$\exp(-i \infty)$ is "Not a Number" according to MATLAB. Why?

Question

I ran into this problem while trying to understand the Laplace transform via MATLAB.

exp(-i*Inf)

NaN + NaNi

But,

exp(-Inf)

0

Furthermore,

syms t
fun= exp(-(0+i)*t)
answer=int(fun,0,Inf)

returns

NaN

But it is able to solve it if no limits are given.

answer =
exp(-t*1i)*1i

Something about Infinity is not quite right with MATLAB. Any suggestions on how to make it right?

Answer 1

The reason is simple. The limit of $$e^{-i\phi}$$ as $\phi\to\infty$ simply doesn't exist.

You see this since we have $$e^{-i\phi}=\cos\phi-i\sin\phi.$$ Letting $\phi$ become infinite, we cannot assign any definite value to the trigonometric functions on RHS.

On the other hand, we have that $e^{-x}$ is vanishing when $x\to\infty$ since $e^{-x}=1/e^x.$