Let T be a Mobius transformation, T doesn't equal to identity. Show that a Mobius transformation S commutes with T if S and T have the same fixed points.

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Definition:
Mobius transformation can be defined as

T(z) = a(z+b)/(z+c), where a, b, and c are some complex numbers.

Let us parametrize the other Mobius transformation as

S(z) = A(z+B)/(z+C), where A, B, and C are some other complex numbers.

We are to prove that if T and S have the same fixed points, then T(S(z)) = S(T(z))

Let us now see what are the fixed points.

For T, if z is a fixed point, then

z = T(z) = a(z+b)/(z+c).

We can solve this by multiplying the equation by the denominator and getting the quadratic equation:
z^2 +(c-a)z - ab = 0.

There are two ways to proceed now, depending on whether you are familiar with Vieta formulae.

1). Let D = {z: |z| < 1 } and find all Mobiustransformations T such that T(D) = D.
2). Show that a Mobiustransformation T satisfies T(0) = infinity and T ( infinity) = 0 if and only if Tz = az^-1 for some a in C ( C is complex plane).

I need a solution for the attached Mobius problem.
(a) Find the most general Mobiustransformation that maps the right half-plane to the unit disc carrying the point 17 to the origin.
(b) Find a Mobiustransformation that maps the right half-plane to the upper half-plane carrying the point 7 + 5i to 3i.

(1) For j = 1,2 let R_j be the circle of diameter j/2 and center at (j/4)i. Also, let p(z) = 1/z be the inversion map.
(a) If G is the region outside R_1 and inside R_2 then prove that p(G) = {z : -2 < Im z < -1}.
(b) Prove that e^(pie*z) maps the strip {z : -2 < Im z < -1}onto the upper half-plane H_u.
(c) Use the pr

See attached a pdf file.
You can find more Mobiustransformation information from John Conway - Functions of One Complex Variable I or Serge Lang - Complex Analysis or other Complex Analysis books.
Problem of Schwarz's Lemma
1. Suppose f: D → ¢ satisfies Re f(z) ≥ 0 for all z in D and suppose that f is analytic and

Prove: For any given circles R and R' in C_oo, there is a mobiustransformation T such that T(r)=R'. Further, we can specify that T take any 3 points on R onto any 3 points of R'. If we do specify Tz_j for j=2,3,4 (distinct z_j in R), then T is unique.

See the attachment for the problems
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- Find the bilinear transformation that maps...
- Show the disposition of two linear fractional transformations...
- A fixedpoint of transformation w = f(z)...
- Show that there is only linear fractional transformation that maps...
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(See attached file for full problem descri

Interpret the following SPSS computer output for the chi-square test. Variable COMMUTE is "How did you get to work last week?" Variable GENDER is "Are you male or female?"
GENDER

The points on the 3-dimension space is deformed with the following equations:
(x1,y1,z1) (x2,y2,z2)
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