16.7. EXERCISES 417

5. Suppose you have f is analytic and has two independent periods. Show that f is aconstant. Hint: Consider a parallelogram determined by the two periods and applyLiouville’s theorem. Functions having two independent periods which are analyticexcept for poles are known as elliptic functions.

6. Suppose f is an entire function, analytic on C, and that it has two periods w1,w2.That is f (z+w1) = f (z) and f (z+w2) = f (z). Suppose also that the ratio of thesetwo periods is not a real number so vectors, w1 and w2 are not parallel. Show, usingLiouville’s theorem, that f (z) equals a constant. Hint: Consider the parallelogramdetermined by the two vectors w1,w2 and tile C with similar parallelograms. Ellipticfunctions are those which have two periods like this and are analytic except for poles.These are points where | f (z)| becomes unbounded. Thus the only analytic ellipticfunctions are constants.

7. You can show that if r is a real irrational number the expressions of the form m+nrfor m,n integers are dense in R. See my single variable advanced calculus book ormodify the argument in Problem 4. This is due to Dirichlet also in the 1830s. (Let PNbe everything of the form m+nr where |m| , |n| ≤ N. Thus there are (2N +1)2 suchnumbers contained in [−N (1+ |r|) ,N (1+ |r|)] ≡ I. Let M be an integer, (2N)2 <

M < (2N +1)2 and partition I into M equal intervals. Now argue some interval hastwo of these numbers in PN etc.) In particular, |m+nr| can be made as small asdesired. Now suppose f is a non constant meromorphic function and it is periodichaving periods w1,w2 where if, for m,n integers, mw1 +nw2 = 0 then both m,n arezero. Show that w1/w2 cannot be real. This was also done by Jacobi.

8. Suppose you have a nonconstant meromorphic function f which has two periodsw1,w2 such that if mw1 + nw2 = 0 for m,n integers, then m = n = 0. Let Pa be aparallelogram with lower left vertex at a and sides determined by w1 and w2 suchthat no pole of f is on any of the sides. Show that the sum of the residues of f foundinside Pa must be zero.

9. Let f (z) = az+bcz+d and let g(z) = a1z+b1

c1z+d1. Show that f ◦g(z) equals the quotient of two

expressions, the numerator being the top entry in the vector(a bc d

)(a1 b1c1 d1

)(z1

)and the denominator being the bottom entry. Show that if you define

φ

((a bc d

))≡ az+b

cz+d,

then φ (AB) = φ (A)◦φ (B) . Find an easy way to find the inverse of f (z) = az+bcz+d and

give a condition on the a,b,c,d which insures this function has an inverse.

10. The modular group2 is the set of fractional linear transformations, az+bcz+d such that

a,b,c,d are integers and ad − bc = 1. Using Problem 9 or brute force show thismodular group is really a group with the group operation being composition. Alsoshow the inverse of az+b

cz+d is dz−b−cz+a .

2This is the terminology used in Rudin’s book Real and Complex Analysis.