Recently, I am dealing with an integral $$\int^\infty_{a+bi}f(z)dz$$ where $f(z)$ is a meromorphic function on $\mathbb C$.
The $\infty$ shall be understood as the real positive infinity, and conventionally the integral is along a straight line connecting the two endpoints.
When I draw out the integral path, I got 
The path is the hypotenuse of a right angled triangle with an infinitely long base and a finite height. Thus, the slope of the integral path is zero.
So, intuitively, $$\int^\infty_{a+bi}f(z)dz=\int^{\infty+bi}_{a+bi}f(z)dz$$ and the integral path can be described as $\Im(z)=b, \Re(z)\ge a$.
Is the equality (trivially) true? If yes, how can one prove it mathematically?
