Complex numbers Definition and 724 Threads

Hello everyone, I'm new to this forum. I have this Linear Algebra question that I have no clue how to solve. Any help would be much appreciated. :) The question goes as follows: The polynomial p(x) = x3 + kx + (3 - 2i) where k is an unknown complex number. It is given to you that if p(x) is...

Homework Statement . Prove that, given constants ##A_1,A_2, \phi_1## and ##\phi_2##, there are constants ##A## and ##\phi## such that the following equality is satisfied: ##A_1\cos(kx+\phi_1)+A_2\cos(kx+\phi_2)=A\cos(kx+\phi)## The attempt at a solution. I've tried to use the...

Homework Statement Show that (a^b)^c can have more values than a^(bc) Use [(-i)^(2+i)]^(2-i) and (-i)^5 or (i^i)^i and i^-1 to show this. Homework Equations The Attempt at a Solution I'm writing out the second one as the first one is long: i^i = e^ilni lni = i (\pi...

I am studying complex numbers and, hard as I try, I cannot see great difference between them and the conjugate numbers known and used since 500 B.C. (http://en.wikipedia.org/wiki/Quadratic_formula#Historical_development) to solve a quadratic equation p/2 \pm \sqrt(p/2 ^2\pm q) where the sum...

Homework Statement z=1-i e^{iz} = ? I have to solve this problem and than picture it. Homework Equations The Attempt at a Solution e^{iz} =e^{i(1-i)}=e^{i+1}=e^i*e I don't really understand how to picture this result. I assume their is an other way, in which the result has a...

Homework Statement If m and x are two real numbers where m ε Integers, then e2micot-1x{(xi+1)/(xi-1)}m, (where i=√(-1)) is equal to : (a) cos(x) + isin(x) (b) m/2 (c) 1 (d) (m+1)/2 Homework Equations The Attempt at a Solution I seriously have no clear cut idea of how to...

I am confused if complex numbers really are vectors. They seem to behave as vectors in addition, but not in multiplication. So why are the complex numbers defined to be vectors although they don't follow the same principles always. Another confusing thing for me is the "complex vector"...

Which is the best book on complex numbers? I'm new to this topic so I would like to begin my study with the basics. I prefer books that deal with concepts in a great detail for a better understanding. The book must also contain good problem sets(high order thinking) for practise. I'm aiming to...

Homework Statement Given: sin(x) = \frac{e^{ix}-e^{-ix}}{2} Show that sin(x) can be written as: sin(x) = \sum_{n=0}^n \frac{x^{(2n+1)}}{(2n+1)!} Homework Equations e^x = \sum_{n=0}^n \frac{x^{n}}{(n)!} The Attempt at a Solution I'm unsure how to treat the imaginary number in...

I was thinking that if you could use quaternions to rotate an object using quaternion algebra that there might be a way to rotate an object using complex numbers in some fashion. I was looking at quaternion rotation of a vector and the amount of operations seemed to be a lot. Of course it levels...

Why do we represent waves as complex numbers? Why won't real suffice? Thanks for any help.

Homework Statement Show that (1+i) is a root of the equation z4=-4 and find the other roots in the form a+bi where (a) and (b) are real.Homework Equations Using De Moivre's Theorem zn=[rn,nθ] Modulus(absolute value of z) = 4 Argument = ? The Attempt at a Solution r4=4 → r = (4)^(1/5)...

Homework Statement Prove for complex number z1, z2, ..., zn that: \mathbb{R}e\left \{ \sum_{k=1}^{N} z_{k}\right \} = \sum_{k=1}^{N}\mathbb{R}e\left \{ z_{k} \right \} Homework EquationsThe Attempt at a Solution Not sure how to setup this problem. I was thinking: \mathbb{R}e\left \{...

what is the defination of complex no?

Lets say z!=0, and zeC(is complex). So for example is z=2+3i. z^0=1 => (2+3i)^0=1. I am correct? I know that all numbers in zero make us one,but it works with complex numbers too?

Complex numbers? Since the system is not an ordered pair, how then is it defined using the complex system as an ordered system to plot the z axis (Plane) to use a function? At the point we input each point of the Real and imaginary plane into a function to get out an answer in the Z plane...

I have a view of complex numbers and the way they are taught. I think the whole concept of i as the sqrt(-1) is a terrible place to start. And calling it "imaginary" is worse yet. They should be called blue numbers, or vertical numbers, or something. They are anything but imaginary. It is...

Hi, I'm doing a mini-project in java that involves some nasty calculations with complex numbers- particularly with complex numbers in exponents. Thus far, I've had success using this class: Complex.java . The problem that I'm encountering involves taking the natural logarithm of a complex number...

