1.Elliptic curve:

It wasn't so hard.

Let have a cartesian plane with coordinates xOy.

The following equation is given:

If it's "true" we plot (x,y). Otherwase, we don't plot it.

Here's a plot of elliptic curve(picture 1).

Because the equation was 2 roots:y=sqrt(...) and y=-sqrt(...) the elliptic curve is a Riemann surface.]]>

A set of values for which f is defined.

For example the function f=n!, defined over Natural numbers (

I'm following the proof that siva(thank you) gave me.

There you can download the full solution. Or you can do this by clicking:

here for zipped .pdf file

here for PostScript

and here for .dvi

(I've tested only the .pdf format)]]>

we use the function f(x)=x^2.(picture 1)

f-¹(y) = {sqrt(y) (picture 2) OR -sqrt(y) (picture 3)}

Then the Riemann surface of the function f-¹(y) is the union of the plots(picture 4)]]>

But how to explain it?

A Riemann surface is a surface-like configuration that covers the complex plane with "sheets.". When we have a functin over the complex plaine C, which is not "single valued":

∃ z1,z2 ∈

What will we get for the inverse function of (z)?

Here's "logical" answer:

-¹(z) =={1-¹(z) OR 2-¹(z)}.

In the general case -¹(z) may be union of k "single valued" functions.

The riemann surface of -¹(z) is the union of the graphs of these functions.

We can use Euclidean plane, too.

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