# 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).
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## 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 ?
The filter is implemented by two parallel, quadrature filter branches with each branch derived from a complex modulation of a low-pass-interpolated FIR filter by complex exponentials. The input signal is modulated with a sine/ cosine sequence in order to achieve the desired frequency shift in the frequency response.
Pereira, Using the matrix pencil method to estimate the parameters of a sum of complex exponentials," IEEE Antenna and Propagation Magazine, vol.
Pereira, "Using the matrix pencil method to estimate the parameters of a sum of complex exponentials", IEEE Antennas Propagat.
This is done by first expanding the electric field spectrum in a Fourier series and next approximating the obtained Fourier series coefficients through a sum of complex exponentials using the generalized pencil-of-function (GPOF) method [15], which leads to a final sum of conical beams.
ESPRIT is a frequency estimation technique which is based on a harmonic model; that means that the considered signal consists of complex exponentials in noise.
From a circuit implementation perspective, two appealing choices for the basis functions [[PHI].sub.i] (t) are (i) those that consist of binary waveforms, and (ii) complex exponentials. Tones and binary signals are straightforward to produce in dedicated simple circuits, (3) with relatively low power consumption, avoiding the use of general purpose digital to analog conversion (DAC) to produce the [[PHI].sub.i] (t) waveforms.
The beam expansion of an aperture-radiated field is obtained by expanding the field spectrum in the aperture plane in terms of complex exponentials. The latter are obtained by the 2D Generalized Pencil of Function 2D-GPOF [2].
of M complex exponentials with complex coefficients [c.sub.j] [not equal to] 0 and distinct frequency vectors [f.sub.j] = [[[f.sub.j,l]].sup.d.sub.l=1] [member of] [T.sup.d] [congruent to] [[-[pi], [pi]).sup.d].
[4] developed a generalized ESPRIT algorithm for estimation of parameters of a signal modeled by the Polynomial Amplitude Complex Exponentials model.
can be proved easily using complex exponentials. The left side is the real part of
For that reason, the idea of the Gabor expansion is to express a signal S(k) as a weighted summation of elementary functions formed from Gaussian weighted complex exponentials that exhibit a corresponding Gaussian-shaped spectrum.
In the original GPOF method, the discrete temporal data f([t.sub.n]) are approximated by using a set of p complex exponentials

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