Complex Analysis Answers

Determine the modulus and argument of z1=1+cos θ+isin θ/1-cos θ+isin θ

Given two complex numbers z1=1+i and z2= √(3)-1

a) Write z1/z2 in algebraic and polar forms.

b) Deduce the exact values of Cos 5pie/12 and Sin 5pie/12

c) What is the lowest positive value of integer n such that (z1/z2) is real

 Consider the four points ABC , , and D , on a complex plane with affixes 2 - 3i , 1/2 , 1+ 4i and 4 + 2i respectively.

a) Plot these points on complex plane

b) Calculate the affixes of vectors AB and BC

c) Determine the affix of point E such that ABCE is a parallelogram

Determine, using complex numbers, the magnitude and direction of the resultant of the coplanar forces given below, which are acting at a point. Force A , 5N acting horizontally, Force B , 9N acting at an angle of 1350 to force A, Force C , 12N acting at an angle of 2400 to force A.  

This problem features the art of finding complex square roots as well as solving a quadratic equation using a resulting formulation based on completing the quadratic square.

(a) Given that u²=-60+32i , express u in the form a+bi where a,b=R

(b) Hence, solve the equation z²-(3_2i)z+5-5i=0

Consider the polynomial of a complex variable z given by f(z)=z³-8z²+25z-26

(a) Show that z-3+2i is factor of f(z)

(b) Develop the quadratic factor of f(z)

(c) Hence, completely solve the equation f(z)=0

If z = 1+i, find z^2 and 1/z. Also, plot the Argand diagram for both.

1. Comparing coefficients in the Laurent developments of cot (pi*z) and of its expression as a sum of partial fractions, find the values of sum 1 to infinity(1/n^ 2) ,

sum 1 to infinity(1/n^ 4) , sum 1 to infinity (1/n^ 6) .

3. Use pi/sin(pi.z) to find the partial fraction development of 1 / (cos pi* z) and show that it leads to pi/4=1- 1/3 + 1/5 - 1/7 +... .

2. Express in closed form. sum =-infinity to infinity{1/ {z^ 3 -n^ 3}}