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 It might be outdated or ideologically biased.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.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
He starts with Cauchy-Riemann equations in the introduction, then proceeds to power series, results on holomorphic functions, logarithms, winding numbers, Couchy's theorem, counting zeros and the open mapping theorem, Eulers formula for sin(z), inverses of holomorphic maps, conformal mappings, normal families and the Riemann mapping theorem, harmonic functions, simply connected open sets, Runge's theorem and the Mittag-Leffler theorem, the Weierstrass factorization theorem, Caratheodory's theorem, analytic continuation, orientation, the modular function, and the promised Picard theorems.