Complex numbers Definition and 724 Threads

Homework Statement Find the modulus and the principal value of the argument for the complex number \sqrt{3} - i The Attempt at a Solution I know the modulus is just 'square both, add, and square root of sum', so r = \sqrt{2}, but I don't know how to find the second part. I know vaguely...

1. Find the real part of z=ii by using De Moivre's formula. Homework Equations z= r(cos\theta + i sin\theta) zn= rn(cos(n\theta) + i sin(n\theta)) I tried using n=i to solve and got the ans 1i, but somehow feel that its not that simple. And the resultant argument I got from this...

Hi PFs i want to know how to evaluate the complex number, and what are the meaning of the evaluating a complex number Let Z be the complex number, Z = (3-4i)^5 what i have to do, just give me hint

Homework Statement Obtain z^10 for z=-1+i Homework Equations z=re^i(theta) The Attempt at a Solution Theta is 3pi/4. So z^10 = 32e^i(15pi/2). The answer in my book is 32e^i(3pi/2) but I'm pretty sure that's wrong, can anyone confirm?

Homework Statement Find the cartesian equation of the locus |z+3+2i|=Re(z) Homework Equations The Attempt at a Solution You let z= x +iy therefore Re(z)=x?

I have the equation: (.462-.32094i)*[(7.2+9.6i)/(4-8i)] the answer is: .107+.748i Now, I need to know how to enter this in Excel, because the 4 will be varying and I will need many rows and then my answer to be changing with the change in the equation. I have tried...

I have a couple questions regarding Euler's formula. First I'm confused about the notation ei\vartheta. To me the notation implies that we are raising e to the exponent i multiplied by \vartheta. Is this correct? If so, how would you do that? Also, my second question regards the second part...

Hi! We started doing complex numbers in maths class a couple of weeks ago, and I'm not fully understanding sketching the locus of points. Homework Statement Sketch the locus of z: arg\left(\frac{z-2}{z+2}\right) = \frac{\pi}{3} The Attempt at a Solution I've rewritten as...

Homework Statement Why do you need to use Cauchy-Riemann in the derivate of x^3 + i(1-y)^3, while not in the derivate of \frac { 1 } {z^2 + 1} in using the quotient rule? where z = x + iy. I derivated the first expression implicitly in the exam which resulted in zero points of the...

I have two ways of evaluating (e^{i 2 \pi}) ^{1/2}, and they give me different answers. Which one is correct, and why is the other wrong? Method 1: (e^{i 2 \pi}) ^{1/2} = e^{i \pi} = -1 Method 2: (e^{i 2 \pi}) ^{1/2} = 1^{1/2} = 1

Hi, I've used complex numbers before for contour integration, circuit theory, and analysis; the only problem is, I have no idea how to think of complex numbers, physically - no teacher, book, or website seems to offer me any adequate solution. Can anyone please help me ? I hate doing...

Hi, I would be grateful of any advice on how to solve the problem below. My aim is to find the voltages at nodes 2, 3, and 4, by means nodal analysis and thus creating simultaneous equations and solving them using matricies or matrix in order to prove that these theories work. I...

Homework Statement Find the eigenvectors of the matrix A Homework Equations 3. The attempt at the solution \[ \left( \begin{array}{cc}4 & -5 \\1 & 0 \end{array} \right)\] First I find the characteristic equation A - \lambda I \[ \left( \begin{array}{cc}4-\lambda & -5...

Why is |z|^2 = z*z? z = a + ib z*z = (a - ib)(a + ib) = a^2 + b^2 z^2 = (a + ib)^2 = a^2 + 2iab - b^2 So it must have something to do with the absolute value, but I don't understand what or why.

Homework Statement b)show that if |z|>=3 then the following inequality holdsHomework Equations|z|>=3 now for this to hold b must be <=0 which gives the following: |e^iz|=e^b(cosa+isina) and taking the magnitude yeilds (e^2b)^1/2 I don't know where to go next.

hey again, im have a problem with 1 of the questions I am doing. Z^3 = 2+2i, and it asks to solve for Z. does it want me to actually get a number for Z? or does it simply want me to write... z = (2+2i)^(1/3) but ^^^^ this seems way to easy

hey I am doing some questions outta a txt book, i sort of understand complex numbers, like multiplying and dividing, .. The question asks to rearrange for z, e^(iz) = i - 1 im not sure what to do with the exponential function. thanks for the help

Homework Statement Prove that \left( {\frac{{1 + \cos x + i\sin x}}{{1 + \cos x - i\sin x}}} \right)^n} = \cos nx + i\sin nx The Attempt at a Solution I thought it would be a good idea calling z = 1 + cos x + i*sen x, because then 1 + cos x - i*sen x would be \bar{z}, and then I would...

