Basics in Complex Analysis I am learning complex analysis and having doubts in the basics as follows:-

*

*I read that the derivative of a complex valued function $f(z)$
behaves exactly in the similar fashion as if it were a real valued
function $f(x)$.I would like to know why that happens. If i guess
that it is because the definition of differentiability in both the
worlds, the complex as well as the real is $'identically'$ similar .
If there's more to it ,please help me here.

*The integral of a complex-valued function is defined along a contour as the sum of the product of the function value at $z$ times the displacement $dz$ .What does this represent if it's not area under the curve ? $$\int_\gamma f(z)\,dz .$$

*Following from above ,in complex integrations , we essentially multiply two complex numbers (i.e. $f(z)$ and $dz$) , simplify them to a single number and then integrate. How is this action inside an integral is justifiable ?

*In Cauchy's theorem of analytic functions in a simple closed curve , is it necessary to have
function $f(z)$ being $'extending ...smoothly'$ towards the contour. Is it necessary as well as suffficient if the function is only analytic on and inside the contour and outside the contour it can behave in a rather funny way.Can that happen ?

If there's some text that people here can share on complex analysis , that would be really helpful.
Thanks for helping.
 A: Is this one question or four questions?
(1). The derivative of a complex function is not at all (other than superficially) like that of a real function.  For example, if a complex function has one derivative, it has infinitely many derivatives!
(2). A contour integral could represent various physical entities (for example work) depending on the context.  Green's theorem and Stokes' theorem relate a contour integral to an "area" or "volume" integral.  In any case, the integral needs no physical analogue.
(3).  The term $dz$ is a differential, similar to the role of differential in a real integral.
(4).  In Cauchy's theorem, $f(z)$ is holomorphic (it has a derivative).
Please read "Chapter zero" of Stein and Shakarchi, Complex Analysis for more motivation.
A: I think the source of the problem is thinking about the integral primarily as "the area under a curve".  That's one application of integration, but it isn't what integration is.  Integration is the infinite sum of infinitely small values.  That's why complex integration works just like real integration.  $f(z)\,dz$ is just an infinitely small value that gets summed.  It works for multivariable equations just like single-variable equations.  It is literally just an infinite sum of infinitely small values.
The reason why, in the reals, it represents an area under a curve, is that if you divide the curve $f(x)$ into infinitely small rectangles, each rectangle will be $f(x)$ tall and $dx$ wide.  So, $\int f(x)\,dx$ is just the sum of all those areas.  But the function that the integration itself is performing is not area, but summation.  It doesn't matter to the integral that $f(x)\,dx$ represents an area or anything else.  Therefore, since addition works the same in the complex numbers as it does in the reals, integration also works the same, because it is just a summation.
If you realize that integration is just the infinite sum of infinitely small values, then I think that answers all of your questions, at least implicitly.
