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Absolute Value Function

Plots

Plots

Plots

Alternate form assuming x>0

x

Alternate form assuming x is real

sqrt(x^2)

Root

x = 0

Properties as a real function

R (all real numbers)

{y element R : y>=0} (all non-negative real numbers)

even

Derivative

d/dx(abs(x)) = x/abs(x)
(assuming a function from reals to reals)

Indefinite integral assuming all variables are real

integral abs(x) dx = (x^2 sgn(x))/2 + constant

Global minimum

min{abs(x)} = 0 at x = 0

Alternative representations

abs(x) = x/sgn(x) for x!=0

abs(x) = sqrt(x x^*)

abs(x) = sqrt(-x^2 + 2 x Re(x))

Series representations

abs(x) = 2/π - (4 sum_(k=1)^∞ ((-1)^k T_(2 k)(x))/(-1 + 4 k^2))/π for (x element R and -1<x<1)

abs(x) = sum_(k=0)^∞ ((-1)^k (1/2 + 2 k) P_(2 k)(x) (-1/2)_k)/((1 + k)!) for (x element R and -1<x<1)

abs(x) = ( sum_(k=0)^∞ ((-1)^k H_(2 k)(x) (-1/2)_k)/((2 k)!))/sqrt(π) for (x element R and -1<x<1)

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