What is Euler formula: Definition and 14 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.

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  1. J

    B Question about Euler's formula

    Euler gave us the below equations: But this doesn't actually give me a number value for where the y value is when you plug in a number for x. For example, if i plug in 2pi for x, i know cosx should be 1. But that equation gives me (e^2i*pi +e^-2i*pi)/2. This doesn't give me 1. So what really...
  2. M

    Using Euler's formula to prove trig identities using "sum to product" technique

    Hello, This is actually not homework. I was google searching for "proving trig identities from geometric point of view), found one of the result which proves trig identities using Euler formula. I really liked it. Easier, quicker & simple. But when the author speak about sum to product formulas...
  3. ke7ijo

    B Differentiating Euler formula vs. multiplying by i

    I differentiated both sides of Euler's formula with respect to x : e^ix = sin x + i cos x => ie^ix = cos x - i sin x Then for comparison I multiplied both sides of Euler's formula by i: e^ix = sin x + i cos x => ie^ix = i sin x - cos x Each of these two procedures seems to yield the...
  4. Ibidy

    Struggling to find solution to 1D wave equation in the following form:

  5. Brandon Trabucco

    B Complex Integration By Partial Fractions

    Hello, I am enrolled in calculus 2. Just having started a section in our textbook about integration by partial fractions, I eagerly began trying to use this integration technique wherever I could. After messing around for multiple days, I ran into this problem: ∫ 1/(x^2+1)dx I immediately...
  6. I

    B Can some explain why e^i(x) = cos(x) + isin(x)?

    Can anyone provide me with a simple explanation for why e^i(x) acts like this.
  7. P

    Euler Formula: Understanding (4.25) to (4.26)

    can someone explain how you go from (4.25) to (4.26) using Euler's formulas in the attachment?
  8. M

    Exploring the Euler Formula: Is e(-iπ) + 1 = 0?

    For Euler formula : e(iπ) + 1 =0 is it the same if we say e(-iπ)+1 = 0 or not ? (minus sign is included in the exponent)
  9. G

    What is the result of using Euler's equation for Fourier transform integrals?

    when I am using Euler equation for Fourier transform integrals of type \int_{-\infty}^{\infty} dx f(x) exp[ikx] I am getting following integrals: \int_{-\infty}^{\infty} dx f(x) cos(kx) (for the real part) and i* \int_{-\infty}^{\infty} dx f(x) sin(kx) (for its imaginary part) I am...
  10. SalfordPhysics

    Comp Sci Efficient Fortran Method for Trajectory Plotting with 4d Vector-First Order ODE

    Homework Statement I have a code that successfully plots the trajectory of a ball moving under gravity and air resistance, but my method is rather long-winded and I want to use a 4d vector-first order ODE instead - but I don't know how to do it. I've tried writing some simple skeletons but...
  11. AL-Hassan Naser

    Why Is the Wave Function Often Expressed as Psi = Cos(kr - wt) + i Sin(kr - wt)?

    why is psi = cos (k r - w t) + i sin ( k r - w t) = e^ [ i ( k r - w t)]? my question precisely is why not: 1. psi = sin (k r - w t) + i cos ( k r - w t) ? 2. psi = sin (k r - w t) + i sin ( k r - w t) ? 3. psi = cos (k r - w t) + i cos ( k r - w t) ? why not any of these three? is...
  12. K

    What is the equation for my graph and how can I use Eulers formula to help me?

    Homework Statement Since the exercise has a graph I uploded it here :http://imageshack.us/photo/my-images/833/img9845wz.jpg/ fmax(t)=1 T=2Pi h=1,3,5,7,... Also I was told that I could use Eulers formula here. Homework Equations maybe someone could give me some tips how to make the...
  13. S

    The Euler formula how was it developed?

    I'm reading a book called The Road to Reality by Roger Penrose, and I'm on the chapter for complex logarithms. What I don't understand is how the identity e\theta i = cos \theta + i sin \theta is found through the use of complex logarithms. I also don't understand how if w = ez, z = log r +...
  14. M

    Prove Euler Identity without using Euler Formula

    Is it possible to prove Euler's identity (e^i*pi = -1) without simply taking it as a special case of Euler's formula (e^i*x = cos(x) + i sin(x))?
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