Register to reply

Integration of functions of Complex Variables

by Charles49
Tags: complex, functions, integration, variables
Share this thread:
Feb22-12, 12:11 PM
P: 87
We can show that
\int_{0}^\infty e^{-kx}dx=\frac{1}{k}
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.
Phys.Org News Partner Science news on
Bees able to spot which flowers offer best rewards before landing
Classic Lewis Carroll character inspires new ecological model
When cooperation counts: Researchers find sperm benefit from grouping together in mice
Char. Limit
Feb22-12, 01:40 PM
PF Gold
Char. Limit's Avatar
P: 1,945
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]


[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.
Feb24-12, 05:02 PM
P: 87
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.

Register to reply

Related Discussions
Graphing functions of two complex variables. General Math 2
Functions of Complex Variables Calculus & Beyond Homework 7
Integration of complex functions Calculus 3
Complex Variables integration formulas Calculus & Beyond Homework 4
Complex Variables and Integration Introductory Physics Homework 5