Series, Limits, and Residues

Sums and Products	Solving Equations Involving Power Series
Power Series	Summation of Series
Making Power Series Expansions	Solving Recurrence Equations
The Representation of Power Series	Finding Limits
Operations on Power Series	Residues
Composition and Inversion of Power Series	Padé Approximation
Converting Power Series to Normal Expressions

Sums and Products

This constructs the sum

You can leave out the lower limit if it is equal to 1:

This makes i increase in steps of 2, so that only odd‐numbered values are included:

Products work just like sums:

Sum[f,{i,i_min,i_max}]	the sum
Sum[f,{i,i_min,i_max,di}]	the sum with i increasing in steps of di
Sum[f,{i,i_min,i_max},{j,j_min,j_max}]	the nested sum
Product[f,{i,i_min,i_max}]	the product

Sums and products.

This sum is computed symbolically as a function of n:

The Wolfram System can also give an exact result for this infinite sum:

As with integrals, simple sums can lead to complicated results:

This sum cannot be evaluated exactly using standard mathematical functions:

You can nevertheless find a numerical approximation to the result:

The Wolfram Language also has a notation for multiple sums and products. Sum[f,{i,i_min,i_max},{j,j_min,j_max}] represents a sum over i and j, which would be written in standard mathematical notation as

. Notice that in Wolfram Language notation, as in standard mathematical notation, the range of the outermost variable is given first.

This is the multiple sum

. Notice that the outermost sum over i is given first, just as in the mathematical notation:

The way the ranges of variables are specified in Sum and Product is an example of the rather general iterator notation that the Wolfram Language uses. You will see this notation again when we discuss generating tables and lists using Table ("Making Tables of Values"), and when we describe Do loops ("Repetitive Operations").

{i_max}	iterate i_max times, without incrementing any variables
{i,i_max}	i goes from 1 to i_max in steps of 1
{i,i_min,i_max}	i goes from i_min to i_max in steps of 1
{i,i_min,i_max,di}	i goes from i_min to i_max in steps of di
{i,i_min,i_max},{j,j_min,j_max},…	i goes from i_min to i_max, and for each such value, j goes from j_min to j_max, etc.

Wolfram Language iterator notation.

Power Series

The mathematical operations we have discussed so far are exact. Given precise input, their results are exact formulas.

In many situations, however, you do not need an exact result. It may be quite sufficient, for example, to find an approximate formula that is valid, say, when the quantity x is small.

This gives a power series approximation to

for

close to

, up to terms of order

The Wolfram Language knows the power series expansions for many mathematical functions:

If you give it a function that it does not know, Series writes out the power series in terms of derivatives:

Power series are approximate formulas that play much the same role with respect to algebraic expressions as approximate numbers play with respect to numerical expressions. The Wolfram Language allows you to perform operations on power series, in all cases maintaining the appropriate order or "degree of precision" for the resulting power series.

Here is a simple power series, accurate to order

When you do operations on a power series, the result is computed only to the appropriate order in x:

This turns the power series back into an ordinary expression:

Now the square is computed exactly:

Applying Expand gives a result with 11 terms:

Series[expr,{x,x₀,n}]	find the power series expansion of expr about the point x=x₀ to at most n ^th order
Normal[series]	truncate a power series to give an ordinary expression

Power series operations.

Making Power Series Expansions

Series[expr,{x,x₀,n}]	find the power series expansion of expr about the point x=x₀ to order at most (x-x₀)ⁿ
Series[expr,{x,x₀,n_x},{y,y₀,n_y}]
	find series expansions with respect to y, then x

Functions for creating power series.

Here is the power series expansion for

about the point

to order

Here is the series expansion of

about the point

If the Wolfram Language does not know the series expansion of a particular function, it writes the result symbolically in terms of derivatives:

In mathematical terms, Series can be viewed as a way of constructing Taylor series for functions.

The standard formula for the Taylor series expansion about the point

of a function

with

^th derivative

. Whenever this formula applies, it gives the same results as Series. (For common functions, Series nevertheless internally uses somewhat more efficient algorithms.)

Series can also generate some power series that involve fractional and negative powers, not directly covered by the standard Taylor series formula.

Here is a power series that contains negative powers of x:

Here is a power series involving fractional powers of x:

Series can also handle series that involve logarithmic terms:

There are, of course, mathematical functions for which no standard power series exist. The Wolfram Language recognizes many such cases.

