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Euler's formula[′ȯi·lərz ‚fȯr·myə·lə]
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
x = ap + bq + cr + ds
y = aq – bp ± cs ∓ dr
z = ar ∓ bs – cp ± dq
t = as ± br ∓ cq – dp
(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.