Euler's Formula

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Euler's formula

[′ȯi·lərz ‚fȯr·myə·lə]
(mathematics)
The formula e ix = cos x + i sin x, where i = √(-1).

Euler’s Formula

 

any of several important formulas established by L. Euler.

(1) A formula giving the relation between the exponential function and trigonometric functions (1743):

eix = cos x + i sin x

Also known as Euler’s formulas are the equations

(2) A formula giving the expansion of the function sin x in an infinite product (1740):

(3) The formula

where s = 1,2,... and p runs over all prime numbers.

(4) The formula

(a2 + b2 + c2 + d2)(p2 + q2 + r2 + s2) = x2 + y2 + z2 + t2

where

x = ap + bq + cr + ds

y = aqbp ± csdr

z = arbscp ± dq

t = as ± brcqdp

(5) The formula (1760)

Also known as the equation of Euler, it gives an expression for the curvature 1/R of a normal section of a surface in terms of the surface’s principal curvatures 1/R1 and 1/R2 and the angle φ between one of the principal directions and the given direction.

Other well-known formulas associated with Euler include the Euler-Maclaurin summation formula and the Euler-Fourier formulas for the coefficients of expansions of functions in trigonometric series.

References in periodicals archive ?
Employing complex exponentials results in the short-time Fourier transform with N coefficients, and there are many options for binary bases.
The use of complex exponential basis functions is well suited to multi-carrier (OFDM) receivers, and we note that the number of basis elements N may be as small as N = 2 and is not required to be equal to the number of carriers (which may be in the hundreds) [31]; it is only required that condition (5) holds with respect to the entire OFDM signal bandwidth B .
1: Number of precomputed and stored complex exponentials (memory), the order of magnitude for the number of floating point operations (flops), and the number of evaluations for the function cexp() (evaluations).
We follow the general approach of [38, 37] and approximate the complex exponentials in the trigonometric polynomial (2.
pi]] direct calls of the function cexp() to evaluate the complex exponentials [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
4] developed a generalized ESPRIT algorithm for estimation of parameters of a signal modeled by the Polynomial Amplitude Complex Exponentials model.
Richard, High-resolution spectral analysis of mixtures of complex exponentials modulated by polynomials, IEEE Trans.
The approximation of data by finite linear combinations of complex exponentials has a long history; see [19, 20].
PEREIRA, Using the matrix pencil method to estimate the parameters of a sum of Complex exponentials, IEEE Antennas and Propagation Magazine, 37 (1995), pp.
d], since by the periodicity of the complex exponential
i[lambda]t}[lambda][member of][subset]C of complex exponentials
For the sequence of zeros [LAMBDA] of the quasipolynomialL the family of complex exponentials [epsilon]([LAMBDA]) = [{[e.

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