Series sees that

has an essential singularity at

, and does not produce a power series:

Series can nevertheless give you the power series for

about the point

Especially when negative powers occur, there is some subtlety in exactly how many terms of a particular power series the function Series will generate.

One way to understand what happens is to think of the analogy between power series taken to a certain order, and real numbers taken to a certain precision. Power series are "approximate formulas" in much the same sense as finite‐precision real numbers are approximate numbers.

The procedure that Series follows in constructing a power series is largely analogous to the procedure that N follows in constructing a real‐number approximation. Both functions effectively start by replacing the smallest pieces of your expression by finite‐order, or finite‐precision, approximations, and then evaluating the resulting expression. If there are, for example, cancellations, this procedure may give a final result whose order or precision is less than the order or precision that you originally asked for. Like N, however, Series has some ability to retry its computations so as to get results to the order you ask for. In cases where it does not succeed, you can usually still get results to a particular order by asking for a higher order than you need.

Series compensates for cancellations in this computation, and succeeds in giving you a result to order

When you make a power series expansion in a variable x, the Wolfram Language assumes that all objects that do not explicitly contain x are in fact independent of x. Series thus does partial derivatives (effectively using D) to build up Taylor series.

Both a and n are assumed to be independent of x:

a[x] is now given as an explicit function of x:

You can use Series to generate power series in a sequence of different variables. Series works like Integrate, Sum, and so on, and expands first with respect to the last variable you specify.

Series performs a series expansion successively with respect to each variable. The result in this case is a series in x, whose coefficients are series in y:

The Representation of Power Series

Power series are represented in the Wolfram System as SeriesData objects.

The power series is printed out as a sum of terms, ending with O[x] raised to a power:

Internally, however, the series is stored as a SeriesData object:

By using SeriesData objects, rather than ordinary expressions, to represent power series, the Wolfram System can keep track of the order and expansion point, and do operations on the power series appropriately. You should not normally need to know the internal structure of SeriesData objects.

You can recognize a power series that is printed out in standard output form by the presence of an O[x] term. This term mimics the standard mathematical notation

, and represents omitted terms of order

. For various reasons of consistency, the Wolfram System uses the notation O[x]^n for omitted terms of order

, corresponding to the mathematical notation

, rather than the slightly more familiar, though equivalent, form

Any time that an object like O[x] appears in a sum of terms, the Wolfram System will in fact convert the whole sum into a power series.

The presence of O[x] makes the Wolfram System convert the whole sum to a power series:

Series objects can involve fractional powers:

Here is the internal representation of the series:

Series can involve logarithmic terms:

The logarithmic factors appear explicitly inside the SeriesData coefficient list:

Operations on Power Series

The Wolfram Language allows you to perform many operations on power series. In all cases, the Wolfram Language gives results only to as many terms as can be justified from the accuracy of your input.

Here is a power series accurate to fourth order in

When you square the power series, you get another power series, also accurate to fourth order:

Taking the logarithm gives you the result 2x, but only to order

The Wolfram Language keeps track of the orders of power series in much the same way as it keeps track of the precision of approximate real numbers. Just as with numerical calculations, there are operations on power series which can increase, or decrease, the precision (or order) of your results.

Here is a power series accurate to order

This gives a power series that is accurate only to order

The Wolfram Language also allows you to do calculus with power series.

Here is a power series for

Here is its derivative with respect to x:

Integrating with respect to x gives back the original power series:

When you perform an operation that involves both a normal expression and a power series, the Wolfram Language "absorbs" the normal expression into the power series whenever possible.

The 1 is automatically absorbed into the power series:

The x^2 is also absorbed into the power series:

If you add Sin[x], the Wolfram Language generates the appropriate power series for Sin[x], and combines it with the power series you have:

The Wolfram Language also absorbs expressions that multiply power series. The symbol a is assumed to be independent of x:

The Wolfram Language knows how to apply a wide variety of functions to power series. However, if you apply an arbitrary function to a power series, it is impossible for the Wolfram Language to give you anything but a symbolic result.

The Wolfram Language does not know how to apply the function f to a power series, so it just leaves the symbolic result:

Composition and Inversion of Power Series

When you manipulate power series, it is sometimes convenient to think of the series as representing functions, which you can, for example, compose or invert.

ComposeSeries[series₁,series₂,…]	compose power series
InverseSeries[series,x]	invert a power series

Composition and inversion of power series.

