so really all theorems about integrals are not applicable in this situation...

    Can I simply treat -(2+3i) as a scalar and use the exact same rules of integration from calculus? i.e. is integrating complex-valued functions exactly the same as integrating real-valued functions if we treat complex numbers as scalars?

    2. Relevant equations
    N/A

    3. The attempt at a solution
    N/A

    Thanks for your help!
     
  2. jcsd
  3. Nov 11, 2009 #2
    In this case, the fact that you have an i in your equation makes no difference at all. If the integration path has complex values, then you should use more caution; however the kind of calculations you usually want to do in a complex plane are a fair bit different from the usual kind of integrals.
     
  4. Nov 11, 2009 #3
    1) I think for complex-valued functions, we need to integrate the real and imaginary parts separately, so to integrate f where f is complex-valued, we use f = Re(f) + i Im(f). Now to integrate iIm(f), can the "i" be pulled out of the integral? (i.e. can we treat "i" as a constant?)


    2) Can we say that for any nonzero complex a,
    d
    ∫ eax dx
    c
    is equal to
    ead/a - eac/a ??
    (which is exactly the same as the case where a is a real number)

    Do all the integration formulas for real-valued functions carry over analogously in the obvious way to the complex-valued function case?
    Are there any examples in which the real-valued formulas does not hold in the complex-valued case?

    Thanks!
     
  5. Nov 11, 2009 #4
    For a complex a

    [tex] \int dx e^{ax} = \frac{1}{a} e^{ax} = \frac{a^*}{|a|^2} e^{ax}. [/tex]

    You really really do not have to care at all whether a is real or not, unless ofcourse if you're asked to prove that something holds for a complex number as well.
     
  6. Nov 12, 2009 #5
    So keeping in mind that the scalars/constants can now be complex numbers, all the formulas for integration of complex-valued functions are exactly the same as the fomrulas for integration of real-valued functions, am I right?
    So is there really no real difference between integrating real or complex valued functions? Are there any examples in which this is not the case?
     
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