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type/series

series data structure

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

type(expr, series)

Parameters

expr

-

expression

Description

• 

The function type/series returns true if the value of expr is Maple's series data structure, explained below.

• 

The series data structure represents an expression as a truncated series in one specified indeterminate, expanded about a particular point. It is created by a call to the series function.

• 

op(0, expr), with expr of type series, returns x-a where x denotes the ``series variable'' and a denotes the particular point of expansion. op(2*i-1, expr) returns the ith coefficient (a general expression) and op(2*i, expr) returns the corresponding integer exponent.

• 

The exponents are ``word-size'' integers, in increasing order.

• 

The representation is sparse; zero coefficients are not represented.

• 

Usually, the final pair of operands in this data type are the special order symbol O(1) and the integer n which indicates the order of truncation. However, if the series is exact then there will be no order term, for example, the series expansion of a low-degree polynomial.

• 

Formally, the coefficients of the series are such that

k1xaeps<|coeffi|<k2xaeps

  

for some constants k1 and k2, for any 0<eps, and as x approaches a. In other words, the coefficients may depend on x, but their growth must be less than polynomial in x. O(1) represents such a coefficient, rather than an arbitrary constant.

• 

A zero series is immediately simplified to the integer zero.

Examples

aseriessinx&comma;x&equals;0&comma;5

a:=x16x3&plus;Ox5

(1)

typea&comma;&apos;series&apos;

true

(2)

typea&comma;&apos;taylor&apos;

true

(3)

op0&comma;a

x

(4)

opa

1&comma;1&comma;16&comma;3&comma;O1&comma;5

(5)

bseries1sinx&comma;x&equals;0&comma;5

b:=x1&plus;16x&plus;Ox3

(6)

typeb&comma;&apos;series&apos;

true

(7)

typeb&comma;&apos;taylor&apos;

false

(8)

op0&comma;b

x

(9)

opb

1&comma;1&comma;16&comma;1&comma;O1&comma;3

(10)

typex3&comma;&apos;series&apos;

false

(11)

seriessinx&comma;x&equals;0&comma;4

x112x5&sol;2&plus;Ox9&sol;2

(12)

type&comma;&apos;series&apos;

false

(13)

whattype

`+`

(14)

cseriesxx&comma;x&equals;0&comma;3

c:=1&plus;lnxx&plus;12lnx2x2&plus;Ox3

(15)

typec&comma;&apos;series&apos;

true

(16)

typec&comma;&apos;taylor&apos;

false

(17)

op0&comma;c

x

(18)

opc

1&comma;0&comma;lnx&comma;1&comma;12lnx2&comma;2&comma;O1&comma;3

(19)

dseriessinx&plus;y&comma;x&equals;y&comma;2

d:=sin2y&plus;cos2yxy&plus;Oxy2

(20)

typed&comma;&apos;series&apos;

true

(21)

op0&comma;d

xy

(22)

opd

sin2y&comma;0&comma;cos2y&comma;1&comma;O1&comma;2

(23)

See Also

op

series

taylor

type

type/laurent

type/taylor

 


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