HOMEWORK
Assignment 1
Problems 1-4. Gamelin P.356 7,11,13,16
Problems 5-7. Gamelin P.360. 2,5,6
Problem 8. Conway P.170. 10. Suppose D is an open set and {f_n} is a sequence in H(D) such that the infinite product f(z) = product(f_n(z)) converges in H(D). a) Show that the sum from k=1 to infinity of (derivative of f_k(z)) times (product of f_n(z) over all n except omit n=k) converges in H(D) and equals the derivative of f(z). b).Assume f is not identically zero Assume K is a compact subset of D on which f does not vanish. Show that (log derivative of f) = the sum of (log derivaatives of f_n) and the convegence is uniform on K.
Problem 9. Cf. Conway P.170. 11.
Assignment 2
Problems 1-5. Gamelin Page 364. 1,3,4,6,7
Problem 6. Prove for any positive integer p, the product of Gamma((z+i)/p) {i goes from 0 to p-1} is equal to (2(pi))^(p-1)/2 times p^((1/2)-z) times Gamma(z)
Problem 7. Prove Gamma(1/6) = 2^(1/3) times (3/(pi))^(1/2) times Gamma(1/3)^2
Problem 8. For a a complex number, n a natural number, let (a/n) = a(a-1)...(a-n+1)/n factorial. Let (a/0) =1. Prove (a/n) = {(-1)^n times Gamma(a-n)}/{Gamma(-a) times Gamma(n+1)} for a a complex number, which is not a natural number.
Assignment 3
Problem 1-6. Gamelin Page 351. 1, 5, 6, 7, 8, 11
Assignment 4
Problem 1-3. Gamelin Page 287. 1, 2, 8
Problem 4. Conway Page 215. 2
Problem 5-11 Conway Page 257. 1,2,4,5,6, 7, 10
Assignment 5
Problem 1. Gamelin. Page 396. 2. That is prove that for a continuous function on an open connected set in R^2, the small-circle mean value inequality is equivalent to the mean value inequality.
Problems 2-6. Gamelin. Page 396. 3-7
Assignment 6
Problem 1-3. Gamelin. Page 401. 2,3,4
Problem 4-5. Gamelin. Page 405. 1,2
Problem 6. Which of the following functions are harmonic? subharmonic? superharmonic? none of the above? a)x^2 + y^2. b)x^2 - y^2. c)x^2 + y. d)x^2 - y.
Problem 7. Let W = slit complex plane (say remove the negative real axis). Construct a barrier for each point of the boundary of W.
Problem 8. Show that the solution of the Poisson problem in a bounded simply connected domain is unique (assuming it exists).
Assignment 7
Problem 1-4. Gamelin Page 423. 3,5,7c,13
Problem 5. Show that the number of non-zero lattice points of minimal modulus is 2, 4 or 6 and give examples of each possibility.
Problem 6. Let f and g be elliptic functions with respect to the same lattice. Assume f and g have the same zero set and the same pole set and for each zero or pole f and g have the same multiplicity and order. Show f = cg for some complex constant c.
Problem 7. Let f be an elliptic function of order m. Then its derivative is an elliptic function of order n. Show m+1 is less than or equal to n which is less than or equal to 2m and give examples where n = m+1 and n=2m occur.
Problem 8. Prove for any n an integer the function z goes to Pay(nz) is a rational function of Pay(z). Here Pay(z) is the Weierstrass Pay-function.
Problem 9. Show the second derivative of Pay(z) evaluated at w_1 (where the lattice L is the rank two abelian group generated by w_1 and w_2) is 2(e_1 - e_2)(e_1 - e_3). Here e_i is Pay(w_i) for i = 1, 2 and e_3 = Pay(w_1 + w_2).
Assignment 8
Problem 1. Let L and M be two lattices. Assume v is a non-zero complex number such that vL is contained in M. Show that multiplication by v induces a holomorphic map from C/L to C/M. Show this map is biholomorphic iff vL = M.
Problem 2. Show that every torus C/L is isomorphic to a torus C/T where T is a lattice in C with basis of the form 1 and w with Im(w) greater than 0.
Problem 3. Let A = a 2 by 2 matrix with integer coefficients and determinant = 1 first row = (a, b) and second row = (c,d). Then the associated fractional linear transformation takes the Upper half plane to itself. Let f(w) = (aw + b)/(cw +d) = w'. Let L have basis 1 and w. and M have basis 1 and w'. Show C/L and C/M are somorphic.
Problem 4. Let f be an entire function and L a lattice in C. Assume for each w in L, there is a polynomial P_w ...(possibly depending on w)...such that f(z+w) = f(z) + P_w(z) for all z in C. Show f is a polynomial.
Problem 5. For two lattices L and M show the following two conditions are equivalent: 1) L intersection M is a lattice. 2)L+M (which is defined to be w+v for all w in L and v in M) is a lattice.
Problem 6.Show the following recursion formulas for the Eisenstein series G_(2m) m greater than or equal to 4. (2m+1)(m-3)(2m-1)G_2m = 3 sum of (2j-1)(2m-2j-1)(G_2j)(G_2m-2j) with j going from 2 to m-2. Any Eisenstein series G_2m is thus representable as a polynomial in G_4 and G_6 with non-negative coefficients.
Problem 7. Let L be a lattice with the property that g_2(L)=8, and g_3(L)=0. The point (2,4) lies on the affine elliptic curve y^2 = 4x^3 - 8x. Let + denote addition on the corresponding projective curve. Show that 2(2,4) = (9/4, -21/4).
Problem 8. Call a meromorphic function f on C real if f(conjugate of z) = conjugate of f(z). A lattice L is real iff w in L implies conjugate of w is in L. Show the following 4 properties are equivalent: 1) g_2(L) and g_3(L) are real. 2) G_2m is real for m greater than or equal to 4. 3) pay function for L is real 4) the lattice L is real.
Assignment 9
Problems 1-5. Gamelin P.302. 5-9