surd - Maple Help
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surd

non-principal root function

 Calling Sequence surd(x, n)

Parameters

 x - any algebraic expression n - any algebraic expression, understood to be an integer

Description

 • For a complex number x and integer n, surd(x, n) computes the nth root of x whose (complex) argument is closest to that of x.  Ties are broken in such a way that the function x -> surd(x,n) is continuous in a counter-clockwise manner onto its branch cuts (that is, continuous in the direction of increasing complex argument).
 • In particular, if n is odd then if x>=0 then surd(x,n) = x^(1/n) and if x<0 then surd(x,n) = -(-x)^(1/n).

Examples Using surd

 > surd(-1, 3);
 ${-1}$ (1)
 > surd( 8, 3);
 ${2}$ (2)
 > surd(-8, 3);
 ${-2}$ (3)
 > surd(-1, 2);
 ${I}$ (4)
 > surd(1+2*I,3);
 ${\left({1}{+}{2}{}{I}\right)}^{{1}}{{3}}}$ (5)
 > surd( x, n);
 $\sqrt[{n}]{{x}}$ (6)
 > convert((6), power);
 ${{x}}^{\frac{{1}}{{n}}}$ (7)
 Maple simplifies the expression before converting. Constants will still be written with fractional exponents.
 > convert((9*x)^(1/3), surd);
 ${{3}}^{{2}}{{3}}}{}\sqrt[{3}]{{x}}$ (8)
 > convert(3^(1/3)*x^(1/2)*a^b+f((-2)^(1/5)*x^(1/n)), surd);
 ${{3}}^{{1}}{{3}}}{}\sqrt{{x}}{}{{a}}^{{b}}{+}{f}{}\left({-}{{2}}^{{1}}{{5}}}{}{{x}}^{\frac{{1}}{{n}}}\right)$ (9)
 > int(surd(x^2,3)*(3*x^3-2*x^2+x-1), x=-2..2);
 ${-}\frac{{612}{}{{2}}^{{2}}{{3}}}}{{55}}$ (10)
 Note the differences among the outputs of the surd, ^, and root commands.
 > (8)^(1/3); root(8, 3); surd(8, 3);
 ${{8}}^{{1}}{{3}}}$
 ${2}$
 ${2}$ (11)
 > (8.0)^(1/3); root(8.0, 3); surd(8.0, 3);
 ${2.000000000}$
 ${2.000000000}$
 ${2.000000000}$ (12)
 > (-8)^(1/3); root(-8, 3); surd(-8, 3);
 ${\left({-8}\right)}^{{1}}{{3}}}$
 ${2}{}{\left({-1}\right)}^{{1}}{{3}}}$
 ${-2}$ (13)
 > (-8.0)^(1/3); root(-8.0, 3); surd(-8.0, 3);
 ${1.000000000}{+}{1.732050807}{}{I}$
 ${1.000000000}{+}{1.732050807}{}{I}$
 ${-2.000000000}$ (14)

 See Also