- Chebyshev cube root
In
mathematics , in the theory ofspecial function s, the Chebyshev cube root is a particularinverse function of theChebyshev polynomial of third degree. It is analogous to thecube root function is the inverse of the third power. It can be used to solve thecubic equation .Motivation
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 one must take the roots of complex numbers, which introduces complex numbers into a situation which does not, in fact, require them.One can get around these problems by using "Chebyshev cube roots" in place of ordinary cube roots. The polynomial 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 : The polynomial satisfies the third order addition relations:
and (as )
:
If t is represented as , then the polynomial equation can now be transformed into :The function is then defined as (a branch of) the
algebraic function of the third order which transforms into . It is given (inverting the relation to ) as:
if t lies in the real interval [−2, 2] . If t lies in the interval , then the Chebyshev root is given as
:
The branch is uniquely defined by the value at , which is , corresponding to the positive solution of .
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 and is the unique branch of the algebraic function with this property. In the domain it can be defined as :where 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 it can be defined as :, where Both D1 and D2 are simply-connected domains in on which the functions and are well-defined analytic functions (because the square roots exist as analytic functions on D1 resp. D2 and the argument functions and of the logarithm do not vanish on each domain). Both (partially overlapping) definitions of the Chebyshev cube root on the domains D1 and D2 can be put together to define the Chebyshev cube root unambiguously as an analytic function on the larger domain . In fact, if one approaches the critical value from either the left or the right on the real axis the value of each representative will tend to 2. Because is asimple root of the polynomial the branch of the Chebyshev root (defined as the algebraic function F(t)=2+G(t) satisfying :and exists locally as an analytic function in a (sufficiently small) neighbourhood "U" of (according to the (complex-analytic )inverse function theorem ) and takes real values if . Then it must coincide (on the intersection and ) with each of the two representatives (in terms of "arccos z" resp. "arccosh z") constructed above. Therefore the Chebyshev cube root is in fact an analytic function on the whole of the domain D.An alternative construction of the Chebyshev cube root in terms of hypergeometric functions is sketched in the next subsection.
Representation as hypergeometric function
The expression : can be transformed (using the difference-to-product
trigonometric identity for thecosine ) into the representation:For general complex parameter the functions and are two linearly independent solutions of the second-orderlinear differential equation :which can be obtained by differentiating the functional relations resp. twice with respect to "x". The differential equation :is equivalent (under the affine substitution ) to thehypergeometric differential equation :with parameters . 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 . It is given by the hypergeometric series (seehypergeometric function ) :Transforming back to the original differential equation one finds a solution of the differential equation :which is analytic at (unique up to scalar multiple). The representation:obtained above shows that the Chebyshev cube root is a solution of the differential equation:for which is analytic at . It must be proportional to the argument-shifted hypergeometric series and thus: where the last series converges if . All three roots of the equation are linear combinations of the two functions and By construction :the other two roots are :and :One derives the further relations :which can be verified independently by calculating the other two roots ( here ) given one root (here ) by means of the relation: solving thequadratic equation for "y", given "x".
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