Homework Statement Hi there, you can see from my nickname that I am a noob in maths :D. So, here should is one problem that I cannot solve, even though I know some basics of complex numbers. Its the 2nd problem from the revision exercises, so please be gentle :) Homework Equations Find...

Basically I don't know anyone in real life that can help me with this, so I need help checking to see if my answers are correct :) PART A 11) Find the vertex of f(x) = -2x^2 - 8x + 3 algebraically. My Answer: (-2,0) 12) Multiply and simplify: (6 - 5i) (4 + 3i) My Answer: 39 - 2i

Prove that all polynomials with real coefficients, having complex roots can occur in complex conjugates only. It's easy to prove in a quadratic equation... ## ax^{2} + bx + c = 0 ## ## \displaystyle x = \frac{-b \pm \sqrt(b^2 - 4ac)}{2a} ## But how to prove the same in general? Please...

"Simple" Line Integral in Complex Numbers If anyone could please double-check my final result for this question it would be greatly appreciated. Rather than write out each step explicitly, I'll explain my approach and write out only the most important parts. "[E]valuate the given...

Homework Statement Given the equivalent impedance of a circuit can be calculated by the expression: Z = Z1 X Z2 / Z1 + Z2 If Z1 = 4 + j10 and Z2 = 12 - j3, calculate the impedance Z in both rectangular and polar form. Homework Equations Multiplication and division of complex...

I'll write down what i know and point it out if I'm wrong.So we normalize the wave function because -∫|ψ(x,t)|^2dx should always be equal to 1 right? Has this anything to do with transition from ψ to ψ^2?

I can find for example Tan(2x) by using Euler's formula for example Let the complex number Z be equal to 1 + itan(x) Then if I calculate Z2 which is equal to 1 + itan(2x) I can find the identity for tan(2x) by the following... Z2 =(Z)2 = (1 + itan(x))2 = 1 + (2i)tan(x) -tan(x)2 = 1...

I guess the best way to start this is by admitting that my conceptual understanding of the Cauchy-Schwarz Inequality and the Lagrange Identity, as the title suggests, is not as deep as it could be. I'm working through Marsden's 3e "Basic Complex Analysis" and it contains a proof of the Cauchy...

Can the arithmetic mean of a data set of complex numbers be calculated? if so, can the method be demonstrated?

draw on a argand diagram |arg(z + 1)| = \dfrac{\pi}{2} I got the correct drawing... but I'm not sure why it's correct. What I thought was arg(z + 1) = \dfrac{\pi}{2} and that's a half line from the point (-1,0) going upwards, and arg(z + 1) = -\dfrac{\pi}{2} and that's a half life...

I'm working on an assignment that is due in roughly two weeks and I'm stuck on a problem. I have what I believe may be a solution but am unsure whether or not it is 'complete'. Here is the problem: "Let C be a circle or a straight line. Show that the same is true of the locus of points...

Let's start with: $$ \int \frac{dx}{1+x^2} = \arctan x + C $$ This is achieved with a basic trig substitution. However, what if one were to perform the following partial fraction decomposition: $$ \int \frac{dx}{1+x^2} = \int \frac{dx}{(x+i)(x-i)} = \int \left[ \frac{i/2}{x+i} -...

matrix inversion with complex numbers?? or faster way? Homework Statement The Attempt at a Solution i managed to get the answer, but it took me like 30min. to work this by hand. i probably worked it differently than my instructor's method above, but wat i did was get the coefficients of V...

Hi guys, I've been trying to help a friend with something that I learned in class but I'm now finding it hard to solve myself. The problem goes as follows: Use geometry to show that |z3-z-3| = 2sin3θ For z=cisθ, 0<θ<∏/6 Now, I chose ∏/12 as my angle and plotted all this on an Argand diagram...

Homework Statement Solve z^5 + 16 conjugate(z) = 0 for z element of C. z^5 + 16z' = 0 http://puu.sh/2EBqC.png Homework Equations The Attempt at a Solution My first thought was to use z = a+bi and z' = a-bi So: (a+bi)5 + 16*(a-bi) = 0 + 0i And then expand and simplify to the real and non real...

Homework Statement Prove the following statements about the inner product of two complex vectors with the same arbitrary numbers of components. (a)<u|w>=<w|u>* (b)|<u|w>|^2=|<w|u>|^2Homework Equations 1. <u|w>=(u*)w 2. (c_1+c_2)*=c_1*+c_2* 3. c**=c 4. ((c_1)(c_2))*=(c_1*)c_2*The Attempt at a...

