Riemann surface question

In summary, the conversation is about a complex analysis assignment and the question of describing the Riemann surface associated with the function f(z) = sqrt((z - x1)(z - x2)...(z - xn)) where xi is a real number. The hint is to consider n as odd and even separately and the suggested answer is a double cover of a sphere with n branch points, such as a torus for n=4. The person asking for help has only done this topic at an undergraduate level and has no experience in topology.
  • #1
Petroz
5
0
Hi guys,

My gf is doing honours and is having some trouble with one question on her assignment for complex analysis. She is really stuck and I've only done this topic at an undergraduate level so I have no idea. Neither of us have done any subjects in Topology so we don't know what to do.

She is required to describe the Riemann surface associated with a function
f(z) = sqrt( (z - x1)(z - x2)...(z - xn) )
where xi is an element of the real number set.
It comes with the hint: Consider n odd and even seperately.

If anyone could shed some light on this it would be greatly apprecicated.

Thanks for your time,

-Petroz
 
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  • #2
imagine a double cover of a sphere with n branch points.

e.g. if n=4 you get a torus.
 
  • #3


Hi Petroz,

The Riemann surface associated with the function f(z) = sqrt( (z - x1)(z - x2)...(z - xn) ) is a complex surface that represents the function on the complex plane. It is a way to visualize and understand the behavior of the function in a more geometric manner.

To start, let's consider n to be odd. In this case, the function has an odd number of square roots, which means that there will be a branch point at z = 0. This branch point will connect two sheets of the Riemann surface, with each sheet representing one of the possible square roots of the function. As z approaches 0 from the positive real axis, the function will approach the positive square root, and as z approaches 0 from the negative real axis, the function will approach the negative square root. This creates a "branch cut" on the Riemann surface, which is a line connecting the two sheets.

Now, for n even, the function has an even number of square roots, which means there will be no branch point at z = 0. Instead, there will be n/2 branch points located at the roots of the function (x1, x2, ..., xn). These branch points will also connect two sheets of the Riemann surface, with each sheet representing one of the possible square roots of the function. As z approaches one of these branch points, the function will switch between the positive and negative square roots, creating a "branch cut" on the surface.

Overall, the Riemann surface for this function will have n/2 branch points for even n and one branch point for odd n. The branch cuts will be lines connecting the sheets of the surface, and the sheets will represent the different possible square roots of the function.

I hope this helps your girlfriend with her assignment. If you need more clarification or have any other questions, please don't hesitate to ask. Good luck to both of you!


 

1. What is a Riemann surface?

A Riemann surface is a mathematical concept that extends the idea of a two-dimensional surface to higher dimensions. It is named after the mathematician Bernhard Riemann and is used to study complex functions and their properties.

2. How are Riemann surfaces used in mathematics?

Riemann surfaces are used in various fields of mathematics, including complex analysis, algebraic geometry, and topology. They are also used in physics, specifically in string theory and quantum field theory.

3. What is the significance of the Riemann surface question?

The Riemann surface question, also known as the Riemann hypothesis, is one of the most famous unsolved problems in mathematics. It deals with the distribution of prime numbers and has important implications in many areas of mathematics, including number theory and analysis.

4. How does the Riemann surface question relate to the Riemann zeta function?

The Riemann surface question is closely related to the behavior of the Riemann zeta function, which is a mathematical function that plays a central role in number theory. The Riemann zeta function is closely linked to the distribution of prime numbers and is used in the formulation of the Riemann hypothesis.

5. Why is the Riemann surface question important?

The Riemann surface question is important because its solution would have far-reaching consequences in mathematics and science. It has connections to many other mathematical problems and has implications in fields such as cryptography and computer science. Its solution would also help us better understand the fundamental properties of the prime numbers.

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