Mobius transformation (Complex Analysis)

1). Let D = {z: |z| < 1 } and find all Mobius transformations T such that T(D) = D. 2). Show that a Mobius transformation 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).

Analytic functions as mappings (Complex)

1). Let G be a region and suppose that f : G -> C ( C is complex plane) is analytic such that f(G) is a subset of a circle. Show that f is constant. 2). If Tz = (az + b)/(cz + d), find necessary and sufficient conditions that T(t) = t where t is the unit circle { z: |z| = 1}. My solution for number 2 is : T(t) = t , which implies that | (az+b/cz+d| = 1 then we have |az+b| = |cz+d| then by solving for z we get |d-b|= |a-c| or |d+b| = |a+c|. Am I right? If not, please provide the correct solution. ( I believe z here is a complex number) Please work these problems only if you are an expert complex math person.

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.

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.

Line Integral

(See attached file for full problem description with equations and diagram) --- Compute where is a square with side = 4, centered at the origin and traced counterclockwise once ---

Composite Function

(See attached file for full problem description with equations) --- real numbers complex numbers Suppose , are continuous functions with nonvanishing first partial derivatives. Let , Show that . ---