1/sqrt(1-x^2)

1/sqrt(1-x^2)

Input
1/sqrt(1 – x^2)
Alternate form
1/(sqrt(1 – x) sqrt(x + 1))
Roots
(no roots exist)
Series expansion at x=-1
1/(sqrt(2) sqrt(x + 1)) + sqrt(x + 1)/(4 sqrt(2)) + (3 (x + 1)^(3/2))/(32 sqrt(2)) + (5 (x + 1)^(5/2))/(128 sqrt(2)) + O((x + 1)^(7/2))
(Puiseux series)
Series expansion at x=0
1 + x^2/2 + (3 x^4)/8 + O(x^5)
(Taylor series)
Series expansion at x=1
1/sqrt(2 – 2 x) – (x – 1)/(4 sqrt(2 – 2 x)) + (3 (x – 1)^2)/(32 sqrt(2 – 2 x)) – (5 (x – 1)^3)/(128 sqrt(2 – 2 x)) + (35 (x – 1)^4)/(2048 sqrt(2 – 2 x)) + O((x – 1)^5)
(generalized Puiseux series)
Series expansion at x=∞
1/sqrt(-x^2) + 1/(2 sqrt(-x^2) x^2) + 3/(8 sqrt(-x^2) x^4) + O((1/x)^5)
(generalized Puiseux series)
Derivative
d/dx(1/sqrt(1 – x^2)) = x/(1 – x^2)^(3/2)
Indefinite integral
integral 1/sqrt(1 – x^2) dx = sin^(-1)(x) + constant
Global minimum
min{1/sqrt(1 – x^2)} = 1 at x = 0
Limit
lim_(x-> ± infinity) 1/sqrt(1 – x^2) = 0

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