What kind of integral is the one in Laplace transform?

Question

In a book about circuits I found a chapter where it is defined the Laplace transform of a function $f:\mathbb{R}\rightarrow\mathbb{R}$ as $$L[f(t)]=\int_0^{+\infty}f(t)e^{-st}dt$$ where $s\in\mathbb{C}$. This is the first time I find an integral where the function to integrate is a complex function and I would like to know what kind of integral this is (how it is defined) because I could not ind it anywhere else.

Answer 1

There are two main cases ; if one wants to integrate a function $f:\mathbb{R} \to \mathbb{C}$, then the problem can be reduced to integrating two functions $x \mapsto \Re(f(x))$ and $x \mapsto \Im(f(x))$. Indeed, we have the following equality :

$$\int_I f(t) dt = \int_I \Re(f(t)) dt + i \int_I \Im(f(t)) dt.$$

Since the integral of real functions is often taught as measuring the "aera under a curve", the integral of complex valued function of real variable is almost the same, except that we are considering two aeras, the one "under the real part curve" of the image and "under the imaginary part".

However, do not use that equality everytime. Some complex valued functions (and a lot of them are usual functions) can be integrated using the same rules as real ones. For example, the antiderivative of $x \mapsto e^{ix}$ is $x \mapsto \frac{e^{ix}}{i}$, just as one would have $x \mapsto \frac{e^{ax}}a, a \in \mathbb{R}$ being the antiderivative of $x \mapsto e^{ax}$, we just had to consider $i$ as a number and apply the usual rule.

The things get more complex if we want to integrate over some domain on the complex plane and is part of a major theory in mathematics constitued by complex analysis.