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Motivational Argument for the Expression: eix = cos x + i sin xProblem:
Show that eix = cos x + i sin x, where i = Solution: Let us turn to the theory of differential equations. The equation y'' + y = 0 (where the prime notation
symbolizes differentiation with respect to x) has a solution of the form
y = We may now equate
We wish to solve for eix. Differentiating once and multiplying the result by -i gives (-
i) [(- Adding eqs. 1 and 2 gives 2 where 1 =
Equating the real and imaginary parts immediately gives
With these values, eq. 3a yields the required expression eix
= cos x + i sin x Q.E.D. |
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