Cubic equation

In mathematics, a cubic equation is a polynomial equation in which the highest occurring power of the unknown is the third power. An example is the equation

Chebyshev radicals

The cube root function is in some respects not a well-behaved function, or one convenient for the purposes of finding the roots of a cubic equation. While cube roots are well-known and traditional, it is possible to use other algebraic functions to determine the roots, and avoid some of the problems of cube roots. The cube root function has a branch singularity at zero, as a result of which the real cube root function does not extend nicely to a complex cube root function. Moreover, when using cube roots to find the roots of a polynomial with three real roots we must take the roots of complex numbers, which introduces complex numbers into a situation which does not, in fact, require them.

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We can get around these problems by using Chebyshev cube roots in place of ordinary cube roots. The polynomial C_3 = x^3 - 3x is the third Chebyshev polynomial normalized to obtain a monic polynomial. The Chebyshev cube root is then defined as a (suitably chosen) root (depending on t) of the polynomial equation

Related Topics:
Chebyshev - Chebyshev polynomial

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: x^3 - 3x = t .

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The polynomial C_3(x) satisfies the third order addition relations

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:, 2,cos(3x)= C_3(2 cos x)

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and (as , 2cos(ix)=2cosh(x) )

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: 2,cosh(3x)=C_3(2cosh x) .

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If t is represented as t=2cos y , then the polynomial equation x^3 - 3x = t can now be transformed into

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: t=2,cos y=C_3(2cos (y/3)) .

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The function C_{1over3}(t) is then defined as (a branch of) the algebraic function of the third order which transforms 2cos(x) into 2cos(x/3) . It is given (inverting the relation t=2cos(x) to x=rccos(t/2)) as

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:C_{1over3}(t) = 2 ,operatorname{cos}left(operatorname{arccos}left({tover2} ight)/3 ight) ,

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if t lies in the real interval . If t lies in the interval , then the Chebyshev root is given as

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:C_{1over3}(t) = 2 ,operatorname{cosh}left(operatorname{arccosh}left({tover2} ight)/3 ight) .

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The branch is uniquely defined by the value at t=0, which is 2, operatorname{cos}left(operatorname{arccos}(0)/3 ight)=2, operatorname{cos}(pi/6)=sqrt{3}, corresponding to the positive solution of x^3-3x=x(x^2-3)=0 .

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This procedure is precisely analogous to the definition of the cube root in terms of logarithms and exponentials, with arccosh(x/2) resp. arccos(x/2) in the place of ln(x), and 2cosh(x) resp. 2cos(x) in the place of exp(x). The Chebyshev cube root can be constructed as an analytic function on the cut plane mathbb{C}setminus and is the unique branch of the algebraic function

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C_{1over3}(t) with this property. In the domain D_1 := {z in mathbb{C}, | , Re{z}>2} it can be defined as

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: C_{1over3}(t)= 2,operatorname{cosh}left(operatorname{arccosh}left({tover2} ight)/3 ight)

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where operatorname{arccosh}(z/2)=ln{{z+sqrt{z^2-4}}over 2},

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using the branch of the logarithm which is real on the positive real line and the branch of the square root which is positive on the real axis. On the domain D_2 :=mathbb{C}setminus{{ cup }}

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it can be defined as

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: C_{1over3}(t)= 2 ,operatorname{cos}left(operatorname{arccos}left({tover2} ight)/3 ight) ,

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where operatorname{arccos}(z/2)={pi over 2}+iln{{iz+sqrt{4-z^2}}over 2} .

