Series
Usage Message:
Series[f, {x, x0, n}] generates a power series expansion for f about
the point x = x0 to order (x - x0)^n. Series[f, {x, x0, nx}, {y, y0, ny}]
successively finds series expansions with respect to y, then x.
Attributes[Series] = {Protected, ReadProtected}
Options:
Analytic ->
True
Notes:
Results from Series are normally returned in the form of a
SeriesData expression.
Series Other Than Power Series
The Series function and the underlying SeriesData
expressions are only designed for representing power series, including series
with negative or fractional exponents. Essential singularities, logarithmic
singularities, and expansions other than power series expansions are handled
only in a few special cases.
Expansion Order
The order specification n in
Series[f, {x, x0, n}] specifies the
order of expansion that will be used in intermediate calculations, and
will not necessarily give the order that will appear in the result.
For example:
In[1]:= Series[1/(1 - Cos[x^25]), {x, 0, 25}]
2 -43
Out[1]= --- + O[x]
50
x
You can normally get the necessary order in the result by increasing the
order specified in the input. For example:
In[2]:= Series[1/(1 - Cos[x^25]), {x, 0, 50}]
50
2 1 x 51
Out[2]= --- + - + --- + O[x]
50 6 120
x
Multivariate Series
Multivariate series expansions are handled as iterated single variable
expansions. Results of multivariate expansions are represented as
SeriesData expressions in which the series
coefficients
are also SeriesData expressions.
In many applications, multivariate expansions are actually expansions
in terms of single parameter that characterizes the common scale of
the variables.
For example, here is an expansion to second order in two variables
x and y. The result contains terms
such as x y^2 and x^2 y^2 for which
the combined order is greater than two:
In[1]:= Series[2 Exp[x + y], {x, 0, 2}, {y, 0, 2}]
2 3 2 3
Out[1]= 2 + 2 y + y + O[y] + (2 + 2 y + y + O[y] ) x +
2
y 3 2 3
> (1 + y + -- + O[y] ) x + O[x]
2
There are several ways to get a result that only includes terms for
which the combined order in x and y
is second order. One method is to explicitly insert a parameter
that describes the common scale of the two variables, and do the
expansion with respect to that parameter:
In[2]:= Series[2 Exp[t x + t y], {t, 0, 2}]
2 2 3
Out[2]= 2 + 2 (x + y) t + (x + y) t + O[t]
The nature of this result can be seen more explicitly by applying
Normal and Expand to the result and
replacing the parameter t by 1:
In[3]:= Expand[Normal[
Series[2 Exp[t x + t y], {t, 0, 2}]]] /. t -> 1
2 2
Out[3]= 2 + 2 x + x + 2 y + 2 x y + y
Known Bugs
Multivariate Series (Version 3 and eariler only)
There is an error in the treatment of multivariate expansions that can
cause the order of the expansion to be higher or lower than intended.
Here is a typical example in which the result, although mathematically
correct, gives the expansion in x only up to order
1, instead of the requested order 3.
In[1]:= Series[Exp[x + y], {x, 0, 3}, {y, 0, 1}]
2 2 2
Out[1]= 1 + y + O[y] + (1 + y + O[y] ) x + O[x]
You can correct this error by adding a rule for Series
which changes the way that iterated series expansions are computed.
In[2]:= Unprotect[Series]
Out[2]= {Series}
In[3]:= Series[p_SeriesData, q__] :=
(Print["using multivariate Series rule"];
MapAt[Series[#, q] &, p, {3}])
In[4]:= Series[Exp[x + y], {x, 0, 3}, {y, 0, 1}]
using multivariate Series rule
2 2 1 y 2 2
Out[4]= 1 + y + O[y] + (1 + y + O[y] ) x + (- + - + O[y] ) x +
2 2
1 y 2 3 4
> (- + - + O[y] ) x + O[x]
6 6
If you decide to use this rule, you may wish to remove the Print
expression after verifying that the rule has been installed correctly.
Note also that this rule does not do any error checking, and so may have
unintended consequences under non-standard conditions.
You can have this workaround automatically loaded by placing
http://support.wolfram.com/mathematica/kernel/Symbols/System/Series.m
at the location given by
In[7]:= ToFileName[{$TopDirectory,"AddOns", "Autoload",
"Series", "Kernel"}, "init.m"]
Additional Online Documentation:
Mathematica 3.0
http://documents.wolfram.com/v3/RefGuide/Series.html
Mathematica 4.0
http://documents.wolfram.com/v4/RefGuide/Series.html
Questions or comments? Send email to support@wolfram.com.
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