Complex exponential Definition and 76 Threads

Homework Statement prove that: 1-exp(-iwt)= 2i*sin(wt/2) Homework Equations exp(iwt)= cos (wt) + i*sin(wt) The Attempt at a Solution I attempted to express the exponential into sum of cos and sin and considering t=2*t/2 in order to obtain an argument like (t/2) (using...

I'm reading that if you have a complex exponential exp(iω0n) where n is in the set of integers, then unlike for the case of a continuous independent variable, the set of complex exponentials that is harmonically-related to this one is finite. I.e. there is only a finite number of distinct...

Homework Statement find the Fourier transform of complex exponential multiplied to a unit step. given: v(t)=exp(-i*wo*t)*u(t) Homework Equations ∫(v(t)*exp(-i*w*t) dt) from -∞ to +∞ The Attempt at a Solution ∫([v(t)]*exp(-i*w*t) dt) from -∞ to +∞...

I need to prove that ez1 x ez2 = e(z1 + z2) using the power series ez = (SUM FROM n=0 to infinity) zn/n! (For some reason the Sigma operator isn't working) In the proof I have been given, it reads (SUM from 0 to infinity) z1n/n! x (SUM from 0 to infinity)z2m/m! = (SUM n,m)...

Homework Statement Let z=|z|e^{\alpha*i} Using the fact that z*w=|z||w|e^{i(\alpha+\beta)}, find all solutions to z^4 = -1 The Attempt at a Solution Not quite sure how to proceed, except for the obvious step i=z^2=|z*z|e^{i(2\alpha)}= |z*z|[cos(2\alpha)+isin(2\alpha)] Kinda stuck here :s...

Homework Statement According to the Inverse Function Theorem, for every z_0 \in C there exists r > 0 such that the exponential function f(z) = e^z maps D(z0; r) invertibly to an open set U = f(D(z_0; r)). (a) Find the largest value of r for which this statement holds, and (b) determine the...

Homework Statement find three independent solutions using complex exponentials, but express answer in real form. d^3(f(t))/dt^3 - f(t) = 0 Homework Equations The Attempt at a Solution after taking the derivative of z = Ce^(rt) three times I put it in the following form...

Homework Statement write e^z in the form a +bi z = 4e^(i*pi/3) --------------------------------------- My guess: z = 4*(cos(pi/3) + i*sin(pi/3)) e^z = e^(4*(cos(pi/3) + i*sin(pi/3))) = e^(4*cos(pi/3))*(cos(4*sin(pi/3)) + i*sin(4*sin(pi/3))) but the solution says...

Hi, I need to solve the equation e^z = -3 The problems arises when i set z to a+bi e^a(cos(b) + isin(b),~b = 0 Then I am left with e^a = -3 However you're not allowed to take the log of a negative number. Also i know that cos(\pi) + isin(\pi) = -1 Obviously 3e^{(\pi*i)} is a solution, but...

Are there any general methods to solve the following complex exponential polynomial without relying on numerical methods? I want to find all possible solutions, not just a single solution. e^(j*m*\theta1) + e^(j*m*\theta2)+e^(j*m*\theta3) + e^(j*m*\theta4) + e^(j*m*\theta5) = 0 where...

what is e^{jw\infty} ?

Homework Statement let be A_{i,j} a Hermitian Matrix with only real values then \int_{V} dV e^{iA_{j,k}x^{j}x^{k}}= \delta (DetA) (2\pi)^{n} (1) Homework Equations \int_{V} dV e^{iA_{j,k}x^{j}x^{k}} = \delta (DetA) (2\pi)^{n} The Attempt at a Solution the idea is that...

Hi guys, I lurk here often for general advice, but now I need help with a specific concept. Ok, so I started a classical and quantum waves class this semester. We are beginning with classical waves and using Vibrations and Waves by A P French as the text. So in the second chapter he discusses...

I'm trying find the 15th derivative of exp[(1 + i(3^.5))theta] with respect to theta To do this do i need to split it into two exponentials, (e^theta).(e^i(3^.5)theta) ??

good evening all! Homework Statement Determine the exact values of j^j Homework Equations j = sauare root of -1 The Attempt at a Solution stuck :cry: :cry: :cry:

I have two homework problems that have been driving me nuts: 1.) evaluate the indefinite integral: integral(dx(e^ax)cos^2(2bx)) where a and b are real positive constants. I just don't know where to start on it. 2.) Find all values of i^(2/3) So far I have: i^(2/3) =...

I have two homework problems that have been driving me nuts: 1.) evaluate the indefinite integral: integral(dx(e^ax)cos^2(2bx)) where a and b are real positive constants. I just don't know where to start on it. 2.) Find all values of i^(2/3) So far I have...

I am trying to find the polar notation for 1 + e^(j4) I know that e^(jx) = cos x + jsin x = cos(4) + jsin(4) I can then find the magnitude and angle. This is nowhere close to the answers below. 1) cos(2) + 1 2) e^(j2)[2cos(2)] 3) e^(-j4)sin(2) 4)...

Hey! I was wondering, is it merely a definition that e^{ix}= cos(x) + i sin(x) or is it actually important that it is the number e which is used as base for the exponential? Thanks!

Express the following in the form z=Re[Ae^{i(\omega t+\alpha)}] z=cos(\omega t - \frac{\pi}{3}) - cos (\omega t) and z=sin(\omega t) - 2cos(\omega t - \frac{\pi}{4}) + cos(\omega t) I got a few of the problems correct by using trig. identities but it was pretty tough and two I can't get...

I just want to be sure I understand this correctly, usually L[f(t)] = 1/(s-a), where f(t) = e^{at}, but if it is a complex number would still be 1/(s - complex_number)? techinically, i think it should be, since every number can be reprsented as complex number. Just want to be sure about this...

Hi, I am solving a second order ODE. the result I got is an exponential to the power of a real and an imaginary part, both of them inside a square root. I need to brake this result into an imaginary and a real part because in this particular case just the imaginary part of the solution is my...

I am stuck on this... Given a circuit: current source (Is(t)), R , C - all parallel; Is(t) = e^jt, Vs(t) = 223.6e^j(t - 63.43), Vs(t) is voltage across the current source, which I assume to be the same across R and C since they are ||. Find R and C. (ans: 500 Ohm, 4mF) My attemp was to...

e^{ix}=cosx + isinx I know this can be easily proven using the Taylor series, but I recall seeing a proof which doesn't use the Taylor series. I'm pretty sure it has something to do with derivatives, but the problem is I don't remember how it went and I can't find it anywhere. So if anyone...

How would one use the complex exponential to find something like this: \frac{{d^{10} }}{{dx^{10} }}e^x \cos (x\sqrt 3 ) I'm guessing we'd have to convert the cos into terms of e^{i\theta } but the only thing I can think of doing then is going through each of the derivatives. I am guessing...

This is an easy question but can some else show/tell me how to do it: "use the complex exponential series to express cos(2x) in terms of sin(x)" I also don't quite understand the 'complex exponential series'. :redface: