Complex analysis: mapping a hyperbola onto a line

In summary, the conversation discusses creating a conformal and one-to-one mapping from (x,y) to (u,v) in which the positive x side of the hyperbola x^2 - y^2 = 1 is mapped onto the line v = 0 and all points to the left of the hyperbola are mapped above the line. The suggested solution involves using the complex function z=x+iy and the Cauchy-Riemann equations to determine the mapping function f(z). However, there is uncertainty about how to disregard the left parabola in the mapping.
  • #1
Grothard
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0

Homework Statement



We want to create a map from (x,y) to (u,v) such that the right side (positive x) of the hyperbola x^2 - y^2 = 1 is mapped onto the line v = 0 AND all the points to the left of that hyperbola are mapped to above the line. The mapping should be one-to-one and conformal


Homework Equations



I think (but am not 100% sure) the map has to be from z = x+iy to w = u+iv, so it's important to define x and y in terms of z. I figured out that x = (z+conj(z))/2 and y = (z-conj(z))/2

The Attempt at a Solution



We can map y directly to u through u = y. That should be sufficient I think.
v = x^2 - y^2 - 1 does map the parabola onto the line (it makes v = 0) but it maps the points to the right of the parabola onto above the line, so we multiply by -1 to get v = y^2 - x^2 + 1. We can write everything as:

w = y + i(y^2 - x^2 + 1) and then substitute the x = (z+conj(z))/2 and y = (z-conj(z))/2 into make it a function of z.

This seems to work for when x >= 0, but it fails to map some points on the left side of the y-axis (all of which should be mapped to above the line v = 0). This is because the function x^2 - y^2 = 1 describes two parabolas, and we have to somehow mathematically disregard the left one. I'm not sure how to do that. Could somebody please help?
 
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  • #2
You aren't really dealing with this as a complex function yet. You have got that v(x,y)=x^2-y^2-1 if f(z)=u(x,y)+i*v(x,y) where z=x+iy.That's a start. Now shouldn't you think about using the Cauchy-Riemann equations to figure out what u(x,y) might be and then try to deduce what f(z) might be?
 

What is complex analysis?

Complex analysis is a branch of mathematics that deals with the study of complex numbers and their functions. It is used to understand and describe complex phenomena, such as mapping a hyperbola onto a line.

What is a hyperbola?

A hyperbola is a type of conic section, similar to a parabola or ellipse, that is defined by the equation x²/a² - y²/b² = 1. It has two symmetrical branches that curve away from each other.

How can a hyperbola be mapped onto a line using complex analysis?

A hyperbola can be mapped onto a line using a complex function called a Möbius transformation. This transformation takes points on the hyperbola and maps them onto points on a line, preserving certain properties such as angles and circles.

What are some applications of mapping a hyperbola onto a line using complex analysis?

Complex analysis and Möbius transformations have many applications in physics, engineering, and other fields. For example, they can be used to model and analyze electromagnetic fields or fluid flow, to design efficient computer algorithms, or to map physical systems onto simpler models for better understanding.

Are there any limitations to mapping a hyperbola onto a line using complex analysis?

While complex analysis and Möbius transformations are powerful tools, their application may be limited by the complexity and nonlinearity of the systems being studied. Some phenomena may require more sophisticated techniques or multiple transformations to accurately model and analyze.

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