Understanding Complex Integrals: Interpretation and Visualization

[Jhenrique]

hello everybody

I'd like to understand what mean the result of a complex integral. For example, integrate f(z) = z² from 0 to 2+i results 2/3 + 11/3 i. But, what is this? What 2/3 + 11/3 i represents geometrically? Is it possivel view this result?

Thx!

 

[hilbert2]

There are many different paths through which you can go from 0 to 2+i on the complex plane. You have to specify which path you use in the integration. As far as I know, complex integrals don't have a simple geometrical interpretation like real integrals do (area under a curve).

 

[Erland]

There are many different paths through which you can go from 0 to 2+i on the complex plane. You have to specify which path you use in the integration.

But in this case, f(z)=z² is analytic in the entire complex plane, and then the result of the integration is independent of the path, only the endpoints matter. This is follows from the Cauchy Integral Theorem.

 

[Jhenrique]

But in this case, f(z)=z² is analytic in the entire complex plane, and then the result of the integration is independent of the path, only the endpoints matter. This is follows from the Cauchy Integral Theorem.

A) If a function is analytic in the entire his dominion, this means that no exist values ​​for which the function is undefined, right?

So, for example, f(z)=1/z is not analytic, because it is not defined for z=0, correct?

B) 2/3+11/3i has geometric interpretation?

 

[Erland]

A) If a function is analytic in the entire his dominion, this means that no exist values ​​for which the function is undefined, right?

So, for example, f(z)=1/z is not analytic, because it is not defined for z=0, correct?

It is not analytic in the entire plane (i.e. it is not entire), but it is analytic in any region which does not contain 0. The Cauchy integral theorem holds for this function only for paths which does not encircle 0. For two paths in the plane with the same endpoints, the results of integrating 1/z along these paths will differ if they go on opposite sides of 0.

B) 2/3+11/3i has geometric interpretation?

No obvious geometric interpretation which I know of. Perhaps the complex integral can be interpreted as work if one makes some changes...

 

[Axiomer]

Analytic (or holomorphic) means that that the function is complex differentiable on it's (open) domain.

I would be interested in learning more about the motivation behind the complex line integral. To me, the complex line integral was presented as this dry definition from which all these incredible results come from, like Cauchy's Integral Theorem(s), open mapping, maximum modulus, residues etc. It all still seems a little mysterious to me.

 

[Erland]

I don't really know, but it seems to be the natural way to define complex integrals. If we interprete $\int f(z)dz$ as a limit of Riemann sums and $dz$ can be any infinitesimal difference of complex numbers, this leads to our definition of the complex integral.