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Both D1 and D2 are simply-connected domains in mathbb{C} on which the functions operatorname{arccos}(z) and operatorname{arccosh}(z)

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are well-defined analytic functions (because the square roots sqrt{pm (z^2-4)} exist as analytic functions on D1 resp. D2 and the argument functions

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{z+sqrt{z^2-4}}over 2 and {iz+sqrt{4-z^2}}over 2 of the logarithm do not vanish on each domain). Both (partially overlapping) definitions of the Chebyshev cube root on

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the domains D1 and D2 can be put together to define the Chebyshev cube root unambiguously as an analytic function on the larger domain D= mathbb{C}setminus . In fact, if one approaches the critical value t=2 from either the left or the right on the real axis the value of each representative will tend to 2. Because x=2 is a simple root of the polynomial x^3-3x-2 the branch of the Chebyshev root (defined as the algebraic function F(t)=2+G(t) satisfying

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:, F(t)^3-3F(t)-2=9G(t)+6G(t)^2+G(t)^3=0

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and F(2)=2 exists locally as an

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analytic function in a (sufficiently small) neighbourhood U of t=2 (according to the (complex-analytic ) inverse function theorem) and takes real values if t=2pm epsilon, ,epsilon >0 . Then it must coincide (on the intersection U cap D_1 and U cap D_2 ) with each of the two representatives (in terms of arccos z resp.

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arccosh z) constructed above. Therefore the Chebyshev cube root is in fact an analytic function on the whole of the domain D.

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An alternative construction of the Chebyshev cube root in terms of

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hypergeometric functions is sketched in the next subsection.

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The Chebyshev cube root as a hypergeometric function

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The expression

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: 2 ,operatorname{cos}left(operatorname{arccos}left({tover2} ight)/3 ight)=2,operatorname{cos}left({piover 6}-operatorname{arcsin}left({tover2} ight)/3 ight)

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can be transformed (using the difference-to-product trigonometric identity for the cosine) into the representation

Related Topics:
Trigonometric identity - Cosine

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: sqrt{3} ,operatorname{cos}left(operatorname{arcsin}left({tover2} ight)/3 ight)+ operatorname{sin }left(operatorname{arcsin}left({tover2} ight)/3 ight) .

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For general complex parameter lambda e 0 the functions 2,operatorname{cos},(lambda, operatorname{arcsin}(x/2)) and 2,operatorname{sin},(lambda ,operatorname{arcsin}(x/2)) are two linearly independent solutions of the second-order linear differential equation

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:, (4-x^2)y-xy'+lambda^2 y=0

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which can be obtained by differentiating the functional relations , f(2sin x)=2sin(lambda x) resp. , f(2sin x)=2cos(lambda x) twice with respect to x. The differential equation

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:, (4-x^2)y-xy'+lambda^2 y=0

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is equivalent (under the affine substitution x mapsto (2-4x) ) to the hypergeometric differential equation

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: x(1-x) ,y+{{1-2x}over 2},y'+lambda^2 y=0

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with parameters c={1over 2},, a=lambda,, b=-lambda . According to the general theory of the hypergeometric equation it has (unless c is zero or a negative integer) a uniquely defined solution g which is analytic in x=0 and satisfies ,g(0)=1 . It is given by the hypergeometric series (see hypergeometric function)

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: ,F(a,b,c;z):=,_2F_1 (a,b;c;z) = sum_{n=0}^infty

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rac{(a)_n(b)_n}{(c)_n} , rac {z^n} {n!} .

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Transforming back to the original differential equation one finds a solution g(x)=F(lambda,-lambda,{1over2} ;{{2-x}over 4}) of the differential equation

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:, (4-x^2)y-xy'+lambda^2 y=0

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which is analytic at x=2 (unique up to scalar multiple). The representation

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:C_{{1over 3}}(t)= sqrt{3} ,operatorname{cos}left(operatorname{arcsin}left({tover2} ight)/3 ight)+ operatorname{sin}left(operatorname{arcsin}left({tover2} ight)/3 ight)

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obtained above shows that the Chebyshev cube root is a solution of the differential equation

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:, (4-x^2)y-xy'+lambda^2 y=0

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for lambda={1over 3} which is analytic at x=2 . It must be proportional to the argument-shifted hypergeometric series and thus

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:C_{{1over 3}}(t)=2F({1over 3},-{1over 3},{1over2} ;{{2-t}over 4}) = sum_{n=0}^infty rac{2}{1-3n} {3n choose n}left(rac{2-t}{27} ight)^n ,

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where the last series converges if |t-2|

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f_1(t)=2operatorname{sin},left({1over 3} operatorname{arcsin}{tover2} ight) and f_2(t)=2operatorname{cos},left({1over 3} operatorname{arcsin}{tover2} ight) .

