What is Eulers formula: Definition and 12 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:






{\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.

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  1. C

    Capstan equation, Euler's formula, power law friction

    Dear colleagues, I am dealing with rope friction and the so-called Capstan equation. Situation: A rope wraps around a cylinder with a wrap angle. It depends on the input force. There are very comprehensive approaches by other colleagues, where the friction value depends on the normal force or...
  2. S

    Simplifying expressions using Euler's formula

    The following is the questions given. I solved the first one, which steps are shown below. But I am not sure if this is how the question wants me to solve the problem. Would you tell me if the way I solved the problem is the proper way of simplifying the expression using euler's formula...
  3. CalcExplorer

    A How to Convert a Complex Logarithm to a Complex Exponential

    Okay, so I'm working with a rather frustrating problem with a calculus equation. I'm trying to solve a calculus equation which I conceptualized from existing methods involving complex number fractal equations. I'm very familiar with pre-calculus, while being self-taught in portions of calculus...
  4. TheDemx27

    Euler's Formula Contradiction?

    I've been using euler's formula now more than I have in the past, (using it for circuit analysis stuff), and so its been floating around in my head a bit more. Say you have e^{2πi}=1 and you take the natural log of both sides. \log_e( e^{2πi})=\log_e(1) 2πi=0 uhhhhh... :confused:
  5. C

    Integration with eulers formula.

    Is it possible to do integrals like this with eulers formula \int e^{-x^2}cos(-x^2) and this integral is over all space. then we write e^{-x^2}e^{-ix^2} then can we square that integral and then do it in polar coordinates, and then we will eventually take the square root...
  6. C

    Integration with Eulers formula.

    I want to integrate \frac{e^x}{cos(x)} with eulers formula. I start by writing \frac{e^x}{e^{ix}} then I integrate that as usual. So after I integrate I get \frac{e^{(1-i)x}}{1-i} Normally I would multiply and divide by the complex conjugate and then back substitute in e^(ix)...
  7. C

    Solving an ODE with eulers formula.

    If I have y''+y'+2y=sin(x)+cos(x) can I just say y=Ae^{ix} and then find y' and y'' and then plug them in and solve for A. so I get that A= \frac{1}{1+i} then i multiply and divided by the complex conjugate. then I back substitute in Eulers formula. now since I have my...
  8. M

    Exploring Eulers Formula for Trigonometric Identities

    The question is: Using Eulers formula for e^±iθ , obtain the trigometric identities for cos(θ1, θ2) and sin(θ1, θ2) I think I have completed the real and imaginary solutions for the base e^+iθ using the real part cos(θ1+θ2)and imaginary isin(θ1+θ2) Gaining cos(θ1 + θ2) = cosθ1 cosθ2 -...
  9. S

    Is this a variant of Euler's Formula? Understanding e^iy = cosy + isiny

    Eulers formula says that e^ix = cosx + isinx but in my textbook there's another formula its e^ikx = coskx + isinkx i still can't figure out how they got that. Is this still eulers formula? and how do u get it in that form
  10. C

    Integral of (secx)^3 with eulers formula

    is it possible to integrate (secx)^3 with eulers formula could we use that cosx = (e^(ix) + e^(-ix)) /(2) then take it to the -3 power and multiply it out and try to integrate sec(x)^3 this way. this is not a homework ?
  11. C

    Integration using eulers formula and complex numbers

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

    What is the significance of the number e and Euler's formula?

    hi, can anyone tell me what is the meaning of number e,i mean how it is discovered ?why derivative and antiderivative of this function same?i know it is very practical property of it but where did we get this number? and another thing eulers formula, which is e^(i*pi)+1=0, i also can't...