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):

e^ix = 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 = 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.