{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

notes Eulers Formula

# Thomas' Calculus: Early Transcendentals

This preview shows pages 1–2. Sign up to view the full content.

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 = n =0 z n n ! . Since | z n | = | z | n , this series converges absolutely: 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 = 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! + · · · + i θ - θ 3 3! + θ 5 5! - θ 7 7! + · · · = 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:

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online