Different integrals for $\mathbb{C} \to \mathbb{C}$ functions I am a little confused with the different integrals on the complex numbers. So lets start


*

*$\mathbb{R} \to \mathbb{R}$
Standard Lebesgue integration

*$\mathbb{R}^2 \to \mathbb{R}$
We can either integrate along a path in $\mathbb{R}^2$, then on 
this path we have a $\mathbb{R} \to \mathbb{R}$ or we use measure theory
and the two dimensional Lebesgue measure.

*$\mathbb{R} \to \mathbb{C}$ (i.e.$\mathbb{R} \to \mathbb{R}^2$)
Just integrate the real and imaginary part separately.
Now we are at the core of my problem
4. $\mathbb{C} \to \mathbb{C}$  (i.e. $\mathbb{R}^2 \to \mathbb{R}^2$)
So again we have two choices. either do a line integral or we use measure theory
and in each case treat real and imaginary parts differently.
Obviously these integrals are not the same. If we take a constant function and
some domain. The integral over this domain using the path integral will be zero.
The other one will not.
Unlike the case $\mathbb{R} \to \mathbb{R}$ there is not a unique integral
for functions $\mathbb{C} \to \mathbb{C}$. So my questions are:


*

*Why is the path integral the "right" one, as it is the one used in complex analysis.

*Is there a more natural or intrinsic construction of this integral.

*Does the other integral for $\mathbb{C} \to \mathbb{C}$ occur anywhere in mathematics.
Does it have important applications

 A: In my opinion, the question is rather misleading. In complex analysis, mathematicians introduced an "integral along a path", which is most useful because it gives the Cauchy theorem. There is a point that is often hidden in classical treatment of function theory (of one complex variable): in the 21-st century we should probably accept that $\int_{\gamma} f(z) \, dz$ is the integral of the 1-form $\omega = f(z) \, dz$. Complex differentiability is simply real differentiability plus Cauchy-Riemann. On simply connected open subsets of $\mathbb{C}$, analytic functions are simply those for which $\omega$ is exact as a differential form: this is Cauchy's theorem.
I'd say that there is no right integral: forget any comparison with measure theory  in $\mathbb{R}$ o in $\mathbb{R}^2$. In a modern setting, complex analysis is a branch of differential geometry. I do not really understand why books still present complex analysis the way the pioneers did in the 19-th century.
A: There isn't necessarily a right answer, but contour integration certainly has a rich theory that essentially lays the foundation for all elementary complex analysis. The Cauchy Integral Theorem/Formula is a pretty huge deal, and also allows us to extend the theory to things like the holomorphic functional calculus. Interestingly enough, the more general Cauchy Integral Formula for a $C^1$ function $f:\mathbb{C}\to\mathbb{C}$ actually involves both contour integrals and area integrals:
$f(z) = \frac1{2\pi i} \left(\oint_{\partial \Omega}\frac{f(w)}{w-z}dw - \iint_{\Omega}\frac{\partial f}{\partial \overline{w}}\frac1{w-z}d\overline{w}dw\right)$
where the boundary $\partial \Omega$ is piecewise $C^1$. If you're interested in this, I would suggest this article by Steven Krantz.

For a more natural interpretation of the contour integral, Polya associated holomorphic function $f(z)$ with the vector field
$W(z) = \left[Re\left(\overline{f(z)}\right),Im\left(\overline{f(z)}\right)\right] = \left[Re\left(f(z)\right),-Im\left(f(z)\right)\right]$
which is both irrotational (having zero two-dimensional curl) and incompressible (having zero divergent) due to the Cauchy-Riemann equations. Given a path $\Gamma$ in the complex plane, the real part of the contour integral $\int_\Gamma f(w) \ dw$ can be interpreted as the work done by $W$ along $\Gamma$, and the imaginary part as the flux. Tristan Needham gives a fantastic exposition of Polya vectors fields in the final chapters of his book Visual Complex Analysis, and Polya's own Complex Variables makes connections with fluid flows throughout.

The big location where I've seen complex area integrals is the Bergman space. The typical (Hilbert) Bergman space on the disc $\mathbb{D} = \{z \in \mathbb{C} : |z|<1\}$ is the collection of holomorphic functions $f:\mathbb{D}\to\mathbb{C}$ such that
$\iint_\mathbb{D} |f(w)|^2 dA(w) <\infty$
The collection of such functions forms a Hilbert space with inner product given by
$\langle f,g \rangle = \left(\iint_\mathbb{D} f(w)\overline{g(w)}dA(w)\right)^{1/2}$
One object of interest in the Bergman space is a function called the reproducing kernel. On $\mathbb{D}$, the reproducing kernel is given by
$K(z,w) = \frac1\pi \frac1{(1-z\overline{w})^2}$
$K(z,w)$ thought of as a function of $w$ can be used to pull out the $z$-value of functions on the Bergman space, as it has the property that
$f(z) = \iint_\mathbb{D} K(z,w)f(w)dA(w)$
for functions $f$ in the Bergman space. You can also look at Bergman spaces on domains other than the disc $\mathbb{D}$, which have their own reproducing kernels. A good introduction to the Bergman space is Krantz's Geometric Function Theory.
A: My take on this would be as follows.
There are actually many different types of integral involving real spaces, such as  Riemann, Riemann–Stieltjes, Lebesgue, Daniell, Young integral, and probably others which have been tried in the past, but which have died out for one reason or another.
In the case of $\mathbb{C}$, my opinion would be that the standard integral in the complex case is the "natural one" since it is the one from which, in a straightforward development can be deduced a huge number of highly useful and beautiful theorems such as the Cauchy Integral Formula, Liouville's Theorem, the Identity Theorem, Schwarz' Lemma, etc.
In other words, subsequent developments by mathematicians (possibly? probably?) decided which was the "right" definition.