-1-(under root)3 i here we find that r=2(hypotenuse) a=-1(base) b=-(under root)3 when i take sin theat= p/h=-(under root)3 / 2 theat from sin is -60 when i take cos theta = b/h =-1 / 2 which gives 120 now one is -60 and other is 120, which is the angel , i have to follow and what...

Homework Statement If c is any nth root of unity other than 1, then 1 + c + c^2 + \cdots + c^{n-1} = 0 The Attempt at a Solution This is what is done so far and I am at a dead stall for about 2 hours lol. Any ideas on what I should be thinking of next? Should I continue...

Homework Statement Find all complex numbers Z (if any) such that the matrix: (it is 2 by 2) (2)(-1) (4)(2) multiplied by a vector V = ZV has a nonzero solution V. part b) for each Z that you found, find all vectors V such that (the same matrix)*V=ZV. The Attempt at a Solution I'm not...

Homework Statement Supposing that A*B is defined (where A and B are both matrices in the field of the complex numbers), show that the conjugate of matrix A * the conjugate of matrix B is equal to the conjugate of A*B. Homework Equations None. The Attempt at a Solution I'm stuck. I've...

Complex numbers in mod-arg form ("cis") Greetings, I'm learning about the mod-arg form. I find it fairly easy when I come across simple radians that relate to the two special triangles like pie/3, pie/4 and pie/6. But when the radians become a little bit more complicated like 3pie/4 I'm in the...

Homework Statement Show that the equation |z - z_0| = R of a circle, centered at z_0 with radius R, can be written |z|^2 - 2Re(z\bar{z_0}) + |z_0|^2 = R^2. Homework Equations The Attempt at a Solution Honestly, I have no clue where to start with this problem. I know that I...

Homework Statement I need to square the magnitude of psi for each of my integrals Homework Equations for x between 0 and a, \psi(x,0) = A(x/a), where A and a are constants The Attempt at a Solution So I take A(x/a) and square it since it is already positive. so A^2\ast(x^2/a^2) ...

Homework Statement Compute the 4th roots of -16 in both Cartesian and polar form and plot their positions in the complex plane. Homework Equations z^1/n=(r^1/n)(e^i(theta)/n), (r^1/n)(e^i(theta)/n)(e^i2(pi)/n... The Attempt at a Solution How do I find the value of r, and theta??

Homework Statement Evaluate the square 0f 5e^(3(pi)i)/4 without using Cartesian form, and also the three different products. Homework Equations e^i(theta) = cos(theta) + isin(theta)? The Attempt at a Solution I have absolutely no idea here, nothing in my notes even begins to...

Homework Statement Determine all solutions of z^2 = 1+2i in the form z=a+bi, where a and b are real numbers.For this question numerical evaluation is not required. I just don't know how to start.:mad: any clue? thanks!

(1- sqrt 3i) ^3 I am having trouble solving the sqrt 3i part. I think I need to use de moivres theorem but I am unsure. If someone could push me in the right direction that would be a massive help. Thanks.

For instance, if I had a function of a complex variable z, how can I evaluate that function using a program?

1. Statement: The Real Part of a "Complex Number is expressed as the following: Real(A) = \frac{1}{2}(A + A*) = \frac{1}{2}(|A|e^{j\alpha} + |A|e^{-j\alpha}) = \frac{1}{2}|A|(2cos(\alpha)) = |A|cos(\alpha). (#1) The Imaginary Part of a "Complex Number" is expressed as the following...

I was wondering if someone could recommend a good text that explains the construction of complex from real, real from rational, rational from integers, and integers from natural numbers. Thanks

Homework Statement (z^3)-1 = 0. Solve the equation and show that the roots are represented in an Argand Diagram by the vertices of an equilateral triangle.Homework Equations The Attempt at a Solution i can only found 1 as the answer.By transferring the -1 to the other side to become 1 and cube...

