MTH632 Quiz by Saman

(Assignment No 1) · Mth632 · Supervisor: Sir Mansour · Student ID:

Q1. Express the complex number

)i in exponential form.

To express the complex number )i in exponential form, we first find the polar form of

)i and then square it.

1. Find the modulus (r):

3) r z

2. Find the argument :

We use the fact that the complex number is in the second quadrant (since ( 3) is negative and

( )i is positive). The argument ( )  is given by:

arg( ) tan z

Thus, we can write:

2 cos sin z i i

3. Square the complex number:

Now square (z):

2 cos sin z i

Using the formula for powers of complex numbers in polar form: cos 2 sin 2 z i

4 cos sin z i

Thus, the exponential form of z is:

i

z e

Q2. Solution:

To express ( ( ) Im( ) Re( ) f x z z z z zz in the form f z u x y iv x y where

z x iy and z x iy let’s analyze each term and simplify.

1. Write ( z ) and ( z in terms of ( x ) and ( y ):

z x iy

z x iy

The imaginary part of ( ), (Im z z y

The real part of ( ), (Re z z x

2. Substitute and simplify each term:

First term: (

Im( ) z z

Im( ) z z x iy y xy iy

Second term: (

Re( ) z z

Re( ) z z x iy x x ixy

Third term: (

zz

zz x iy x iy x y

3. Combine all terms to form ( f(z) ):

Now, we add up the three expressions we obtained:

Im( ) Re( ) f z z z z z zz f z xy iy x ixy x y

Group real and imaginary parts:

Real part ( u(x, y) ):

u x y xy x x y x xy y

Imaginary part ( v(x, y) ):

v x y y xy xy y

Q3. Find 4th roots of -1.

Solution:

Step 1: Express (-1) in Polar Form

The complex number (-1) has a modulus of (1) (since (|-1| = 1)) and an argument of ( )  (since it

lies on the negative real axis).

So, we can write:

1 1 cos( ) sin( ) i or in exponential form,

ie 

Step 2: Apply De Moivre's Theorem for the 4th Roots

The fourth roots of a complex number ( i z re  are given by:

for 0,1,2,3 k

i

k w r e k

In our case, (r = 1) and ( , so the 4th roots are:

for 0,1,2,3 k k i i

k w e e k

Step 3: Calculate Each Root

Now, we calculate each root by substituting (k = 0, 1, 2, 3):

1. For (k = 0):

cos sin i

w e i i

2. For (k = 1):