Exponential integral

1. Jul 31, 2010

Skullmonkee

Hi i was wondering if anyone could help me with a fairly simple integration.

$$\int exp(iab/c)exp(-iaz) da$$

Where the integral is from 0 to infinity

Im not sure how to proceed with this and am basically teaching myself
Thanks in advance for any help.

2. Jul 31, 2010

Pengwuino

Do you have any conditions on b,c, and z? This integral is not going to be solvable as far as I can tell. Remember, $$e^{ix} = cos(x) + isin(x)$$ and you'll end up trying to take a limit at infinity which does not exist for cosines and sines.

Last edited: Jul 31, 2010
3. Jul 31, 2010

ACPower

I say the answer is 0
First consolidate the integrand to exp(iaK) where K = (b/c) - z, then expand the exponential to cos(aK) +i*sin(aK)
then $$\int cos(aK) da + i\int sin(aK) da = 0$$
since integrating a sine or cosine gives zero, since there are equal positive and negative areas for every cycle.

4. Jul 31, 2010

Pengwuino

unless the limit is at infinity, which means the limit is not defined.

5. Aug 1, 2010

JJacquelin

In the wording of the problem, Skullmonkee forgot to specify if the coefficients b, c, z are real or complex.
Suppose b, c real and z complex. If z=x+i y , with x , y real and y<0
then, the integral (from 0 to infinity) is convergent = 1/(-y+i(x-b))

6. Aug 1, 2010

Let $k = i(b/c-z)$. Then the integral is the same as
$$\int_0^\infty \exp(ka) \;da$$
which converges precisely when $\operatorname{Re}(k) < 0$. In this case, the value of the integral is
$$\frac{1}{k} \exp(ka) \biggr|_{a=0}^\infty = -\frac{1}{k} = \frac{i}{b/c-z}.$$

Pengwuino and ACPower assume that $b/c - z$ is real, but if that is not the case, the integrand may not be purely sinusoidal (that is, $k$ above may have nonzero real part). JJacquelin gives an incorrect value for the integral (there is a missing c).

Last edited: Aug 1, 2010
7. Aug 1, 2010

JJacquelin

Adriank is right, I forgot c in my equation. Thanks for bringing the mistake to our attention.
If b and c are real and z complex ( i.e.: z=x+i y , with x and y real), the condition of convergence is y<0 as already said.
Of course, if all b, c, z are complex a more general condition has to be derived from :
[Real part of i((b/c)-z)] < 0