This function computes the Chebyshev series expansion of f, with respect to the variable x on the interval $a..b$, valid to accuracy eps.

If the second argument is simply a name x then the equation $x=\mathrm{-1}..1$ is implied.

If the second argument is a range then the first argument is assumed to be a Maple operator and the result will be returned as an operator. Otherwise, the first argument is assumed to be an expression and the result will be returned as an expression.

If the third argument eps is present then it specifies the desired accuracy;  otherwise, the value used is $\mathrm{eps}={10}^{-\mathrm{Digits}}$. It is an error to specify eps less than 10^(-Digits).

The expression or operator f must evaluate to a numerical value when x takes on a numerical value.  Moreover, it must represent a function which is analytic in a region surrounding the interval $a..b$.

The resulting series is expressed in terms of the Chebyshev polynomials $T\left(k\,x\right),...$ with floating-point series coefficients. If 'ser' is the Chebyshev series then conversion to ordinary polynomial form can be accomplished via eval(ser, T=orthopoly[T]).

The series computed is the ``infinite'' Chebyshev series, truncated by dropping all terms with coefficients smaller than eps multiplied by the largest coefficient.

Note:  The name T used in representing the Chebyshev polynomials is a global name, so the user must ensure that this name has no previous value.

The command with(numapprox,chebyshev) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{numapprox}\right)\:$

$\mathrm{Digits}≔5\:$

$\mathrm{chebyshev}\left(\cos \left(x\right)\,x\right)$

$0.76520T\left(0\,x\right)-0.22981T\left(2\,x\right)+0.0049533T\left(4\,x\right)-0.000041877T\left(6\,x\right)$

$\mathrm{chebyshev}\left(\exp \left(x\right)\,x=0..1\,0.001\right)$

$1.7534T\left(0\,2x-1\right)+0.85039T\left(1\,2x-1\right)+0.10521T\left(2\,2x-1\right)+0.0087221T\left(3\,2x-1\right)$

$\mathrm{chebyshev}\left(\exp \,0..1\,0.001\right)$

$x\mapsto 1.7534\cdot T\left(0\,2\cdot x-1\right)+0.85039\cdot T\left(1\,2\cdot x-1\right)+0.10521\cdot T\left(2\,2\cdot x-1\right)+0.0087222\cdot T\left(3\,2\cdot x-1\right)$

$\mathrm{chebyshev}\left(\sin +\cos \,-1..1\right)$

$x\mapsto 0.76520\cdot T\left(0\,x\right)+0.88010\cdot T\left(1\,x\right)-0.22981\cdot T\left(2\,x\right)-0.039127\cdot T\left(3\,x\right)+0.0049533\cdot T\left(4\,x\right)+0.00049952\cdot T\left(5\,x\right)-0.000041877\cdot T\left(6\,x\right)$