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By construction

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: r_1=C_{{1over 3}}(t)=sqrt{3}

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,operatorname{cos}left(operatorname{arcsin}left({tover2} ight)/3 ight)+ operatorname{sin},left(operatorname{arcsin}left({tover2} ight)/3 ight), ,

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the other two roots are

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: r_2=-C_{{1over 3}}(-t)=-sqrt{3}

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,operatorname{cos}left(operatorname{arcsin}left({tover2} ight)/3 ight)+ operatorname{sin},left(operatorname{arcsin}left({tover2} ight)/3 ight)

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and

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: r_3=-r_1-r_2= -2 ,operatorname{sin}left(operatorname{arcsin}left({tover2} ight)/3 ight) .

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One derives the further relations

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: r_1={sqrt{3}over 2}sqrt{4-r_3^2}-{r_3over 2} , qquad r_2=-{sqrt{3}over 2} sqrt{4-r_3^2}- {r_3over 2}

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which can be verified independently by calculating the other two roots ( here r_1, r_2 ) given one root (here r_3 ) by means of the relation

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: t=x^3-3x=y^3-3y Longrightarrow (y-x)(y^2+xy+x^2-3)=0,

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solving the quadratic equation , y^2+xy+(x^2-3)=0 for y, given x.

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Solving a general cubic equation using Chebyshev cube roots

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If we have a cubic equation which is already in depressed form, we may write it as ,x^3 - 3px - q = 0. Substituting x = sqrt{p} z we obtain z^3 - 3z - p^{-rac{3}{2}}q = 0 or equivalently

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:z^3 - 3z = p^{-rac{3}{2}}q .

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From this we obtain solutions to our original equation in terms of the Chebyshev cube root as

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:r_1 = sqrt{p},C_{1over3}(p^{-rac{3}{2}}q),,

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:r_2 = -sqrt{p},C_{1over3}(-p^{-rac{3}{2}}q),,

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:r_3 = -r_1 - r_2 .

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If now we start from a general equation

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:x^3 + ax^2 + bx +c = 0 qquad (1)

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and reduce it to the depressed form under the substitution x = t − a/3, we have

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, p = (a^2-3b)/9 and , q = -(2a^3-9ab+27c)/27, leading to

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:t_{a;b;c} = p^{-rac{3}{2}}q = -rac{2a^3-9ab+27c}{(a^2-3b)^{3/2}}.

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This gives us the solutions to (1) as

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:r_1 = sqrt{p},C_{1over3}(t_{a;b;c})-{aover 3} ,,

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:r_2 = -sqrt{p},C_{1over3}(-t_{a;b;c})-{aover 3},,

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:r_3 = -r_1 - r_2 - {aover 3} .

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The case of a real equation

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Suppose the coefficients of (1) are real. If s is the quantity q/r from the section on real roots, then s = t2; hence 0 < s < 4 is equivalent to −2 < t < 2, and in this case we have a polynomial with three distinct real roots, expressed in terms of a real function of a real variable, quite unlike the situation when using cube roots. If s > 4 then either t > 2 and C_{1over3}(t) is the sole real root, or t < −2 and -C_{1over3}(-t) is the sole real root.

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If s < 0 then the reduction to Chebyshev polynomial form has given a t which is a pure imaginary number; in this case iC_{1over3}(-it)-iC_{1over3}(it) is the sole real root. We are now evaluating a real root by means of a function of a purely imaginary argument; however we can avoid this by using the function

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:S_{1over3}(t) = iC_{1over3}(-it)-iC_{1over3}(it) = 2 operatorname{sinh}left(operatorname{arcsinh}left({tover2} ight)/3 ight),,

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which is a real function of a real variable with no singularities along the real axis. If a polynomial can be reduced to the form x^3 + 3x -t with real t, this is a convenient way to solve for its roots.

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