Analytic mapping of unit disc onto itself with two fixed pts.

In summary, the problem asks to show that an analytic mapping of the unit disc onto itself with two fixed points must be equal to the identity function. The Schwarz lemma can be used to prove this.
  • #1
d2j2003
58
0

Homework Statement



let f(z) be a 1-1 analytic mapping of unit disc |z|<1 onto itself with two fixed points in |z|<1 Show that f(z)=z


Homework Equations



none

The Attempt at a Solution



I'm thinking there has to be a theorem or something that I am missing for this.. But I'm not really sure where to get started..

To start, let a and b be the two fixed points in |z|<1, then f(a)=a and f(b)=b and I am sure there is a way to show that the only function that does this is f(z)=z not sure how though...
 
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  • #2
Try thinking about the Schwarz lemma.
 
  • #3
Got it, Thanks!
 

Related to Analytic mapping of unit disc onto itself with two fixed pts.

1. How does analytic mapping of a unit disc work?

Analytic mapping of a unit disc is a mathematical technique used to transform a unit disc onto itself. This means that the points on the unit disc are rearranged in a specific way while still remaining within the boundaries of the original disc.

2. What is the significance of having two fixed points in this mapping?

The presence of two fixed points in analytic mapping of a unit disc is crucial as it ensures that the transformation preserves the shape and size of the disc. This means that the distance between any two points on the unit disc will remain the same before and after the mapping.

3. Can this type of mapping only be applied to unit discs?

No, analytic mapping can be applied to any type of shape or structure. However, in the case of unit discs, the mapping is easier to visualize and understand.

4. How is this mapping different from other types of transformations?

Analytic mapping is different from other types of transformations, such as translations or rotations, because it involves complex mathematical functions that can change the shape of the disc in a more intricate and precise manner.

5. What are the practical applications of analytic mapping of a unit disc?

Analytic mapping has many practical applications in fields such as engineering, physics, and computer science. It can be used to model and solve complex problems, as well as to create visual representations of data and structures.

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