Sqrt(1+x^2)

Sqrt(1+x^2)

Input
sqrt(1 + x^2)
Series expansion at x=0
1 + x^2/2 – x^4/8 + O(x^5)
(Taylor series)
Series expansion at x=-i
sqrt(2 – 2 i x) + (i sqrt(1 – i x) (x + i))/(2 sqrt(2)) + (sqrt(1 – i x) (x + i)^2)/(16 sqrt(2)) – (i sqrt(1 – i x) (x + i)^3)/(64 sqrt(2)) – (5 sqrt(1 – i x) (x + i)^4)/(1024 sqrt(2)) + O((x + i)^5)
(generalized Puiseux series)
Series expansion at x=i
sqrt(2 + 2 i x) – (i sqrt(1 + i x) (x – i))/(2 sqrt(2)) + (sqrt(1 + i x) (x – i)^2)/(16 sqrt(2)) + (i sqrt(1 + i x) (x – i)^3)/(64 sqrt(2)) – (5 sqrt(1 + i x) (x – i)^4)/(1024 sqrt(2)) + O((x – i)^5)
(generalized Puiseux series)
Series expansion at x=∞
x + 1/(2 x) – 1/(8 x^3) + O((1/x)^4)
(Laurent series)
Derivative
d/dx(sqrt(1 + x^2)) = x/sqrt(x^2 + 1)
Indefinite integral
integral sqrt(1 + x^2) dx = 1/2 (sqrt(x^2 + 1) x + sinh^(-1)(x)) + constant
Global minimum
min{sqrt(1 + x^2)} = 1 at x = 0

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