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# Sqrt

 Sqrt[z]or gives the square root of z.
• Mathematical function, suitable for both symbolic and numerical manipulation.
• can be entered using Ctrl+2 z or .
• Sqrt[z] is converted to .
• Sqrt is not automatically converted to z.
• These conversions can be done using PowerExpand, but will typically be correct only for positive real arguments.
• For certain special arguments, Sqrt automatically evaluates to exact values.
• Sqrt can be evaluated to arbitrary numerical precision.
• Sqrt automatically threads over lists.
• z can also be used for input. The character is entered as Esc sqrt Esc or \[Sqrt].
Evaluate numerically to any precision:
Negative numbers have imaginary square roots:
Sqrt threads element-wise over lists:
is not automatically replaced by :
It can be simplified to if one assumes :
Enter using Ctrl+2:
 Out[1]=

Evaluate numerically to any precision:
 Out[1]=

Negative numbers have imaginary square roots:
 Out[1]=

Sqrt threads element-wise over lists:
 Out[1]=

 Out[1]=

is not automatically replaced by :
 Out[1]=
It can be simplified to if one assumes :
 Out[2]=

Enter using Ctrl+2:
 Out[1]=
 Scope   (2)
Exact roots are factored out when possible:
Find square roots of complex numbers:
 Applications   (2)
Roots of a quadratic polynomial:
Generate periodic continued fractions:
Reduce combinations of square roots:
Evaluate power series involving square roots:
Factor polynomials with square roots in coefficients:
Simplify handles expressions involving square roots:
There are many subtle issues in handling square roots for arbitrary complex arguments:
PowerExpand expands forms involving square roots:
It generically assumes that all variables are positive:
Take limits accounting for branch cuts:
Square root is discontinuous across its branch cut along the negative real axis:
Sqrt cannot automatically be reduced to x:
With x assumed positive, the simplification can be done:
Use PowerExpand to do the formal reduction:
Along the branch cut, these are not the same:
Approximation to GoldenRatio:
Riemann surface for square root:
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