Math 121A Spring 2007 Homework#2 1. Let A 2 M3 (R). Denote by v1 ; v2; v3 2 R3 the rows of A: Show that det(A) = V ol(v1 ; v2; v3 ) the volume of the parallelogram dened by v1 ; v2; v3 : 2. In M. L. Boaz, page 123 problems 13, 15. 3. Let M2 (R) be the alg
2. Page 699 of the M.L. Boas book. Problems: 1, 3, 6, 10, 11, 12, 13, 15, 17, 19, 20. Remarks You are very much encouraged to work with other students. However, submit your work alone. I will be happy to help you with the homeworks. Please visit me if you
Math 121A Spring 2007 Homework#9 Poles, Laurent series, Residue theorem. 1. Write the Laurent series of f around z0 and compute its radius of convergence for (a) f (z) = (b) f (z) = (c) f (z) = (d) f (z) =
1 z(z 1)(z+1)
and z0 = and z0 =
1; 0; 1: 1; 0; 1:
Math 121A Spring 2007 Homework#8 Integration, Cauchy theorem. s 1. Recall the denition of the integral: Let f : U C ! C be a "nice" function and consider a curve C : [a; b] ! U: We dene the integral of f along C by
C
Show that the Newton-Leibniz formula h
Math 121A Spring 2007 Homework#7 How to think on analytic function? Taylor series. 1. (Topology) A subset U C is called open if for every z0 2 C there exists a ball B = B(z0 ; r) = fz 2 C; jz z0 j < rg with z0 2 B ( U: Draw the following sets and decide i
Math 121A Spring 2007 Homework#6 Derivative, C-R equations. 1. Compute from the denition f 0 (z) = lim that (z n )0 = nz n 1 : 2. Use linearity, the quotient rule, the chain rule and the Leibniz rule to calculate the derivatives of (a) ez (b)
3
f (z + w)
Math 121A Spring 2007 Homework#5 Complex functions 1. Use the ratio test to test for convergence of the following series (a) 1 + 1 P (b)
n=0 1+i 2
+
1+i p 3
(1+i)2 4 n
+
(1+i)3 8
+ : +
(1+i)n 2n
+ :
:
2. Use the ratio test to nd the radios of convergence
Math 121A Spring 2007 Homework#3 Linear Algebra 1. Consider the vector space V = R2 and the basis B =f(1; 1); (1; 2)g and C = f(1; 0); (1; 1)g. 1 Compute the transition matrix T = TC;B . Suppose v 2 V with [v]B = . Compute 1 [v]C . 2. Consider the vector
Math 121A Spring 2007 Homework#11 Fourier transform, Series and Parseval theorem. s 1. Consider the following sets L2 (T ) of all functions from [ ; ] to C that satisfy R 1 jf (x)j2 dx < 1 and the set l2 (Z) of all sequences (an )n2Z of complex numbers 2