Imaginary Definition and 359 Threads

I recently read the book "A Brief History of Time"by Stephen Hawking, and in it he described the concept of imaginary time. It had something to do with the squares of numbers being equal to negative numbers, which were called imaginary numbers. Also, he mentioned being able to travel in...

In solutions to a problem I was working on, I saw that when an expression such as e^{i\alpha} , alpha being an angle (in polar coordinates), was squared, the expression goes to unity, ie (e^{i\alpha})^2=1 But I see no reason to think that \alpha is a multiple of \pi. Could there be any other...

Hello everyone, In Fermi Liquid Theory, I'm trying to understand what the relationship is between the Green's function and the average occupancy of levels. In my lecture they gave the relation \left\langle n_k \right\rangle = G(k,\tau\rightarrow 0^+) Anyone know where this comes from...

So I have a control systems mid-semester exam coming up and the lecturer has posted up a formula sheet for us. However it is different to past years exams and has a geometry section with the following equations: e^(±jθ)=cos(θ)±jsin(θ) cos(θ)=(e^jθ+e^-jθ)/2 sin(θ)=(e^jθ-e^-jθ)/2j Now I've...

Homework Statement Solve sin z=2 by (a) equating the real and imaginary parts (b) using the formula for arcsin z. Homework Equations (a) sin z = sin x * cosh y + i * cos x * sinh y arccosh z = log[z + sqrt(z^2 - 1)] (b) arcsin z = -i * log [i * z + sqrt(1 - z^2)] The...

Well I have developed a number system which allows the existence of imaginary numbers. Please visit it at : http://www.scribd.com/doc/46064105/Math-Paper. An intro of these ideas is presented at :http://www.scribd.com/doc/46117043/Introduction-to-My-Research-Paper Please provide me feedback...

so i have the function z=(2+i)/(i(-3+4i)) and i need to linearize it to find the Im(z) and Re(z) I get down to z= (-6 +8i -3i -4 )/ (9i +12 +12 -16i) which i then simplify down to z= (5i -10)/(-5i+24) However when solve it i get a different answer from wolfram (from when i plugged...

does the behavior the imaginary part behave in anyway similar to the real part of a holomorphic function. say if the real part if bounded or positive, what can you conclude about the imaginary part.

In my linear algebra text it says it's possible to define (for nxn matrix A) A_1^* =\frac{A+A^*}{2} A_2^* =\frac{A-A^*}{2i} so A=A1+iA2 It then asked if this was a reasonable way to define the real and imaginary parts of A. Is there a specific convention to define the real and imaginary parts...

Can someone please explain the concept of optical losses and its correlation with the imaginary part of the dielectric function in elementary terms. I am confused.

Hello, I did the integral of a Fourier Transform which resulted in this: A(je^(-jwe^(To+t/2) - je^-jw(T0-t/2))/(1/w) Where A is the amplitude, j the imaginary number, and w is omega or 2*pi*f. My question is, how can this be further simplifier. I am forgetting how to simplify...

A tachyon is a hypothesized particle that has imaginary mass (imaginary numbers) and moves faster than light speed. I don't believe in it because it can't have imaginary mass, what about you?

Consider mass m_{1}and m_{2}with position vector (from an inertial frame) \overrightarrow{x_{1}} and \overrightarrow{x_{2}} respectively and distance between them be x_{0}. m_{1}\frac{d^{2}}{dt^{2}}\overrightarrow{x_{1}}=\overrightarrow{F} \Rightarrow...

Homework Statement given the following diagram : http://www.freeimagehosting.net/uploads/… a)mention for every step if it is isothermal / adiabatic / else. does the system receive heat or emit heat. b)given this gas is ideal gas, sketch a diagram with respect to P and V c)calculate work done...

I'm really confused with how to prove this result...could anybody help please? Let I_{1} be the line segment that runs from iR (R>0) towards a small semi-circular indentation (to the right) at zero of radius epsilon (where epsilon >0) and I_{2} a line segment that runs from the indentation...

We can easily comment the result of a root operation just by the information if the degree of the root is odd or even. But what if the degree of the root (or power) is irrational? For example; -64 ^ \frac{1}{2} \, = \, j8 \,\,\,\,\, (imaginary) -64 ^ \frac{1}{3} \, = \, -4 \,\,\,\,\...

