Math 621 Homework Assignment 5 Spring 2000

Due: Tuesday, April 11

1.   Lang, page 171 Problem 10: Show that any function, which is meromorphic on the extended complex plane, is a rational function. (You may use Problem 7 in Homework assignment 4).
2. Lang, page 171 Problem 11: Define the order of a meromorphic function f at a point p to be

   Above, m could be zero, meaning that f is analytic at p and .

   Let f be a meromorphic function on the extended complex plane (so a rational function by problem 1).

   1. Prove that

      In other words, the number of points in the fiber , counted with multiplicity, is equal to the number of points in , counted with multiplicity.

   2. Prove that all fibers , of f consist of the same number of points, provided they are counted with multiplicity,
3. Ahlfors, page 130 Problem 5: Let be an isolated singularity of an analytic function f. Prove that if is bounded from above or below, then is a removable singularity. Ahlfors' Hint: Apply a linear l.f.t. Note: Personally, I find it easier to avoid using a l.f.t (which does not seem to help rule-out the case of a pole). Instead, a short proof can be obtained using both the Casorati-Weirstrass and the Open Mapping Theorems.
4. Let be a complex number and assume that . A function f is said to be doubly periodic with periods 1 and if

   

   Show that every entire function, which is doubly periodic with periods 1 and , is necessarily constant. (We will see that there exist non-constant, doubly periodic, meromorphic functions ).

5. Jan 96 Basic Exam, Problem 5: Find the maximum value of the function on the disk . Justify your answer!
6. Lang page 213 Problem 1: Let f be analytic on the unit disc D, and assume that on the disc. Prove that if there exist two distinct points a, b in the disc, which are fixed under f (that is f(a)=a and f(b)=b), then f(z)=z.
7. Lang, page 219 problem 8: Use Schwarz's Lemma to prove that is the group of holomorphic automorphisms of the upper half plane. ( is naturally identified with the group of fractional linear transformations which are associated to invertible matrices with real coefficients and determinant 1). Hint: You may use the fact (proven in class when we showed that l.f.t preserve the notion of symmetry with respect to lines and circles) that, if a l.f.t. f(z) takes to itself, then it can be written in the form , with real coefficients a, b, c, d.
8. Lang page 213 Problem 2: Let be a holomorphic map from the disc into itself. Prove that, for all , we have

   

   Moreover, equality for some a implies that f is a linear fractional transformation. Hint: Let g be an automorphism of D such that g(0)=a, and let h be the automorphism which maps f(a) on 0. Let . Compute F'(0) and apply the Schwarz Lemma.

9. Ahlfors, page 136 Problem 2: Let f(z) be analytic and for all z in the upper half plane. Show that

   

   and, writing z=x+iy,

   

   Moreover, equality, in either one of the two inequalities above, implies that f is a linear fractional transformation.

10. Ahlfors, page 136 Problem 6: If is a path, piecewise of type , contained in the open unit disc D, then the integral

    

    is called the non-euclidean length or hyperbolic length of . Let be an analytic function from the disc into itself. Show that f maps every on a path with smaller or equal non-euclidean length. Deduce that a linear fractional transformation from D onto itself preserves non-euclidean lengths.

11. Ahlfors, page 136 Problem 7: (Modified) It can be shown, that the path of smallest non-euclidean length, joining the origin 0 to a point , is the straight line segment between them.
    1. Use this fact to show that the path of smallest non-euclidean length, that joins two given points in the unit disk, is the piece of the circle C which is orthogonal to the unit circle . The shortest non-euclidean length is called the non-euclidean distance.
    2. Show that the non-euclidean distance between and is

       

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