How to integrate in s-domain We know that a simple integral in the time domain for example, is put like that:
$$\int_a^bf(t)dt$$
So $t$ will vary from $a$ to $b$ and all values in between (like a straight line).
But when I do something like this:
$$\int_{z_1}^{z_2}F(s)ds$$
which I suppose that can be re-written by:
$$\int_{a+jb}^{c+jd}F(s)ds  , s\in \mathbb{C} $$
since $s$ is a complex number. If so, is that the same thing of doing this: (?)
$$\int_{b}^{d}\int_a^cF(p+jq)dpdq$$
If not, what does it really mean to make $s$ vary from a point to another in the complex plane? A straight line connecting both points?
 A: To evaluate, or indeed to define, an integral such as $\int_{z_1}^{z_2} F(s) ds$ in the complex domain, a path joining $z_1$ and $z_2$ must specified, and for $ds$ to make sense, it must be a differentiable path; that is, a map $\gamma:[t_1, t_2] \to \Bbb C$ such that 
$\gamma(t_1) = z_1$ and $\gamma(t_2) = z_2$ and $\gamma'(t)$ exists for all $t \in [t_1, t_2]$.  Then writing $s = \gamma(t)$, we have $ds = \gamma'(t)dt$ and hence
$\int_{z_1}^{z_2} F(s) ds = \int_{t_1}^{t_2} F(\gamma(t)) \gamma'(t) dt; \tag{1}$
the real and imaginary parts of $ F(\gamma(t)) \gamma'(t)$ occurring on the right-hand side of (1) may thus be integrated as ordinary (real) functions of $t$.  Note that we needn't perform a double integral such as $\int_{b}^{d}\int_a^cF(p+jq)dpdq$.  Also, a straight line segment joining $z_1$ and $z_2$ is perhaps the simplest path admissible under the conditions I have stated, and it proves very useful in very many applications; but there are many other possible paths joining $z_1$ and $z_2$; different paths suit different problems.
Hope this helps.  Cheers,
and as always,
Fiat Lux!!!
