What is Complex exponential: Definition and 77 Discussions

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number x:





e

i
x


=
cos
⁡
x
+
i
sin
⁡
x
,


{\displaystyle e^{ix}=\cos x+i\sin x,}
where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively. This complex exponential function is sometimes denoted cis x ("cosine plus i sine"). The formula is still valid if x is a complex number, and so some authors refer to the more general complex version as Euler's formula.Euler's formula is ubiquitous in mathematics, physics, and engineering. The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics".When x = π, Euler's formula evaluates to eiπ + 1 = 0, which is known as Euler's identity.

View More On Wikipedia.org
Recent contents Top users
  1. H

    Mapping of complex exponential

    Homework Statement Determine the image of the line segment joining e^(i*2*pi/3) to -e^(-i*2*pi/3) under the mapping f = e^(1/2*Log(z)). Homework Equations The Attempt at a Solution The line joining the two points: {z | -0.5 < x 0.5, y = sqrt(3)/2} f = the principle branch of...
  2. K

    Reduce complex exponential expression

    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...
  3. cepheid

    Discretized complex exponential

    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...
  4. C

    Fourier transform of complex exponential multiplied to unit step

    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 +∞...
  5. R

    Complex exponential proof using power series

    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)...
  6. M

    Solving Complex Exponential Equations: Finding Solutions to z^4 = -1

    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...
  7. R

    Finding Largest r in the Complex Exponential Function Mapping D(z0,r)

    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...
  8. P

    Third Order DE Using Complex Exponential

    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...
  9. M

    Exponential Form of e^z for z = 4e^(i*pi/3)

    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...
  10. James889

    The complex exponential function

    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...
  11. B

    Solving complex exponential polynomials

    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...
  12. T

    Complex Exponential: What is e^jw∞?

    what is e^{jw\infty} ?
  13. M

    Integral of a complex exponential

    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...
  14. D

    Wave Superposition - Complex Exponential

    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...
  15. F

    Derivative using complex exponential

    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) ??
  16. S

    What is the Exact Value of j^j?

    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:
  17. G

    Solve Integral & Complex Exponential Problems: Help Needed

    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) =...
  18. G

    Could really use a hand with a complex exponential and integral problem

    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...
  19. E

    Polar notation for a complex exponential function

    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)...
  20. Repetit

    Complex Exponential: Why is e Used?

    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!
  21. M

    How Can I Simplify Trigonometric Expressions Using Euler's Formula?

    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...
  22. E

    Laplace transform of complex exponential

    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...
  23. G

    Complex exponential (properties)

    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...
  24. E

    RC circuit with complex exponential

    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...
  25. E

    Can complex exponential e^{ix} be proven without using Taylor series?

    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...
  26. D

    How to use complex exponential to find higher derivatives of e^x cos(x√3)?

    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...
  27. N

    Learn How to Express cos(2x) as sin(x) Using Complex Exponential Series

    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:
Back
Top