Hi, I have a problem on how to convert the imaginary parts of expression into all real parts. For example: x1 = - (a + ib) x2 = (a + ib) x3 = - (a - ib) x4 = (a - ib) My question is that how to express x1, x2, x3 and x4 in terms of real parts only without imaginary parts. I have used...

Hi all, I am having a hard time solving a partial second order differential equation with an imaginary part. I basically took a much bigger function with real and imaginary parts and simplified it down to this. I also know the solution to a similar equation (shown in image). Any help would...

Why and how does this definite integral result in an imaginary solution ? At wolframalpha ... definite integral 1 / [e^x arcsin x] dx from 1 to 10 = 0.156 + .09i Area under such a function should be positive or negative but how does it become imaginary ? Thanks

Homework Statement Write 2*EXP(i*pi/3) in the form \alpha + i\beta Answer is given = 1 + sqt(3)i Homework Equations The Attempt at a Solution I'm supposed to turn this exponential form of imaginary number into a standard form in order to solve an ODE. I have no idea how they got 1+sqt(3)i...

My (high school, gah!) textbook gives an experiment: I take a straight current carrying conductor, a cardboard sheet is placed perpendicular to the conductor, so that the conductor passes straight through the sheet, remains perpendicular. Then I use a salt sprinkler to sprinkle iron filings on...

Homework Statement Find the real and imaginary part of sin(4+3i) Homework Equations sinx = \frac{e^z - e^(-z)}{2i} cosx = \frac{e^z + e^(-z)}{2} sin(iy) = i\frac{e^y - e^(-y)}{2} cos(iy) = \frac{e^y + e^(-y)}{2} various trig identities The Attempt at a Solution So I used sin(x+y) trig...

Are there "imaginary" numbers other than i? I'm taking a class in complex analysis and the professor wrote the textbook so I'm getting most of it. There is one elephant in the room though, and I haven't been able to make office hours to clear it up. Are there "imaginary" numbers other than...

How is this possible? \int_{i\infty}^\pi e^{ix} dx = i I mean, I understand that the integral of exp(ix) is -i exp(ix) and then you evaluate that from π to i∞ — but that's exactly it, how does one "draw a line" from (π, 0) on the Argand plane to (0, ∞)? (assuming Argand plane tuples (a, b) ↔...

Imagine an empty universe, where nothing exist and time stands still. Then add lots of stars of equal size, distributed in a symmetry around a spot that we call the center of our universe. Since time has not passed, no curvature (gravity) has propagated to affect any of the other stars. No...

So, according to my understanding, m= m_o/√(1-(v^2/c^2 )) gives the mass of an object in respect to the object's original mass and its velocity. I wondered what happened if the mass of an object became lower than the rest mass? [I have no idea how this would happen, but it was a, what if...

-1/1=1/-1 sqrt(-1/1)=sqrt(1/-1) i/1=1/i i*(i/1)=i*1/i i^2/1=i/i -1/1=1 -1=1 <-- Well my conclusion is that properties don't work with imaginary numbers or did i do something wrong?

in ac fields permittivity becomes complex quantity and has real and imaginary parts. in metals (may be few exceptions but i don't know) imaginary part is always positive and represents loss factor or energy absorbed. why the plot of imaginary part of dielectric constant as function of energy is...

i need to find the equation for the capacitator to lower the imaginary power how they got thered marked equation

Is there a more convenient way to multiply and divide imaginary numbers than converting back and forth from phasors? (I guess I should say "when also having to add and subtract them") I always find AC circuit calculations to be tedious and problem filled when I do it that way. For example...

Homework Statement Find the general solution for: y''+2y'+5y=3sin2t The attempt at a solution y''+2y'+5y=3sin2t First step is to find the general solution to the homogenous equation, so skipping 2 steps (letting y=e^rt and dividing) R^2+2r+5 (-2+/- sqroot(4-4*5))/2 =-1 +/- 2i...

Why do we have an imaginary number? I don't see it's usefulness. Why dos it matter if we can make up a number that satisfies this equation (\ x^{2}+1=0 )? It must have real world applications that I'm unaware of.

