Unknown
4.5. The Fundamental Theorem: Functions of a Complex Variable 143 to consider how the image point h(z) moves correspondingly thr ...
144 4. Equations (c) Deduce from (b), that if 1.~1 < l/3, then p(z) is contained within the circle of center 1 and radius 2/3 ...
4.5. The Fundamental Theorem: Functions of a Complex Variable 145 rcosO+irsinO) = r(cosO+isinO)(2rcosO+3)+ (b) Analyze the image ...
146 4. Equations (b) Use (a) to argue that, when ]%I = r, then p(z) is in the annulus with center 0, inner radius (1/2)r” and ou ...
4.6. Consequences of the Fundamental Theorem 147 (a) If si is a nonreal zero of p(t), prove that the polynomial t2 - (2Re si)t + ...
148 4. Equations Let f(t) be a polynomial of degree n over Z. Show that, if f(t) as- sumes prime integer values for 2n + 1 dist ...
4.7. Problems on Equations in One Variable 149 The result we want is that all the zeros of the polynomial f’(t) also lie in the ...
150 4. Equations For which values of a, b, c does the equation x+a&+b+&=c have infinitely many real zeros. Let k be a ...
4.8. Problems on Systems of Equations 151 4.8 Problems on Systems of Equations Solve the following systems of equations: x4 + y ...
152 4. Equations x+y+z=3a x2+y2+z2= 14+2a+5a2 xyz = 6 - lOa - 4a2. a(y - z) + b(z - x) + c(x - y) = 0 (x - y)(y - z)(z - x) = d ...
4.8. Problems on Systems of Equations^153 Prove that (x,y) = (1,2) is the unique real solution of x(x + y)2 = 9 “(Y3 - x3) = 7 ...
154 4. Equations Given that a 2 1 and b are real numbers, prove that the system y=x3+ax+b z=y3+ay+b x=z3+az+b has exactly one ...
4.9. Other Problems 155 4.9 Other Problems For which complex numbers a is the mapping one-one on the closed unit disc (1~1 5 l ...
(^156) 4. Equations Determine a manic cubic polynomial over Q one of whose zeros is 1 _ p/3 + p/3. Hints Chapter 4 1.1. Add th ...
Hints 157 8.1. Begin by eliminating the terms with coefficient 9/8 and determine simple possible relations between t and y. 8.2. ...
158 4. Equations 9.5. Note that the conditions imply that ct + d = a-‘c(at + b). 9.6. Noting that the relation 1 + w + w 2 = 0 i ...
5 Approximation and Location of Zeros 5.1 Approximation of Roots The population of,a biological species in a favorable environme ...
160 5. Approximation and Location of Zeros This quartic equation is easy to solve explicitly. However, if we look at the possibi ...
5.1. Approximation of Roots 161 (c) Show that, if (a+ b)/2 is not a zero of p(t), then there is a zero either between a and (e + ...
162 5. Approximation and Location of Zeros (^1) -3 7 -15 1 2 -2 10 -10 (^1) -1 5 -5 -9 2 2 14 I 1 1 7 9 2 6 I 1 3 13 2 1 w 1 5 1 ...
«
4
5
6
7
8
9
10
11
12
13
»
Free download pdf