Note: Use only in the MuPAD Notebook Interface. This functionality does not run in MATLAB. |
The Taylor series expansion is the most common way to approximate
an expression by a polynomial. However, not all expressions can be
represented by Taylor series. For example, you cannot compute a Taylor
series expansion for the following expression around x =
2
:
taylor(1/(x^3 - 8), x = 2)
Error: Cannot compute a Taylor expansion of '1/(x^3 - 8)'. Try 'series' for a more general expansion. [taylor]
If a Taylor series expansion does not exist for your expression,
try to compute other power series. MuPAD^{®} provides the function series
for computing
power series. When you call series
, MuPAD tries
to compute the following power series:
Taylor series
Laurent series
Puiseux series. For more information, see Series::Puiseux
.
Generalized series expansion of f
around x = x_{0}.
For more information, see Series::gseries
.
As soon as series
computes
any type of power series, it does not continue computing other types
of series, but stops and returns the result. For example, for this
expression it returns a Laurent series:
S := series(1/(x^3 - 8), x = 2); testtype(S, Type::Series(Laurent))
When computing series expansions, MuPAD returns only those
results that are valid for all complex values of the expansion variable
in some neighborhood of the expansion point. If you need the expansion
to be valid only for real numbers, use the option Real
.
For example, when you compute the series expansion of the following
expression for complex numbers, series
returns:
series(sign(x^2*sin(x)), x = 0)
When you compute the series expansion for real numbers, series
returns a simplified
result:
series(sign(x^2*sin(x)), x = 0, Real)
Along the real axis, compute series expansions for this expression
when x
approaches the value 0 from the left and
from the right sides:
series(sign(x^2*sin(x)), x = 0, Left); series(sign(x^2*sin(x)), x = 0, Right)