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Calculus/complex number

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Expert: Alon Mandes - 11/10/2009


Question
I keep having trouble understanding in complex number solving.And i am hopping you could show me how to solve the complex by step by step or even at least point me out what i should read on(from online or books).Thanks in advance.

z^3 = -4*Squareroot3 + 4i
Question is to find all the soultion and sketch them in the complex plane.(is there any formula to easy approach to sketch it?)

Answer

drawing
every complex number z can be represented by either on of two:
Cartesian Rep : a+jb    or simply (a,b)
Polar     Rep : ρCis(θ) or simply (ρ,θ)

z³=-4√3+4i . Therefore z=(-4√3+4i)^(1/3) .

1st of all lets transform -4√3+4i to polar representation :
ρ=√[√3²+4²]=√[3+16]=√19=4.35
θ=Arctang[4/-4√3]=Arctang[-1/√3]=-30°
No , we claim :

Let's find the new r &  Ø after performing Power (1/3) :
r=ρ^(1/3)=4.35^(1/3)=1.63
Ø1=θ/3=-30°/3=-10°
Ø2=(θ+360°)/3=(-30+360°)/3=110°
Ø3=(θ+540°)/3=(-30+540°)/3=170°

So, our answer is 3 numbers :
z1=rcos(Ø1)+irsin(Ø1)=1.63*cos(-10°)+isin(-10°)=  0.984 - 0.173i
z2=rcos(Ø2)+irsin(Ø2)=1.63*cos(110°)+isin(110°)= -0.342 + 0.939i
z3=rcos(Ø3)+irsin(Ø3)=1.63*cos(170°)+isin(170°)= -0.984 + 0.173i

The image or the drawing is aatached.

Alon.  

Alon Mandes

Expertise

Kind of questions I can answer : Limits, Derivatives, Integration, Implicit functions, continuousity, differentiation ,Extremum problems, Lagrange multipliers, Gradients, Surface integrals, Multi variables functions ,Multi variables Integrals,Complex variables ,Complex functions, Curves, Trajectory integrals & Vector analyse,Divergence,Rotor & word problems. Kind of question I can't answer : Economics,Combinatorics,infinite series & convergence ,Statistics & Probabilities .

Experience

1. I'm a team member of mathnerds (math site for answering questions) 2. I'm a team member in the Student's Union of the Technion, helping students who have problems in mathematics. 3. 2 years of experience as a math teacher in college. 4. I give free homework help for high school students in Mathematics & Physics. 5. I teach part time in collage the subjects : "Digital Signal Processing" , "Random Signals & Noise" , "Complex Functions".

Organizations
Hi-Tech company : GSM4VOIP ; job possition : Algorythm developer.

Education/Credentials
M.A in Mathematics & Bs.c in Electronics.

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