# Upper limit gives zero

by neelakash
Tags: limit, upper
 P: 511 OK,I am writing it in LaTeX: $$\int^\infty_{2\pi/3}$$ $$\ e^{(i-s)t}$$ $$\ dt$$
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,682 Upper limit gives zero Well, obviously, the anti-derivative is $$\int e^{(i-s)t}dt= \frac{1}{i-s}e^{(i-s)t}$$ IF s> i, then the limit as t goes to infinity will be 0. If $s\le i$ the integral does not exist.
 Sci Advisor PF Gold P: 1,594 Moreover, what does an expression like $s \leq i$ mean, considering that the complex numbers are not ordered? If $s = \sigma + i\omega$, then $$\begin{array}{rcl}\frac 1 {i - s} e^{(i -s)t} & = & \frac1 {i - \sigma - i\omega} e ^{(i- \sigma - i\omega)t}\\&&\\ &=& \frac 1 {-\sigma + i(1-\omega)} e^{i(1-\omega)t} e^{-\sigma t}\end{array}$$ which has a finite limit at infinity only if $\sigma > 0$, and hence $\Re(s) > 0$. Hmm...maybe that's what was meant originally, then.
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,682 My mistake. I had not realized that "i" was the imaginary unit! In that case, separate it into two parts. $e^{(i-s)t}= e^{it}e^{-st}$. The $e^{it}$ part oscilates (it is a sin, cos combination) while e^{-st} will go to 0 as t goes to infinity because of the negative exponential. The entire product goes to 0 very quickly so the anti-derivative goes to 0. (The "ghost" of eit- I like that.)