Euler's Formula

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 + ...` take 8 (taylorExpansion (e ^ (i * x)) x 0) -- [1, i * x, - x^2 / 2, - i * x^3 / 6, x^4 / 24, i * x^5 / 120, - x^6 / 720, - i * x^7 / 5040] take 8 (taylorExpansion (cos x) x 0) -- [1, 0, - x^2 / 2, 0, x^4 / 24, 0, - x^6 / 720, 0] take 8 (taylorExpansion (i * sin x) x 0) -- [0, i * x, 0, - i * x^3 / 6, 0, i * x^5 / 120, 0, - i * x^7 / 5040] take 8 (map2 (+) (taylorExpansion (cos x) x 0) (taylorExpansion (i * sin x) x 0)) -- [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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