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The TCHEBYCHEFF function
The function TCHEBYCHEFF(
n
) generates the Tchebycheff (or Chebyshev)
polynomial of the first kind, order n, defined as
T
n
(X) = cos(n
⋅
arccos(X)).
If the
integer n is negative (n < 0), the function TCHEBYCHEFF(
n
) generates the
Tchebycheff polynomial of the second kind, order n, defined as
T
n
(X) =
sin(n
⋅
arccos(X))/sin(arccos(X)).
Examples:
TCHEBYCHEFF(3) = 4*X^3-3*X
TCHEBYCHEFF(-3) = 4*X^2-1
Fractions
Fractions can be expanded and factored by using functions EXPAND and
FACTOR, from the ALG menu (‚×). For example:
EXPAND(‘(1+X)^3/((X-1)(X+3))’) = ‘(X^3+3*X^2+3*X+1)/(X^2+2*X-3)’
EXPAND(‘(X^2*(X+Y)/(2*X-X^2)^2’) = ‘(X+Y)/(X^2-4*X+4)’
EXPAND(‘X*(X+Y)/(X^2-1)’) = ‘(X^2+Y*X)/(X^2-1)’
EXPAND(‘4+2*(X-1)+3/((X-2)*(X+3))-5/X^2’) =
‘(2*X^5+4*X^4-10*X^3-14*X^2-5*X)/(X^4+X^3-6*X^2)’
FACTOR(‘(3*X^3-2*X^2)/(X^2-5*X+6)’) = ‘X^2*(3*X-2)/((X-2)*(X-3))’
FACTOR(‘(X^3-9*X)/(X^2-5*X+6)’ ) = ‘X*(X+3)/(X-2)’
FACTOR(‘(X^2-1)/(X^3*Y-Y)’) = ‘(X+1)/((X^2+X+1)*Y)’
The SIMP2 function
Functions SIMP2 and PROPFRAC are used to simplify a fraction and to
produce a proper fraction, respectively. Function SIMP2 takes as arguments
two numbers or polynomials, representing the numerator and denominator of
a rational fraction, and returns the simplified numerator and denominator. For
example: SIMP2(‘X^3-1’,’X^2-4*X+3’) = { ‘X^2+X+1’,‘X-3’}.
The PROPFRAC function
The function PROPFRAC converts a rational fraction into a “proper” fraction,
i.e., an integer part added to a fractional part, if such decomposition is
possible. For example:
