Calculate arccos

Calculate arccos

Input
cos^(-1)(x)
Root
x = 1
Series expansion at x=-1
pi – sqrt(2) sqrt(x + 1) – (x + 1)^(3/2)/(6 sqrt(2)) – (3 (x + 1)^(5/2))/(80 sqrt(2)) – (5 (x + 1)^(7/2))/(448 sqrt(2)) – (35 (x + 1)^(9/2))/(9216 sqrt(2)) + O((x + 1)^5)
(Puiseux series)
Series expansion at x=0
pi/2 – x – x^3/6 – (3 x^5)/40 + O(x^6)
(Taylor series)
Series expansion at x=1
(-1)^(ceiling((arg(x – 1))/(2 pi))) (i sqrt(2) sqrt(x – 1) – (i (x – 1)^(3/2))/(6 sqrt(2)) + (3 i (x – 1)^(5/2))/(80 sqrt(2)) – (5 i (x – 1)^(7/2))/(448 sqrt(2)) + (35 i (x – 1)^(9/2))/(9216 sqrt(2)) + O((x – 1)^(11/2)))
Series expansion at x=∞
1/2 i (2 log(x) + log(4)) – i/(4 x^2) – (3 i)/(32 x^4) + O((1/x)^6)
(generalized Puiseux series)
Derivative
d/dx(cos^(-1)(x)) = -1/sqrt(1 – x^2)
Indefinite integral
integral cos^(-1)(x) dx = x cos^(-1)(x) – sqrt(1 – x^2) + constant
Global maximum
max{cos^(-1)(x)} = pi at x = -1
Global minimum
min{cos^(-1)(x)} = 0 at x = 1

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