Here is the power series for

to order

This replaces the variable

in the power series for

by a power series for

The result is the power series for

If you have a power series for a function

, then it is often possible to get a power series approximation to the solution for

in the equation

. This power series effectively gives the inverse function

such that

. The operation of finding the power series for an inverse function is sometimes known as reversion of power series.

Here is the series for

Inverting the series gives the series for

This agrees with the direct series for

Composing the series with its inverse gives the identity function:

Converting Power Series to Normal Expressions

Normal[expr]

convert a power series to a normal expression

Converting power series to normal expressions.

Power series in the Wolfram Language are represented in a special internal form, which keeps track of such attributes as their expansion order.

For some purposes, you may want to convert power series to normal expressions. From a mathematical point of view, this corresponds to truncating the power series, and assuming that all higher‐order terms are zero.

This generates a power series, with four terms:

Squaring the power series gives you another power series, with the appropriate number of terms:

Normal truncates the power series, giving a normal expression:

You can now apply standard algebraic operations:

SeriesCoefficient[series,n]

give the coefficient of the n ^th order term in a power series

Extracting coefficients of terms in power series.

This gives the coefficient of

in the original power series:

This gives the coefficient for the term

in the Taylor expansion of the function

about zero:

Solving Equations Involving Power Series

LogicalExpand[series₁==series₂]	give the equations obtained by equating corresponding coefficients in the power series
Solve[series₁==series₂,{a₁,a₂,…}]	solve for coefficients in power series

Solving equations involving power series.

Here is a power series:

This gives an equation involving the power series:

LogicalExpand generates a sequence of equations for each power of x:

This solves the equations for the coefficients a[i]. You can also feed equations involving power series directly to Solve:

Some equations involving power series can also be solved using the InverseSeries function discussed in "Composition and Inversion of Power Series".

Summation of Series

Sum[expr,{n,n_min,n_max}]

find the sum of expr as n goes from n_min to n_max

Evaluating sums.

The Wolfram System recognizes this as the power series expansion of

This sum comes out in terms of a Bessel function:

Here is another sum that can be done in terms of common special functions:

Generalized hypergeometric functions are not uncommon in sums:

There are many analogies between sums and integrals. And just as it is possible to have indefinite integrals, so indefinite sums can be set up by using symbolic variables as upper limits.

This is effectively an indefinite sum:

This sum comes out in terms of incomplete gamma functions:

This sum involves polygamma functions:

Taking the difference between results for successive values of

gives back the original summand:

The Wolfram System can do essentially all sums that are found in books of tables. Just as with indefinite integrals, indefinite sums of expressions involving simple functions tend to give answers that involve more complicated functions. Definite sums, like definite integrals, often, however, come out in terms of simpler functions.

This indefinite sum gives a quite complicated result:

The definite form is much simpler:

Here is a slightly more complicated definite sum:

Solving Recurrence Equations

If you represent the n ^th term in a sequence as a[n], you can use a recurrence equation to specify how it is related to other terms in the sequence.

RSolve takes recurrence equations and solves them to get explicit formulas for a[n].

This solves a simple recurrence equation:

This takes the solution and makes an explicit table of the first 10 a[n]:

RSolve[eqn,a[n],n]

solve a recurrence equation

Solving a recurrence equation.

This solves a recurrence equation for a geometric series:

This gives the same result:

This gives an algebraic solution to a recurrence equation:

This solves the Fibonacci recurrence equation:

RSolve can be thought of as a discrete analog of DSolve. Many of the same functions generated in solving differential equations also appear in finding symbolic solutions to recurrence equations.

This generates a gamma function, which generalizes the factorial:

This second‐order recurrence equation comes out in terms of Bessel functions:

RSolve does not require you to specify explicit values for terms such as a[1]. Like DSolve, it automatically introduces undetermined constants C[i] to give a general solution.

This gives a general solution with one undetermined constant:

RSolve can solve equations that do not depend only linearly on a[n]. For nonlinear equations, however, there are sometimes several distinct solutions that must be given. Just as for differential equations, it is a difficult matter to find symbolic solutions to recurrence equations, and standard mathematical functions only cover a limited set of cases.

Here is the general solution to a nonlinear recurrence equation:

This gives two distinct solutions:

RSolve can solve not only ordinary difference equations in which the arguments of

differ by integers, but also

‐difference equations in which the arguments of

are related by multiplicative factors.

