What is the argument of (1+i)^13/(1-i)^7 ?

We'll write the numerator and denominator in polar
form:

z = r(cos t + i*sin t)

The reason
of putting the numerator and denominator in polar form is to use Moivre's
rule.

We'll put the numerator in polar
form:

z1 = 1+i

Re(z1) 
=1

Im(z1) = 1

r1 = sqrt(1^2 +
1^2)

r1 = sqrt2

tan t =
Im(z1)/Re(z1)

tan t = 1/1

t = arctan
1

t = pi/4

z1 = sqrt2(cos pi/4 + i*sin
pi/4)

(1+i)^13 = [sqrt2(cos pi/4 + i*sin
pi/4)]^13

We'll use Moivre's
rule:

(1+i)^13 = 2^(13/2)(cos 13pi/4 + i*sin
13pi/4)

(1+i)^13 = 2^(13/2)(cos 5pi/4 + i*sin
5pi/4)

We'll put the denominator in polar
form:

z2 = 1 - i

Re(z2) 
=1

Im(z2) = -1

r2 = sqrt(1^2 +
(-1)^2)

r2 = sqrt2

t2 = arctan
-1

t2 = -pi/4

z2 = sqrt2(cos -pi/4 +
i*sin -pi/4)

(1-i)^7 = [sqrt2(cos -pi/4 + i*sin
-pi/4)]^7

We'll use Moivre's
rule:

(1-i)^7 = 2^(7/2)*(cos -7pi/4 + i*sin
-7pi/4)

(1-i)^7 = 2^(7/2)*(cos (2pi-7pi/4) + i*sin
(2pi-7pi/4))

(1-i)^7 = 2^(7/2)*(cos pi/4 + i*sin
pi/4)

Now, we'll calculate the
ratio:

(1+i)^13/(1-i)^7 = 2^[(13-7)/2]*[cos (5pi-pi)/4 + i*sin
(5pi-pi)/4]

(1+i)^13/(1-i)^7 = 2^3* (cos pi + i*sin
pi)

Since cos pi = -1 and sin pi = 1, we'll
get:

(1+i)^13/(1-i)^7 =
-2^3

(1+i)^13/(1-i)^7 =
-8

The argument of the ratio
(1+i)^13/(1-i)^7  is pi.