Why is preferable the line integral than the area integral over C plan

[Jhenrique]

Why is preferable to use the line integral than the area integral over the complex plane?

 

[HallsofIvy]

I don't know what you mean by "preferable" here. Both are used for different purposes.

 

[WWGD]

Line integrals are used when you ( or anyone! ) integrates over contours/curves. When you integrate
over a surface, you ( or anyone!) integrates over a region , you use surface integrals, tho you may be able
to use Stokes' thm. to integrate over a curve if the conditions are right.

 

[platetheduke]

When you do a line integral in real euclidean space you it is defined using real number multiplication, when you do a line integral in the complex case it uses complex multiplication, which takes account of angles as well as magnitudes. So there is a different geometric interpretation.

Furthermore, just as in the real case, complex line integrals let you create local inverses of your functions. For example, $e^z$ has period $2\pi i$, so it is not 1-1. If it has an inverse $f(x)$ near $0$, then $f(1)=0$ and $e^{f(z)}=z$ near $z=0$. Taking derivatives gives $1=f^{\prime}(z)e^{f(z)}=f^{\prime}(z)z$. So $f^{\prime}(z)=\frac{1}{z}$. The analogue of the fundamental theorem of calculus shows that $f(z)=\int_{1}^{z}\frac{1}{t}dt$ no matter which path you take in $\mathbb{C}$ from $1$ to $z$ as long as your path doesn't go "too far" away from $1$. In this case, $f$ is one of the branches of the complex logarithm.

As final example, line integrals in $\mathbb{C}$ allow you to extract information about an analytic function (including its derivatives) due to my first point. This is not possible in $\mathbb{R}$.

In summary, analytic functions are very rigid, and continuous paths allow you to take data about them at one location and propagate it elsewhere. This is not possible in the reals, since knowing information about a smooth real function at one location can give you no information about how it behaves elsewhere. Just as a path let's you carry the information, an integral can allow you to see its cumulative effect or average it out.