# Euler's Formula

(redirected from*Complex exponentials*)

## Euler's formula

[′ȯi·lərz ‚fȯr·myə·lə]*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):

*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

(*a*^{2} + *b*^{2} + *c*^{2} + *d*^{2})(*p*^{2} + *q*^{2} + *r*^{2} + *s*^{2}) = *x*^{2} + *y*^{2} + *z*^{2} + *t*^{2}

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/*R*_{1} and 1/*R*_{2} 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.