This solves the

‐difference analog of the factorial equation:

Here is a second‐order

‐difference equation:

RSolve[{eqn₁,eqn₂,…},{a₁[n],a₂[n],…},n]
	solve a coupled system of recurrence equations

Solving systems of recurrence equations.

This solves a system of two coupled recurrence equations:

RSolve[eqns,a[n₁,n₂,…],{n₁,n₂,…}]
	solve partial recurrence equations

Solving partial recurrence equations.

Just as one can set up partial differential equations that involve functions of several variables, so one can also set up partial recurrence equations that involve multidimensional sequences. Just as in the differential equations case, general solutions to partial recurrence equations can involve undetermined functions.

This gives the general solution to a simple partial recurrence equation:

Finding Limits

In doing many kinds of calculations, you need to evaluate expressions when variables take on particular values. In many cases, you can do this simply by applying transformation rules for the variables using the /. operator.

You can get the value of

at 0 just by explicitly replacing

with 0, and then evaluating the result:

In some cases, however, you have to be more careful.

Consider, for example, finding the value of the expression

when

. If you simply replace

in this expression, you get the indeterminate result

. To find the correct value of

when

, you need to take the limit.

Limit[expr,x->x₀]

find the limit of expr when x approaches x₀

Finding limits.

This gives the correct value for the limit of

No finite limit exists in this case:

Limit can find this limit, even though you cannot get an ordinary power series for

The same is true here:

The value of Sign[x] at

is 0:

Its limit, however, is 1. The limit is by default taken from above:

Not all functions have definite limits at particular points. For example, the function

oscillates infinitely often near

, so it has no definite limit there. Nevertheless, at least so long as

remains real, the values of the function near

always lie between

and

. Limit represents values with bounded variation using Interval objects. In general, Interval[{x_min,x_max}] represents an uncertain value which lies somewhere in the interval

Limit returns an Interval object, representing the range of possible values of

near its essential singularity at

The Wolfram Language can do arithmetic with Interval objects:

The Wolfram Language represents this limit symbolically in terms of an Interval object:

Some functions may have different limits at particular points, depending on the direction from which you approach those points. You can use the Direction option for Limit to specify the direction you want.

Limit[expr,x->x₀,Direction->1]	find the limit as x approaches x₀ from below
Limit[expr,x->x₀,Direction->-1]	find the limit as x approaches x₀ from above

Directional limits.

The function

has a different limiting value at

, depending on whether you approach from above or below:

Approaching from below gives a limiting value of

Approaching from above gives a limiting value of

Limit makes no assumptions about functions like f[x] about which it does not have definite knowledge. As a result, Limit remains unevaluated in most cases involving symbolic functions.

Limit has no definite knowledge about f, so it leaves this limit unevaluated:

Residues

Limit[expr,x->x₀] tells you what the value of expr is when x tends to x₀. When this value is infinite, it is often useful instead to know the residue of expr when x equals x₀. The residue is given by the coefficient of

in the power series expansion of expr about the point x₀.

Residue[expr,{x,x₀}]

the residue of expr when x equals x₀

Computing residues.

The residue here is equal to 1:

The residue here is zero:

Residues can be computed at the point at infinity:

Padé Approximation

The Padé approximation is a rational function that can be thought of as a generalization of a Taylor polynomial. A rational function is the ratio of polynomials. Because these functions only use the elementary arithmetic operations, they are very easy to evaluate numerically. The polynomial in the denominator allows you to approximate functions that have rational singularities.

PadeApproximant[f,{x,x₀,{n,m}}]	give the Padé approximation to centered at x₀ of order (n,m)
PadeApproximant[f,{x,x₀,n}]	give the diagonal Padé approximation to centered at x₀ of order n

Padé approximations.

More precisely, a Padé approximation of order

to an analytic function

at a regular point or pole

is the rational function

where

is a polynomial of degree

, and the formal power series of

about the point

begins with the term

. If

is equal to

, the approximation is called a diagonal Padé approximation of order

Here is the Padé approximation of order

This gives another Padé approximation of the same order:

The initial terms of this series vanish. This is the property that characterizes the Padé approximation:

This plots the difference between the approximation and the true function. Notice that the approximation is very good near the center of expansion, but the error increases rapidly as you move away:

In the Wolfram Language PadeApproximant is generalized to allow expansion about branch points.

This gives the diagonal Padé approximation of order