How to integrate COMPLEX-valued functions?

[kingwinner]

Homework Statement

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?

Homework Equations

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The Attempt at a Solution

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Thanks for your help!

 

[clamtrox]

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.

 

[kingwinner]

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
∫ e^ax dx
c
is equal to
e^ad/a - e^ac/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!

 

[clamtrox]

For a complex a

$$\int dx e^{ax} = \frac{1}{a} e^{ax} = \frac{a^*}{|a|^2} e^{ax}.$$

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.

 

[kingwinner]

For a complex a

$$\int dx e^{ax} = \frac{1}{a} e^{ax} = \frac{a^*}{|a|^2} e^{ax}.$$

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.

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?