M361 FALL 2015 UNIQUE NUMBER 53490 HOMEWORK 1

This homework is due on Friday, September 4, 2015, before lecture begins. Latehomework is not accepted. In order to receive credit for this assignment you mustwrite your name and UT eid at the top of first page of the turned assignment, staplefurther pages carefully, label neatly your exercises, show all of your work, and... returnit before the lecture begins. This assignment is graded 0-10 points depending on thequality of your solutions to the problems below. I stress: show all of your work, andfully explain how do you get to your answers. A correct answer without explanationswill be graded 0.

Given a complex number in Cartesian form z = x + i y (so that x, y ∈ R), define themodulus of z and the conjugate of z as

|z| =√

x2 + y2 , z̄ = x − i y .

Exercise 1. Show thatz z̄ = |z|2 .

Exercise 2. Show that if z = r eiθ then

z̄ = r e−iθ .

Comment on the geometric meaning of this formula.******Given a complex number in Cartesian form z = x + i y such that |z| > 0, let us set

1z=

xx2 + y2 − i

y

x2 + y2 .

Exercise 3. Prove the identities

z1z= 1 ,

1z=

z̄|z|2 ,

1z=

1r

e−iθ .

Exercise 4. Write the complex number

z =3 + i

(2 − 4i)(1 + i)3 ,

in Cartesian form; that is, find x and y real numbers such that z = x + iy.

Exercise 5. Write the complex number z39 in Cartesian form, provided

z =1√

2+

i√

2.

Hint: recall that zn = rn einθ for z = r eiθ.

Exercise 6. Find all the solutions to the equation

z4 = 16 ,

and draw their positions on the complex plane.

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