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.

SumWikipedia
` 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

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Updated on 2019 Mar 16.
DISCLAIMER: This is not professional advice. The ideas and opinions presented here are my own, not necessarily those of my employer.