Sqrt(1-x^2)

Sqrt(1-x^2)

Input
sqrt(1 – x^2)
Alternate form
sqrt(1 – x) sqrt(x + 1)
Series expansion at x=-1
sqrt(2) sqrt(x + 1) – (x + 1)^(3/2)/(2 sqrt(2)) – (x + 1)^(5/2)/(16 sqrt(2)) + O((x + 1)^(7/2))
(Puiseux series)
Series expansion at x=0
1 – x^2/2 – x^4/8 + O(x^5)
(Taylor series)
Series expansion at x=1
i sqrt(2) sqrt(x – 1) + (i (x – 1)^(3/2))/(2 sqrt(2)) – (i (x – 1)^(5/2))/(16 sqrt(2)) + O((x – 1)^(7/2))
(Puiseux series)
Series expansion at x=∞
i x – i/(2 x) – i/(8 x^3) + O((1/x)^4)
(Laurent series)
Derivative
d/dx(sqrt(1 – x^2)) = -x/sqrt(1 – x^2)
Indefinite integral
integral sqrt(1 – x^2) dx = 1/2 (sqrt(1 – x^2) x + sin^(-1)(x)) + constant
Global maximum
max{sqrt(1 – x^2)} = 1 at x = 0
Definite integral
integral_(-1)^1 sqrt(1 – x^2) dx = pi/2~~1.5708

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