1-2cosx

1-2cosx

Input
1 – 2 cos(x)
Alternate form
-e^(-i x) – e^(i x) + 1
Series expansion at x=0
-1 + x^2 – x^4/12 + O(x^6)
(Taylor series)
Derivative
d/dx(1 – 2 cos(x)) = 2 sin(x)
Indefinite integral
integral (1 – 2 cos(x)) dx = x – 2 sin(x) + constant
Global minima
min{1 – 2 cos(x)} = -1 at x = 2 pi n for integer n
Definite integral over a half-period
integral_0^pi (1 – 2 cos(x)) dx = pi~~3.14159
Definite integral over a period
integral_0^(2 pi) (1 – 2 cos(x)) dx = 2 pi~~6.28319
Definite integral mean square
integral_0^(2 pi) (1 – 2 cos(x))^2/(2 pi) dx = 3

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