Square Root

Author

Eric W. Weisstein
February 15, 2006

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/SquareRoot.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/SquareRoot.html.

©2006 Wolfram Research, Inc. except for portions noted otherwise

Definition

Solve[r^2==x,r]

Reduce[r^2==x,r]

Recurrence

v4.2

<<DiscreteMath`RSolve`

RSolve[{x[k+1]==(1/2)(x[k]+(n/(x[k]))),x[0]==1},x[k],k]

v5

RSolve[{x[k+1]==(1/2)(x[k]+(n/(x[k]))),x[0]==1},x[k],k]

v5.2

RSolve[{x[k+1]==(1/2)(x[k]+(n/(x[k]))),x[0]==1},x[k],k]

Plots

Real Axis

V5

Plot[Sqrt[x],{x,0,10},
AxesLabel->TraditionalForm/@{x,Sqrt[x]},
PlotStyle->Red]

V6

Plot[Sqrt[x],{x,0,10},AxesLabel->{x,Sqrt[x]},PlotStyle->Red]

Complex Plane

ComplexPlot3D[Sqrt[z],{z,-5,5}]//Timing

ComplexContourPlot[Sqrt[z],{z,-5,5}]//Timing

Complex Identities

(Sqrt[x+i y])==(((x^2)+(y^2))^(1/4)) (cos((1/2) (tan^(-1))(x,y))+i sin((1/2) (tan^(-1))(x,y)))

ComplexExpand[Sqrt[x+I y],TargetFunctions->{Re,Im}]//Simplify

(Sqrt[x+i y])==(1/2) (Sqrt[2]) (i (Sqrt[(Sqrt[(x^2)+(y^2)])-x]) sgn(y)+(Sqrt[x+(Sqrt[(x^2)+(y^2)])]))

FullSimplify[Subtract@@{(Sqrt[x+I y]),(1/2)(Sqrt[2])((Sqrt[(Sqrt[(x^2)+(y^2)])+x])+I Sign[y](Sqrt[(Sqrt[(x^2)+(y^2)])-x]))},
(x|y)∈Reals]

Table[Subtract@@{(Sqrt[x+I y]),(1/2)(Sqrt[2])((Sqrt[(Sqrt[(x^2)+(y^2)])+x])+I Sign[y](Sqrt[(Sqrt[(x^2)+(y^2)])-x]))}/.Thread[{x,y}->Table[Random[Real,5{-1,1}],{2}]],{100}]//Chop//Union