THE HARLEQUIN said:
i thought complex plane is a 2 dimensional plane just like our cartesian plane ... so why can't i just take the area under the curve in complex plane here too ?
and while approaching non complex integration why don't we integrate for any two points on the plane with an arbitrary path ? ( sorry if i am wrong about everything i said , i am still a noob at complex integration )
Yes and no. Yes it is a 2-dimensional cartesian plane when viewed a certain way. No because it is a
generalization of the real line when viewed another way.
Example: In the 2-dimensional cartesian plane you can define
addition and
subtraction of points easily, but there is no straightforward way of defining a way to
multiply two points and get a new point. In the complex plane,
multiplication is defined from the beginning.
As I said, the complex plane is a generalization of the real line. Complex functions, though, have
stricter requirements than real functions. For a real function to be differentiable, the right- and left-hand derivative must both exist and be equal. For a complex function, the derivatives must be equal no matter how we approach the point in question. Such functions are called
analytic, and they have several interesting properties, one of them being that the integral from one point to another is
not dependent on the path used. This again means that the integral of an analytic function along a closed curve is
zero.
Integration is one of the most important tools in complex analysis and the basis for several important theorems.
