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This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
View current documentation (Version 11.2)
Mathematica
>
BUILTIN MATHEMATICA SYMBOL
Symbolic Mathematics: Basic Operations
Power Series
Making Power Series Expansions
Operations on Power Series
The Representation of Power Series
Converting Power Series to Normal Expressions
Tutorials »

SeriesCoefficient
InverseSeries
ComposeSeries
Limit
Normal
InverseZTransform
RSolve
O
SeriesData
PadeApproximant
FourierSeries
See Also »

Analytic Number Theory
Calculus
Manipulating Equations
Multiplicative Number Theory
Series Expansions
New in 6.0: Mathematics & Algorithms
More About »
Series
Series
generates a power series expansion for
f
about the point
to order
.
Series
successively finds series expansions with respect to
x
, then
y
, etc.
MORE INFORMATION
Series
can construct standard Taylor series, as well as certain expansions involving negative powers, fractional powers, and logarithms.
Series
detects certain essential singularities.
On
makes
Series
generate a message in this case.
Series
can expand about the point
.
Series
constructs Taylor series for any function
f
according to the formula
.
Series
effectively evaluates partial derivatives using
D
. It assumes that different variables are independent.
The result of
Series
is usually a
SeriesData
object, which you can manipulate with other functions.
Normal
[
series
]
truncates a power series and converts it to a normal expression.
SeriesCoefficient
finds the coefficient of the
n
order term.
EXAMPLES
CLOSE ALL
Basic Examples
(3)
Power series for the exponential function around
:
Convert to a normal expression:
Power series of an arbitrary function around
:
In any operation on series, only appropriate terms are kept:
Power series for the exponential function around
:
In[1]:=
Out[1]=
Convert to a normal expression:
In[2]:=
Out[2]=
Power series of an arbitrary function around
:
In[1]:=
Out[1]=
In[1]:=
Out[1]=
In any operation on series, only appropriate terms are kept:
In[2]:=
Out[2]=
Scope
(10)
Series
can handle fractional powers and logarithms:
Symbolic parameters can often be used:
Laurent series with negative powers can be generated:
Truncate the series to the specified negative power:
Find power series for special functions:
Find the series for a function at a branch point:
With
x
assumed to be to the left of the branch point, a simpler result is given:
Piecewise functions:
Power series at infinity:
Series
can give asymptotic series:
Series expansions of implicit solutions to equations:
Series expansions of unevaluated integrals:
Generalizations & Extensions
(4)
Power series in two variables:
Series
is threaded elementwise over lists:
Series
generates
SeriesData
expressions:
Series
can work with approximate numbers:
Options
(4)
Series
by default assumes symbolic functions to be analytic:
Use
Assumptions
to specify regions in the complex plane where expansions should apply:
Without assumptions, piecewise functions appear:
Get expansions in Stokes regions:
Applications
(8)
Plot successive series approximations to
:
Find a series expansion for a standard combinatorial problem:
Find Fibonacci numbers from a generating function:
Find Legendre polynomials by expanding a generating function:
Set up a generating function to enumerate ways to make change using U.S. coins:
The number of ways to make change for $1:
Find the lowestorder terms in a large polynomial:
Find higherorder terms in Newton's approximation for a root of
near
:
Plot the complex zeros for a series approximation to
Exp
[
x
]
:
Properties & Relations
(9)
Series
always only keeps terms up to the specified order:
Operations on series keep only the appropriate terms:
Normal
converts to an ordinary polynomial:
Any mathematical function can be applied to a series:
Adding a series of lower order causes the higherorder terms to be dropped:
Differentiate a series:
Solve equations for series coefficients:
Find the list of coefficients in a series:
Use
O
[
x
]
to force the construction of a series:
ComposeSeries
treats a series as a function to apply to another series:
InverseSeries
does series reversion to find the series for the inverse function of a series:
Possible Issues
(7)
When there is an essential singularity,
Series
will attempt to factor it out:
Numeric values cannot be substituted directly for the expansion variable in a series:
Use
Normal
to get a normal expression in which the substitution can be done:
Series must be converted to normal expressions before being plotted:
Power series with different expansion points cannot be combined:
Not all series are represented by expressions with head
SeriesData
:
Some functions cannot be decomposed into series of powerlike functions:
Series
does not change expressions independent of the expansion variable:
SEE ALSO
SeriesCoefficient
InverseSeries
ComposeSeries
Limit
Normal
InverseZTransform
RSolve
O
SeriesData
PadeApproximant
FourierSeries
TUTORIALS
Symbolic Mathematics: Basic Operations
Power Series
Making Power Series Expansions
Operations on Power Series
The Representation of Power Series
Converting Power Series to Normal Expressions
MORE ABOUT
Analytic Number Theory
Calculus
Manipulating Equations
Multiplicative Number Theory
Series Expansions
New in 6.0: Mathematics & Algorithms
RELATED LINKS
Implementation notes: Algebra and Calculus
NKSOnline
(
A New Kind of Science
)
New in 1  Last modified in 3
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