Hello, Can someone help me understand why the Inner Product of a Null vector with itself can be non zero if complex numbers are involved? And why using the complex conjugate resolved this? I may have understood this wrong. It could be that an Inner Product of any non-Null vector with...

quick note, i am not allowed to use a calculator when doing these questions! 1: if Z = -i and W = 3+3i find arg(z) and arg(w) 2: 2Z + z^{-}(that's the conjugate symbol) = a+2i then z = 3: |z-i| <=1, what would it look like?, describe it's position from the point 0,0 my...

Complex numbers - are they the 'ultimate', or are there any "complex complex" numbers When we try to calculate the root of a negative number, we come to the idea to introduce complex numbers. Is there any operation for which complex numbers wouldn't be enough, so there's a need to introduce...

Sorry if this is a question that has been asked before but I've been browsing and not found anything great. What, specifically, is the advantage of using the field of complex numbers over simply R^2 (the real plane). I know that neater/less notation may sometimes help, but this would...

Homework Statement 1. (a) Express -1 + √3i in modulus-argument form. Evaluate (-1 + √3i)^8 expressing your answer in (a + ib) form. Find also the square roots of -1 + √3i in (a + ib) form. (b) Use complex numbers to find (intergral is between 0 and ∞) ∫ e^-x cos2x dx. 2. (a)...

Homework Statement 2iz^2 - (3-8i)z -6 + 7i Homework Equations z = \frac{-b +/- \sqrt{b^2 - 4ac}}{2a} The Attempt at a Solution Right, here goes... a = 2i b = -3 + 8i (is this correct? Or would it be better to leave it as 3-8i?) c = (-6 + 7i) so using these: z =...

Homework Statement Hi all. I have seen a conformal mapping of z = x+iy in MAPLE, and it consists of horizontal and vertical lines in the Argand diagram (i.e. the (x,y)-plane). On the Web I have read that a conformal map is a mapping, which preserves angles. My question is how this...

Homework Statement Solve the equation z^4= -i Homework Equations De Moivre's Theorem The Attempt at a Solution I understand how to find the roots by equating modulus and argument but I wanted to ask how do you know which arguments to take? Because I got up to 4*theta = -Pi/2...

Homework Statement If z = x + ( x+1) i, find the value of x for which Arg (z) = pi/3 Homework Equations The Attempt at a Solution ( x+1/x) = pi/3 x = 3/( pi -3) Answer: ( 3) ^(1/2) + 1 divided by 2

Homework Statement prove: arctanh(z) = 1/2 ln( (1+z)/(1-z) ) Homework Equations cosh z = (ez + e-z)/2 sinh z = (ez - e-z)/2 ez = ex + iy = ex(cosy + siniy) The Attempt at a Solution cosh z / sin hz = ez+e-z/ez-e-z

when indicating the region of the complex plane that corresponds to all points z that satisfy the condition |z|=11 this region would be the circle of radius 11 around the origin because the |z-w| is the distance between z and w, where z and w are both complex numbers in general |z-w|=r would...

Homework Statement \sqrt[n]{Z} has exactly n distinct value for integer n. What can you say about non-integer n's ? Homework Equations \sqrt[n]{Z}={|Z|}^{1/n}.(cos((\theta+2k\pi)/n)+isin((\theta+2k\pi)/n) The Attempt at a Solution I used Euler's formula to see clearly what the...

Homework Statement Resolve z^5-1 into real linear and quadratic factors. Hence prove that cos\frac{2\pi}{5}+cos\frac{4\pi}{5}=-\frac{1}{2} Homework Equations z=cis\theta z\bar{z}=cis\theta.cis(-\theta)=cos^2\theta+sin^2\theta=1 z+\bar{z}=cis\theta+cis(-\theta)=2cos\theta...

Homework Statement The curve on the Argand diagram for which |z-2|+|z-4|=10 is an ellipse. Find the co-ordinates of its centre and the lengths of its major and minor axes. Homework Equations z=x+iy The Attempt at a Solution I could only find the centre by using common sense in a...

Homework Statement Im am having trouble grasping the concept of complex numbers, could anyone please take the time to explain it. I have asked my lecturer but his help just seems to confuse me more. I have also searched the internet looking for explanations but that is giving me mixed...

does anyone know how to integrate e^(-2x)(cos(3x)dx using eulers formula e^(ix)= i(sinx)+cosx

When is it true that is |a|=|b|, then either a=b or a=b*, where a and b are complex numbers and b* is the complex conjugate of b?