HOMEWORK
Assignment 1
Problem 1-4. Gamelin, P.10, 1(fgh),3,5,6d
Problem 5. Gamelin, P.21, 2
Problem 6,7. Gamelin, P.24, 2,4
Problem 8,9. Gamelin, P.31, 3,4
Problem 10. Show that the closure of a totally bounded set is totally bounded.
Problem 11. Let X be the set of all bounded sequences of complex numbers; i.e., x={x_n} belongs to X if there is a real number K such that absolute value of x_n is less than K for all n. For x and y in X define d(x,y) = sup absolute value of x_n – y_n over all n. Show that X is a metric space. Show that for each x in X and all positive real k, the closed disk D(x,k) [consisting of elements of y in X with d(x,y) less than or equal to k] is complete but not totally bounded.
Assignment 2
Problem 1,2. Gamelin, P.41. 13,16
Problem 3,4. Gamelin, P.46. 3,4
Problem 5-7. Gamelin, P.50. 3,4,8
Problem 8. Let x=Re(z), y=Im (z) compute the derivative of x+ (y)^2 with respect to z(bar) where z(bar) is the complex conjugate of z.
Problem 9. If f is a complex valued function on a domain D in the complex plane, then prove that the conjugate of the derivative of f at a is the derivative of the conjugate of f with respect to z(bar) at a.
Problem 10. Assume f is holomorphic on a domain D. Prove that the Laplacian of the square of the absolute value of f is equal to four times the square of the absolute value of the derivative (with respect to z) of f.
Problem 11. Fix n a positive integer. Assume z_1,...z_n are complex numbers satisfying that the absolute value of the sum of (z_i)(w_i) is less than or equal to 1 (here i runs from 1 to n) for all complex numbers w_1,...w_n such that the sum of the absolute values of (w_i)^2 less than or equal to 1 (here i runs from 1 to n). Prove that sum of the absolute values of (z_i)^2 (here i runs from 1 to n) is less than or equal to 1.
Assignment 3
Problem 1-4. Gamelin. P.106. 2, 3a, 5, 6
Problem 5-7. Gamelin. P.109. 1,2,4
Problem 8. Show that the path defined as follows g(t) = t + i(sin(1/t)) from the closed unit interval to C is not rectifiable.
Problem 9. Let g be a closed rectifiable path in domain D. Assume a does not belong to D. Show that for n greater than or equal to 2, the integral over g of (z-a)^(-n) dz is 0.
Problem 10. Let g(t) = 1 + e^(it) for t between 0 and 2(pi). Evaluate the intgral over g of (z^2 - 1)^(-1) dz.
CORRECTION: In problem 8 above it should have read: g(0) = 0, and g(t)= t + itsin(1/t) for t in (0,1].
Assignment 4
Problem 1. Do problem 8 (corrected) from Assignment 3.
Problem 2-3. Gamelin. P.89. 3,4
Problem 4. Gamelin. P.116. 1aceg.
Problem 5. Gamelin. P.116. 2
Problem 6-8. Gamelin. P.119. 2,3,4
Problem 9. Let D be open connected in C. Assume f and g are analytic functions on D and fg = 0 on D. Show f=0 on D or g=0 on D. (So the analytic functions on D form an integral domain).
Problem 10. Find the power series expansion of [exp(z)-1]/z and determine radius of convergence. Consider f(z) = z/{exp(z)-1} and write f(z) = summation (a_k)(z^k)/k! k running from 0 to infinity. What is its radius of convergence? Show that 0=sum of (a_k) times binomial coefficient n+k choose k where k goes from 0 to n (for all n). Note f(z) + (z/2) is even so that a_k = 0 for k odd and k>1. The numbers B_(2n) = (-1)^(n-1) a_(2n) are the Bernoulli numbers. Compute B_2n for n=1,2, 3, 4, 5.
Assignment 5
Problem 1-5. Gamelin. P.137. 2,7,8,9,12
Problem 6. Gamelin P.144. 7
Problem 7-9. Gamelin P.147. 1def, 2, 4
Problem 10-12. Gamelin. P.157. 1abc,9,12
Assignment 6
Problem 1-5. Gamelin P.170. 1c,2c,3,4,6
Problem 6-10 Gamelin P.176 1e,h,i, 2c,8,10 12
Problem 11-13. Gamelin P. 179 1e,h,i, 2(also give examples), 3.
Assignment 7
Problem 1-5 Gamelin. P.202. 2,3,6,7,9.
Problem 6-9 Gamelin. P.207. 2,3,6,8
Problem 10-11 Gamelin P.215. 3,6
Problem 12-13 Gamelin P.218. 2,3
Assignment 8
Problem 1-3. Gamelin. P.228. 2,5,7
Problem 4-7. Gamelin. P.230. 1,2,4,7
Problem 8-9. Gamelin P.235. 6,7
Problem 10. Gamelin P.15. 7
Problem 11-13 Gamelin P.68. 1abcg, 10, 12
Assignment 9
Problem 1-2. Gamelin P.137. 2, 10
Problem 3-8 Gamelin P.293 1,2,5,9, 11
Problem 9. (Dini's Theorem). Consider D an open subset of the complex or real numbers. Consider C(D,R) continuous functions from D to the reals. Assume {f_n} is a sequence of monotonically increasing functions which converges pointwise to f in C(D,R). Show f_n converges to f in C(D,R), that is in the metric space C(D,R). (CORRECTION ADDED: SHOULD READ "f_n IS A MONOTONICALLY INCREASING SEQUENCE OF FUNCTIONS")
Problem 10. Let {f_n} belong to C(D, omega) where omega is a complete metric space, D an open in the complex numbers C. Assume {f_n} is equicontinuous on D. If f in C(D, omega) is such that f_n converges pointwise to f, show that f_n converges to f in C(D,omega)
Problem 11. Consider two circles. The larger circle has center at i and radius 1. The smaller circle has center at i/2 and has radius 1/2. They meet at the origin and have the same tangent line there. Show that the open region between the two circles1 can be mapped onto the upper half-plane (Recall that a conformal map will preserve angles between curves).
Problem 12. Consider two circles which meet orthogonally. One with center at -1 and radius 1 and the other with center i and radius 1. Note they meet orthogonally at the origin and the point (-1,1), i.e. at -1 +i in C. Find a conformal map which takes the union of the interiors of the two circles to the upper half plane.
CORRECTIONS TO FINAL EXAM
Problem 4. Should read "Prove that either f is constant or is one-one"
Problem 1, Can take D properly contained in C (complex numbers).
Problem 7. Go from interior of unit disk to first quadrant (instead of exterior).