Difference between complex and real analysis

[Silviu]

Hello! I see that all theorems in complex analysis are talking about a function in a region of the complex plane. A region is defined as an open, connected set. If I am not wrong, the real line, based on this definition, is a region. I am a bit confused why there are so many properties of the complex functions that we don't have in the real ones, if the real line is just a particular case of the most general one (a region in the complex plane)? Thank you!

 

[fresh_42]

Hello! I see that all theorems in complex analysis are talking about a function in a region of the complex plane. A region is defined as an open, connected set. If I am not wrong, the real line, based on this definition, is a region. I am a bit confused why there are so many properties of the complex functions that we don't have in the real ones, if the real line is just a particular case of the most general one (a region in the complex plane)? Thank you!

The real line is only closed if considered in itself. As part of the complex plane, it is not a region.
You can at best try to draw some parallels between $\mathbb{C}$ and $\mathbb{R}^2$. Now the crucial part here is, that in contrast to the real plane, where you have basically two independent components of the type $\mathbb{R}^1$, the complex plane allows arithmetic operations which relate the real and imaginary part. E.g. differentiability is automatic in all directions, or complex functions like the exponential map have branches due to Euler's identity which isn't available in the real plane.

 

[Silviu]

The real line is only closed if considered in itself. As part of the complex plane, it is not a region.
You can at best try to draw some parallels between $\mathbb{C}$ and $\mathbb{R}^2$. Now the crucial part here is, that in contrast to the real plane, where you have basically two independent components of the type $\mathbb{R}^1$, the complex plane allows arithmetic operations which relate the real and imaginary part. E.g. differentiability is automatic in all directions, or complex functions like the exponential map have branches due to Euler's identity which isn't available in the real plane.

Thank you! But I am not sure why a line is not closed in the complex plane. The definition of connectedness is that any 2 points can be connected by a line lying completely in that region, which is the case for the real line.

 

[fresh_42]

It is closed. It's not a region. My fault. I was too far in my thinking. I should have said open instead.

But I am not sure why a line is not closed in the complex plane.

A region is defined as an open, connected set.

Sorry for that.

 

[Silviu]

It is closed. It's not a region. My fault. I was too far in my thinking. I should have said open instead.Sorry for that.

Wait sorry, I am confused. An interval on the real line is both open and connected (As any 2 points can be connected inside the interval). Why is it not a region?

 

[zwierz]

The point is not only that the complex plane is considered instead of the real line. An analytic function is not the same as $C^\infty-$function

 

[lavinia]

Wait sorry, I am confused. An interval on the real line is both open and connected (As any 2 points can be connected inside the interval). Why is it not a region?

It is a region of the line but it is not a region of the plane. The plane and the line are two different topological spaces. Regions of the plane contain an open disk around each point. There are no open disks contained in the real line around any of its points.

As zwierz pointed out, an analytic function is not the same as a general differentiable function. Its real and complex parts are harmonic i.e. their Laplacian is zero. Complex analysis could be viewed as the study of harmonic functions of two variables.

Real analysis is more the study of integration theory of measurable functions.

 

[WWGD]

And both the Real and Complex parts of the function are Real-Analytic.

 

[FactChecker]

A complex derivative at a point is a single complex number, the same no matter which way the point is approached. That simple requirement is so restrictive that only the nicest functions have it. They are infinitely differentiable; their integrals are independent of path; interior values can be obtained from integrals around a circular path; etc. etc. etc.
There is no such nice result if functions from R² to R² are just required to have partial derivatives.

 

[WWGD]

Ultimately, the specialness is given by Cauchy-Riemann equations.

 

[Stephen Tashi]

Wait sorry, I am confused. An interval on the real line is both open and connected (As any 2 points can be connected inside the interval). Why is it not a region?

A real line is not "open" when considered as a subset of the complex plane. For a set $S$ to be "open" in the complex plain requires that around any point $p$ in $S$ we can find a circle whose interior is a subset of $S$. For a line, the interior of a circle about a point on the line also contains points not on the line.

Doesn't the material you are reading state the definition of an open set? - or are you not reading about complex analysis in a organized fashion?

 

[WWGD]

Wait sorry, I am confused. An interval on the real line is both open and connected (As any 2 points can be connected inside the interval). Why is it not a region?

Just to make explicit a point made explicitly a few times, I would suggest, until this becomes second nature to you, to say: "The set S is open/connected/etc _IN _ the space X" , and not just loosely state " set S is open ". In this case, S is the interval, which is not, strictly speaking " open and connected"; it is " open and connected in the Real line" .