Euler's Formula

Apr 14 2016

Verify Euler's Formula Using MacLaurin Series

The following equation is known as Euler's formula.

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

We can verify that applying MacLaurin expansion to the both side of the above equation.

`e^(i x) = 1 + i x - x^2 / 2 - (i x^3) / 6 + x^4 / 24 + (i x^5) / 120 - x^6 / 720 - (i x^7) / 5040 + ...`

`cos(x) = 1 - x^2 / 2 + x^4 / 24 - x^6 / 720 + ...`

`i sin(x) = i x - (i x^3) / 6 + (i x^5) / 120 - (i x^7) / 5040 + ...`

`cos(x) + i sin(x) = 1 + i x - x^2 / 2 - (i x^3) / 6 + x^4 / 24 + (i x^5) / 120 - x^6 / 720 - (i x^7) / 5040 + ...`

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