Homework Statement Question: Homework Equations Any relevant to complex numbers. The Attempt at a Solution Given, Arg(\frac{z}{w})= Arg(z)-Arg(w) z=x+yi z1 = -1-2i z2 = 2+3i Arg(z-z1)=Arg(z2-z1) LHS: Arg(x+yi+1+2i) Arg((x+1) + i(y+2)) tan(\theta)=\frac{y+2}{x+1}...

let z,w be complex numbers. Prove: 2|z||w| <_ |z|^2 + |w|^2

Hi there, eI have two numbers: z1 = 2 + i z2 = exp(iδ) * z1 i are complex numbers and δ is a real number. I need to answer a question - what does the graphical representation of z2 have in relation to the graphical representation of z1. Thanks for any help!

Homework Statement Question 1: You find an old map revealing a treasure hidden on a small island. The treasure was buried in the following way: the island has one tree and two rocks, one small one and one large one. Walk from the tree to the small rock, turn 90 to the left and walk the same...

arg(\dfrac{z}{z-2}) = \dfrac{\pi}{3} sketch the locus of z and find the centre of the circle I've sketched the locus of z but I can't seem to find the centre of the circle. Is there a way to do it algebraically? I've attempted to use z = x + iy, but to no avail.

Homework Statement Evaluate: Homework Equations sin\frac{\pi }{7}.sin\frac{2\pi }{7}.sin\frac{3\pi }{7} The Attempt at a Solution Using z^{7}-1 got: cos\frac{\pi }{7}.cos\frac{2\pi }{7}.cos\frac{3\pi }{7}=\frac{1}{8}

Author: Titu Andreescu, Dorin Andrica Title: Complex Numbers from A to ...Z Amazon Link: https://www.amazon.com/dp/0817643265/?tag=pfamazon01-20

Homework Statement Sketch the line described by the equation: |z − u| = |z| z = x+jy u = −1 + j√3 The Attempt at a Solution (x+1)^2 + j(y-√3)^2 = (x+jy)^2 I just don't quite get where to go with this please give me a headstart

Homework Statement Let z be a complex number such that z^n=(z+1)^n=1. Show that n|6 (n divides 6) and that z^3=1. Homework Equations n|6 → n=1,2,3,6 The Attempt at a Solution The z+1, I think, is what throws me off. Considering z^n=1 by itself, for even n, z=±1 and for odd n...

Homework Statement Hello everyone :) ok so that is a problem involving complex numbers and its a bit challenging, so i would be really glad if i could get some help with it! The problem is: Consider the complex equation z^n=a+bi , where |a+bi|= 1. I am supposed to generalize and...

Homework Statement Let z be a complex number satisfying the equation ##z^3-(3+i)z+m+2i=0##, where mεR. Suppose the equation has a real root, then find the value of m.Homework Equations The Attempt at a Solution The equation has one real root which means that the other two roots are complex and...

Homework Statement So, I know how to solve multiple equations using the cSolve method on the ti 89, but for some reason when I try to solve the following... 80a + 240b = 0 and (80+J79.975)a-80b = 50 by using the following syntax... cSolve(80a + 240b = 0 and (80+J79.975)a-80b = 50...

Homework Statement Suppose you raise a complex number to a very large power, z^n, where z = a + ib, and n ~ 50, 500, one million, etc. On raising to such a large power, the argument will shift by n*ArcTan[b/a] mod 2*Pi, and this is easy to see analytically. However, is there less numerical...

Let's take \sum_{n=1}^{\infty} (-i)^{n} a_{n} , which is convergent , a_{n} > 0 . What can we say about the convergence of this one: \sum_{n=1}^{\infty} (-1)^{n} a_{n}? What can I do with it?

Homework Statement Show, using complex numbers, that sin(x)+cos(x)=(√2)cos(x-∏/4) Homework Equations cos(x)=(e^(ix)+e^(-ix))/2 sin(x)=(e^(ix)-e^(-ix))/2i e^ix=cos(x)+isin(x) The Attempt at a Solution I was given the hint that sin(x)=Re(-ie^(ix)), but have thus far not been...

Def. Let $\{z_j\}$ be a sequence of non-zero complex numbers. We call the exponent of convergence of the sequence the positive number $b$, if it exists, $$b=inf\{\rho >0 :\sum_{j=1}^{+\infty}\frac{1}{|z_j|^{\rho}}<\infty \}$$ Now consider the function $$f(z)=e^{e^z}-1$$ Find the zeros $\{z_j\}$...