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 |