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Absolute Value of Complex Numbers
Plot
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 representation
abs(x) = x/sgn(x) for x!=0
abs(x) = sqrt(x x^*)
abs(x) = sqrt(-x^2 + 2 x Re(x))
Series representation
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 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 abs(x) = ( sum_(k=0)^∞ ((-1)^k H_(2 k)(x) (-1/2)_k)/((2 k)!))/sqrt(π) for (x element R and -1