- #1
kingwinner
- 1,270
- 0
Homework Statement
The problem is to evaluate:
+∞
∫ exp(-3x- i ωx) dx
0
But I don't understand how to compute
lim exp(-3x- i ωx)
x->+∞
The following is the solution presented, but I don't understand it...
lim exp(-3x- i ωx) =
x->+∞
lim exp(-3x) exp(- i ωx) =
x->+∞
lim exp(-3x) [cos(ωx) - i sin(ωx)]
x->+∞
Now since exp(-3x)->0 as x->+∞ and [cos(ωx) - i sin(ωx)] is bounded, we have that
lim exp(-3x) [cos(ωx) - i sin(ωx)] = 0
x->+∞
Can someone please explain the part in red?
1) What does it mean for a COMPLEX-valued function [cos(ωx) - i sin(ωx)] to be BOUNDED?
2) What is the meaning of a LIMIT of a COMPLEX-valued function?
3) Why exp(-3x)->0 as x->+∞ and [cos(ωx) - i sin(ωx)] bounded => lim exp(-3x) [cos(ωx) - i sin(ωx)] = 0 ?
I am so confused...everything I've learned in calculus is in the field of REAL numbers, and I have no background in complex analysis at all, so please explain in the simplest way...
Homework Equations
N/A
The Attempt at a Solution
N/A
Any help is greatly appreciated! :)