Wave functions are, of course, almost always complex-valued. In all of the examples that I have seen (infinite square well, etc.), the real part of the wave function and the imaginary part of the wave function are basically the same function (except for a phase difference and possibly a sign...

Extract Real and Imaginary Equation with MATLAB ! Hi all, How to write M-files that can extract the real and imaginary components, or the magnitude and phase, of a symbolic expression for a complex signal with MATLAB. x(t) = e^j*2*pi*t/16 + e^j*2*pi*t/8 <<< equation example Thanks In...

Homework Statement C=A*e^(-i*wt)*sin(k*x); A,w,t,k,x are real numbers. Find imaginary part. Homework Equations The Attempt at a Solution Im(C)=cos(wt)-i*sin(wt)

I was just doing some homework, and I got to thinking about this. So if the limit of a function is an imaginary number, does that mean that the limit does not exist? Or that it does not exist on the xy-plane, or what? I mean...imaginary and complex numbers exist, we just can't graph them...

Homework Statement Evaluate \int\int x^{2}e^{x^{2}y} dx dy over the area bounded by y=x^{-1}, y=x^{-2}, x=ln 4 Homework Equations The Attempt at a Solution \int^{1}_{(ln 4)^{-2}}\int^{y^{-1}}_{y^{\frac{-1}{2}}}x^{2}e^{x^{2}y}dx dy I got this far before I realized that this wasn't a...

Homework Statement I came across this expression in homework and for the life of me I can't figure out how this evaluates to 0: 1 - ( -i )^-4 = 1 - 1 = 0 I know that i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. I'm just not sure how to treat the negative on the i. Do I just treat i as if it...

can anyone tell me how to get the real and imaginary parts of the following function : (x+ i y)* Log( a+i b) where x, y a and b are all real numbers and i =sqrt (-1). Thanks very much

i know this must seem real stupid but if 1 x 1 =1 ( square root wise) how can -1x-1=+1 again square root wise. i am reading fermats last theorum to me if you times negative you increase the negative. i don't see why the imaginary numbers need to be invoked. i understand the argument for...

Homework Statement (1+i)i = reiθ Find the real values of r and θ. The Attempt at a Solution Well, after doing a similar(ish) question I decided taking logs would be a good start: i loge(1+i) = loger + iθ From here, I have no idea where to go. Using a power of i is killing me...

I'm doing some practice problems for my mechanics exam tomorrow (good ol' SHM) and I can't solve this for the life of me: Determine: (-1+i)^(1/3) Any help would be greatly appreciated.

A very general question: What do the real and imaginary parts of a wave function correspond to physically? Cheers

Hello, I have a quick question that I imagine anyone who has studied physics or math at a university can answer rather easily. If not, I apologize in advance for the effort! What is the physical significance of imaginary numbers? I have heard repeatedly that imaginary numbers are relevant...

Imaginary numbers are a lot less mysterious than they sound. They are the result from trying to take the square root of a negative number. They are called “imaginary” because they don’t exist in the normal number system, normally you can’t take the square root of a negative number because the...

Steven Hawking writes in A Brief History of Time that time itself must sometimes have an imaginary component in order for Feynman's Sum-Over-Histories approach to work. Why, in a nutshell, is this so? Thanks in advance.

In Misner, Thorne, Wheeler: "Gravitation" it is stated on that "no one has discovered a way to make an imaginary coordinate work in the general curved spacetime manifold" (p.51). Can anyone elaborate on this? Right now, I don't get why it wouldn't work and nothing more is said in the book.

Hi: Does anyone know of an explicit formula for the Real and Imaginary parts of GAMMA(1/2+I*y) as functions of y ? I know about |GAMMA(1/2+I*y)|^2 =Re(GAMMA(1/2+I*y))^2+Im(GAMMA(1/2+I*y))^2= Pi/cosh(Pi*y) but can't find anything about each of the Real and Imaginary terms...

Hi all, What happens when we take the product of the imaginary parts of all the n-roots of unity (excluding 1)? I read somewhere that we get n/(2^(n-1)). How can we prove this? Thanks!

I have trouble understanding the concept of imaginary time. As I understand it (by reading other online sources), imaginary time was developed by Stephen Hawking and Hartle in their theory of quantum cosmology. Quantum cosmology is applying quantum mechanical principles to cosmology as a...