Mathematics - My Favorite Euler Formulas

Euler’s formula in complex analysis


`e^(ix) = cos x + i sin x`


See wikipedia.

Euler’s identity


`e^(ipi) + 1 = 0`


See wikipedia.

Euler’s number


`e = sum_(n = 0)^infty 1/(n!) = 1/1 + 1/1 + 1/(1 * 2) + 1/(1 * 2 * 3) + cdots`


See wikipedia.

Euler’s product formula for the Riemann zeta function


`zeta(s) = sum_(n=1)^infty 1/(n^s) = prod_(p bbb " prime") 1/(1 - p^-s)`


See wikipedia.

See also Riemann hypothesis, one of the Millennium Prize Problems: wikipedia.

Sum Wikipedia
` 1 + 1 + 1 + 1 + cdots = -1/2 = zeta(0) ` link
` 1 - 1 + 1 - 1 + cdots = 1/2 ` link
` 1 + 2 + 3 + 4 + cdots = -1/12 = zeta(-1) ` link
` 1 - 2 + 3 - 4 + cdots = 1/4 ` link
` 1 + 2 + 4 + 8 + cdots = -1 ` link
` 1 - 2 + 4 - 8 + cdots = 1/3 ` link