Quiz+2+Solutions




 * 1. a. Consider the transformation F(z) = 5z + 4/z acting on the punctured complex plane C- {0}.**
 * Find all fixed points of this transformation.**

So a fixed point is one where F(z)=z. ie it leaves all points fixed. So, set f(z) = z ... F(z) = 5z + 4/z = z 4z = -4/z 4z^2= -4 z^2 = -1 z = i, -i


 * 1. b. Prove that the transformation F preserves the real axis. (i.e., if R, a subset of C, denotes the real axis, prove that F(R) = R).**

Define R:

R = { x + 0i is in the complex plane s/t x is a real number, x does not equal zero}

Define F(R)

F(R) = { z is in the complex plane s/t there exists an x that is a real number s/t z = F(x) }

Proof:

Let x be a real number, then F(x) = 5x + (4/x)

By closure of the real numbers, 5x is real, (4/x) is real, and 5x + (4/x) is real, thus F preserves the real axis.

Note: F(R) does not equal 5(R) + (4/R).

Prove this does = NOT = preserve y=x. L={x+iy : x,y Is in R, x= y, x not equal to 0} ={x+iy : x is in R, x is not equal to 0} Provide a counter example. Pick some z=x+xi= re^(i pi/4)                 r is in R and not equal to 0 Try z=1 + i    F(1+i)= 5(1+i) + 4/(1+i) =5+5i+ [4/(1+i) * (1-i)/(1-i)] =5+5i+ (4-4i)/2 =5+5i+2-2 i = 7 +3i    Which is not on the line y=x.
 * 1. c.**

=OR= Try z=e^(i pi/4) F(e^(i pi/4)) = 5e^(i pi/4) + 4/e^(i pi/4) = 5e^(i pi/4) + 4e^(-i pi/4) ^^^^not on the line y=x


 * 2. Consider an equilateral triangle T sitting in the complex plan C, centered at the origin. Write down all the transformations //of the plane// which preserve the triangle. (This means find all transformations of the plane F which satisfy F(T) = T).**

(1) A rotation by 120 degrees (2pi/3).

(2) A rotation by 240 degrees (4pi/3).

(3) Identity (3a) A rotation by 360 degrees (2pi) or 0 degrees (0). (3b) A translation in which b = 0. T(Triangle ABC) = Triangle ABC + b = Triangle ABC + 0 = Triangle ABC

(4) Reflection (4a) A reflection over the bisector of angle A. (4b) A reflection over the bisector of angle B. (4c) A reflection over the bisector of angle C.