# How to integrate COMPLEX-valued functions?

1. Nov 11, 2009

### kingwinner

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

2. Nov 11, 2009

### 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.

3. Nov 11, 2009

### 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
∫ eax dx
c
is equal to
(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!

4. Nov 11, 2009

### 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.

5. Nov 12, 2009

### kingwinner

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?