Q:

Convergence of $\int\limits^0_{-\infty} e^{i a t + b t} dt$

Consider the integral $I :=$ $\int^{0}_{-t_{0}} e^{iat+bt} dt$, then it is true that $$ I = \frac{(1-e^{-(ia+b)t_{0}})}{ia+b} $$
I am now interested on the case where $ {t_{0}\to -\infty} $. Wolfram Alpha says that for $Re(b)>Im(a)$ it converges to $I = \frac{1}{ia+b} $
For my case I even have $a,b \in \mathbb{R}$ but I am still interested why this is the case. 
A hint where or at what to look would be very helpful, thanks!

A:

Lets just consider $z\in\mathbb C$ non-zero and write 
$$I(t_0)=\int_{-t_0}^0e^{zt}dt$$
for $t_0\geq0$. Then, as you noted,
$$I(t_0)=\frac{1-e^{-zt_0}}{z}.$$
Now we have 
$$\left|\frac{1}{z}-\frac{1-e^{-zt_0}}{z}\right|=\frac{|e^{-zt_0}|}{|z|}=\frac{e^{-Re(z)t_0}}{|z|}.$$
So if $Re(z)>0$ the right-hand side goes to zero, hence $I(t_0)$ converges to $\frac{1}{z}$ as $t_0\to\infty$.
Now if we write $z=ia+b$ where $a,b\in \mathbb C$, then $Re(z)=Re(b)-Im(a)$, so the condition $Re(z)>0$ is equivalent to $Re(b)>Im(a)$.

