Complex equation Definition and 68 Threads

The problem says: Find all solutions z of the equation: z^6 - 4z^3 + 4 = i First I factored the equation into (z^3 -2)^2 = i, set w= z^3 -2 and solved w^2 = i for w_1 = sqrt(i) and w_2 = -sqrt(i). I tried setting z^3 - 2 = sqrt(i) and solving but I get stuck there. I really have no idea how...

Homework Statement Draw the set of points in the complex plane satisfying the equation |z + 3i| = 4 Homework Equations The Attempt at a SolutionI don't know what z is supposed to be. In class, we've been using z to stand for a complex number (x + yi). Am I supposed to substitute...

Homework Statement z is a complex number. Find all the solutions of (z+1)^5 = z^5 The Attempt at a Solution Of course one could expand (z+1)^5, but I remeber our professor solving this with roots of unity. Can anyone help?

Homework Statement hi in trying to solve 4z - 2conj(z) + i = 0 Homework Equations The Attempt at a Solution the calculator spits out z = -1/2*i i did 4(a+bi) - 2(a - bi) + i = 0 4a + 4bi - 2a + 2bi + i = 0 2a + 6bi + i = 0 i get z = -1/6*i did i mess up...

Homework Statement Find all complex solutions to the following equation: 3(x^2 + y^2) + (x - iy)^2 + 2(x + iy) = 0 Homework Equations I want to use the quadratic formula, but not sure if it applies here. The Attempt at a Solution This is as far as I can get. What I would like is some idea...

Homework Statement So you basically have to prove this big long equation d=2v2cos(o)sin(O+o)/gcos(O) Homework Equations sin(O+o)=sin(O)cos(o)+sin(o)cos(O) The Attempt at a Solution so i have weeded it down to this...

Homework Statement Let z be a complex number. I want to solve cos(z)= -i*sin(z). The Attempt at a Solution Here's my work: cos(z) = -i*sin(z) implies cos(z) + isin(z) = 0. Therefore exp(i*z) = 0. Now put z= x+iy then i*z = i*(x+iy) = ix - y, hence exp(i*z) = exp(ix-y) =...

Homework Statement Determine all z ∈ C such that z^3-i = 0. Homework Equations z^3-i=0 The Attempt at a Solution z = -(i^3) = -((0+i)^3)

Homework Statement A simple way to factor the left hand side of the given equation. Homework Equations E^2-E^2(\frac{\frac{F^2vt}{c}}{M})+(\frac{\frac{F^4v^2t^2}{c^2}}{M^2})=-p+(\frac{\frac{F^4v^2t^2}{c^2}}{M^2}) The Attempt at a Solution I would, just that i am confused on...

Homework Statement determine the number of roots, counting multiplicities, of the equation z^7-5*z^3+12=0 in side the annulus 1<=|z|<2 Homework Equations use rouche's theorem The Attempt at a Solution

all i want to do is finding all the solution in a area. i know the Newton's method, but the problem is how can i divide the area into much smaller areas that i can judge it whether include a solution and can only have one solution. and how can judge it? any talk will be appreciated...

Z(x)=(50*m+m^2*tanh((a+b*i)*x))/(m+50*tanh((a+b*i)*x)) where i is imaginary unit, x is the independent variable, Z is the dependent variable, and a, b and m are the parameters. We measures variable Z at a discrete set of values of variable x: x=[1, 2, 3, 4, ...20] Z=[57.6286+0.6328*i...

Homework Statement I am reading the book Wiley Signal And Systems-Simon Haykin and at the page 34 i read the followings Consider the complex exponential e^j*theta. Using Euler's identity we may expand this term as e^j*theta=cos(theta)+jsin(theta) This result indicates that we may express the...

Homework Statement Find all roots of the equation cos(z)=2 (z is a complex number)Homework Equations The Attempt at a Solution What do they mean find the roots of this equation? We're just going over trig functions and it doesn't say anything about roots so I'm not sure what they're...

find all solutions of the given equation: (z+1)^4=1-i im not sure if i did this right, but here's what i did the first thing that i did was notice that 1-i = 2^1/2 * (cos (pi/4) + i*sin(pi/4)) then i found z= [2^(1/2*1/4) * (cos (pi/4) + i*sin (pi/4) )^1/4] -1 then using de moivre's thrm...

Solve the following equations for the real variables x and y. (b) \left ( \frac{1 + i}{1 - i} \right )^2 + \frac{1}{x + iy} = 1 + i I reduced this to 2x + 2iy + ix - y - 1 = 0 but I cannot get any further. Have I reduced it correctly?

im not sure i can fully remember the rules for complex numbers but i have to solve an equation that has 3 solutions to equal zero. so far i have (x=2), (x=i) and i think there was a + rule for a solution with complex numbers that there would always be a conjugate solution of it. so i figured...

z^2 - (7+i)z + 24 + 7i = 0 z= \frac{\ 7+i \pm \sqrt{(7+i)^2 - 4(24+7i)} }{ 2 } z= \frac{\ 7+i \pm \sqrt{-47 - 14i }}{ 2 } i stuck here