notes Eulers Formula

# Thomas' Calculus: Early Transcendentals

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Euler’s Formula Where does Euler’s formula e = cos θ + i sin θ come from? How do we even deﬁne , for example, e i ? We can’t multiple e by itself the square root of minus one times. The answer is to use the Taylor series for the exponential function. For any complex number z we deﬁne e z by e z = X n =0 z n n ! . Since | z n | = | z | n , this series converges absolutely: X n =0 | z | n n ! is a real series that we already know converges. If we multiply the series for e z term-by-term with the series for e w , collect terms of the same total degree, and use a certain famous theorem of algebra, we ﬁnd that the law of exponents e z + w = e z · e w continues to hold for complex numbers. Now for Euler’s formula: e = X n =0 ( ) n n ! = 1 + - θ 2 2! - i θ 3 3! + θ 4 4! + i θ 5 5! - θ 6 6! - i θ 7 7! + · · · = ± 1 - θ 2 2! + θ 4 4! - θ 6 6! + · · · ² + i ± θ - θ 3 3! + θ 5 5! - θ 7 7! + · · · ² = cos θ + i sin θ. The special case θ = 2 π gives e 2 πi = 1 . This celebrated formula links together three numbers of totally diﬀerent origins:

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## This note was uploaded on 02/12/2008 for the course MATH 104 taught by Professor Nelson during the Fall '07 term at Princeton.

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notes Eulers Formula - Euler's Formula Where does Euler's...

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