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asympt -- compute an asymptotic series expansion

Introduction

asympt(f, x) computes the first terms of an asymptotic series expansion of f with respect to the variable x around the point infinity.

Call(s)

asympt(f, x)
asympt(f, x <= x0> <, order> <, dir>)

Parameters

f - an arithmetical expression representing a function in x
x - an identifier
x0 - the expansion point: an arithmetical expression; if not specified, the default expansion point infinity is used
order - the number of terms to be computed: a nonnegative integer; the default order is given by the environment variable ORDER (default value 6)

Options

dir - either Left or Right. With Left, the expansion is valid for real x < x0; with Right, it is valid for x > x0. For finite expansion points x0, the default is Right.

Returns

an object of domain type Series::gseries, or an expression of type "asympt".

Side Effects

The function is sensitive to the environment variable ORDER, which determines the default number of terms in series computations.

Overloadable:

f

Related Functions

limit, O, ORDER, series, Series::gseries, taylor, Type::Series

Details

Example 1

We compute an asymptotic expansion for x -> infinity:

>> s := asympt(sin(1/x + exp(-x)) - sin(1/x), x)
                          exp(-x)   exp(-x)    / exp(-x) \
                exp(-x) - ------- + ------- + O| ------- |
                              2          4     |    6    |
                           2 x       24 x      \   x     /

The leading term and the third term are extracted:

>> lmonomial(s), nthterm(s, 3)
                                      exp(-x)
                             exp(-x), -------
                                         4
                                        x

In the following call, only 2 terms of the expansion are requested:

>> asympt(
     exp(sin(1/x + exp(-exp(x)))) - exp(sin(1/x)), x, 2
   )
                             exp(-exp(x))    / exp(-exp(x)) \
              exp(-exp(x)) + ------------ + O| ------------ |
                                  x          |       2      |
                                             \      x       /
>> delete s:

Example 2

We compute a expansion around a finite real point. By default, the expansion is valid ``to the right'' of the expansion point:

>> asympt(abs(x/(1+x)), x = 0)
                         2    3    4    5    6      7
                    x - x  + x  - x  + x  - x  + O(x )

A different expansion is valid ``to the left'' of the expansion point:

>> asympt(abs(x)/(1 + x), x = 0, Left)
                         2    3    4    5    6        7
                  - x + x  - x  + x  - x  + x  + O(- x )

Example 3

The following expansion is exact. Therefore, it has no ``error term'':

>> asympt(x/exp(x), x = -infinity)
                                 x exp(-x)

Example 4

Here is an example where asympt cannot compute an asymptotic series expansion:

>> asympt(cos(x*s)/s, x = infinity)
                           / cos(s x)               \
                     asympt| --------, x = infinity |
                           \    s                   /

Changes




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