sqrt -- the square root
function
Introductionsqrt(z) represents the square root of
z.
Call(s)sqrt(z)
Parametersz |
- | an arithmetical expression |
Returnsan arithmetical expression.
z
Side
EffectsWhen called with a floating point argument, the function is
sensitive to the environment variable DIGITS which determines the numerical
working precision.
Related
Functions
Detailssqrt jump when crossing this cut. Cf.
example 2.sqrt(z) coincides with
z^(1/2) = _power(z,1/2). However,
sqrt provides more simplifications than
_power. Cf. example 5.
Example
1We demonstrate some calls with exact and symbolic input data:
>> sqrt(2), sqrt(4), sqrt(36*7), sqrt(127)
1/2 1/2 1/2
2 , 2, 6 7 , 127
>> sqrt(1/4), sqrt(1/2), sqrt(3/4), sqrt(25/36/7), sqrt(4/127)
1/2 1/2 1/2 1/2
2 3 5 7 2 127
1/2, ----, ----, ------, --------
2 2 42 127
>> sqrt(-4), sqrt(-1/2), sqrt(1 + I)
1/2 1/2
2 I, 1/2 I 2 , (1 + I)
>> sqrt(x), sqrt(4*x^(4/7)), sqrt(4*x/3), sqrt(4*(x + I))
1/2 2/7 / 4 x \1/2 1/2
x , 2 x , | --- | , (4 x + 4 I)
\ 3 /
Example
2Floating point values are computed for floating point arguments:
>> sqrt(1234.5), sqrt(-1234.5), sqrt(-2.0 + 3.0*I)
35.13545218, 35.13545218 I, 0.8959774761 + 1.674149228 I
A jump occurs when crossing the negative real semi axis:
>> sqrt(-4.0), sqrt(-4.0 + I/10^100), sqrt(-4.0 - I/10^100)
2.0 I, 2.5e-101 + 2.0 I, 2.5e-101 - 2.0 I
Example
3The square root of symbolic products involving positive integer factors is simplified:
>> sqrt(20*x*y*z)
1/2 1/2
2 (x y z) 5
Example
4Square roots of squares are not simplified, unless the argument is real and its sign is known:
>> sqrt(x^2*y^4)
2 4 1/2
(x y )
>> assume(x > 0): sqrt(x^2*y^4)
4 1/2
x (y )
>> assume(x < 0): sqrt(x^2*y^4)
4 1/2
- x (y )
Example
5sqrt provides more simplifications than the
_power function:
>> sqrt(4*x), (4*x)^(1/2) = _power(4*x, 1/2)
1/2 1/2 1/2
2 x , (4 x) = (4 x)