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Lecture14 - 1 3 5 7 9 x x x x x x x x x x x x x x x x = = =...

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Proof of Euler's Formula Use Taylor Series: 2 3 4 2 4 3 exp( ) 1 ... 1! 2! 3! 4! 1 ... ... 2! 4! 1! 3! cos( ) sin( ) i i i i i ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ = + - - + + = - + + + - + = + 2 3 ( ) (0) '(0) ''(0) '''(0) ... 1! 2! 3! x x x f x f f f f = + + + + exp( i ϕ ) = cos( ϕ ) + i sin( ϕ ) 2 3 4 2 4 6 8 3 5 7 9 exp( ) 1 ... 1! 2! 3! 4!
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Unformatted text preview: 1! 3! 5! 7! 9! x x x x x x x x x x x x x x x x = + + + + + = -+-+ + =-+-+ + If we substitute x = i into exp( x ) , then:...
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