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series

generalized series expansion

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

series(expr, eqn)

series(expr, eqn, n)

Parameters

expr

-

expression

eqn

-

equation (such as x = a) or name (such as x)

n

-

(optional) non-negative integer

Description

• 

The series function computes a truncated series expansion of expr, with respect to the variable x, about the point a, up to order n. If a is infinity then an asymptotic expansion is given.

• 

If eqn evaluates to a name x then the equation x=0 is assumed.

• 

If the third argument n is present then it specifies the "truncation order" of the series calculations. This does not mean the truncation order of the actual series.  See Order for more information about this. If n is not present, the "truncation order" is determined by the global variable Order. The user may assign any non-negative integer to Order.  The default value of Order is 6.  See Order for more information.

• 

If the series is not exact then an "order term" (for example, Ox6 ) is the last term in the series.

• 

It is possible to invoke series on user-defined functions. For example, if the procedure `series/f` is defined then the function call series(f(x,y),x)  will invoke `series/f`(x,y,x)  to compute the series. Note that this user-defined function `series/f` must return a series data structure, not just a polynomial (see type/series).

• 

If series is applied to an unevaluated integral then the series expansion of the integral will be computed (if possible).

• 

The result of the series function is a generalized series expansion. This could be a Taylor series or a Laurent series or a more general series. Formally, the coefficients in a "generalized series" are such that

k1xaeps<|coeffi|<k2xaeps

  

for some constants k1 and k2, for any 0<eps as x approaches a. In other words, the coefficients may depend on x but their growth must be less than the polynomial in x. The order term may also hide such a coefficient, rather than an arbitrary constant. E.g., series considers x2lnx to be Ox2.

• 

If a=infinity or a=-infinity, respectively, then the series expansion is only guaranteed to be valid for positive real x or negative real x, respectively. Use the substitution x&equals;1x and then call series with a&equals;0 to get an expansion that is valid around the North pole of the Riemann sphere.

• 

Usually, the result of the series function is represented in the form of a series data structure. For an explanation of the data structure, see the type/series help page. However, the result of the series function will be represented in ordinary sum-of-products form, rather than in a series data structure if it is a generalized series requiring fractional exponents.

Examples

seriesx1xx2&comma;x&equals;0

x&plus;x2&plus;2x3&plus;3x4&plus;5x5&plus;Ox6

(1)

If the second argument is x then the equation x&equals;0 is assumed.

seriesx1xx2&comma;x

x&plus;x2&plus;2x3&plus;3x4&plus;5x5&plus;Ox6

(2)

convert&comma;polynom

5x5&plus;3x4&plus;2x3&plus;x2&plus;x

(3)

The third argument, specifies the "truncation order" of the series calculations.

seriesx&plus;1x&comma;x&equals;1&comma;3

2&plus;x12&plus;Ox13

(4)

series&ExponentialE;x&comma;x&equals;0&comma;8

1&plus;x&plus;12x2&plus;16x3&plus;124x4&plus;1120x5&plus;1720x6&plus;15040x7&plus;Ox8

(5)

series&ExponentialE;xx&comma;x&equals;0&comma;8

x1&plus;1&plus;12x&plus;16x2&plus;124x3&plus;1120x4&plus;1720x5&plus;15040x6&plus;Ox7

(6)

series&Gamma;x&comma;x&equals;0&comma;2

x1&gamma;&plus;112&pi;2&plus;12&gamma;2x&plus;Ox2

(7)

seriesx3x4&plus;4x5&comma;x&equals;&infin;

1x4x4&plus;5x5&plus;O1x7

(8)

&int;&ExponentialE;x3&DifferentialD;x

1312&sol;323x11&sol;3&pi;3&Gamma;23x31&sol;3x11&sol;3&Gamma;13&comma;x3x31&sol;3

(9)

series&comma;x&equals;0

x&plus;14x4&plus;Ox7

(10)

pseriesxx&comma;x&equals;0&comma;3

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

(11)

typep&comma;&apos;series&apos;

true

(12)

The result of the series function will be represented in ordinary sum-of-products form, rather than in a series data structure, if it is a generalized series requiring fractional exponents.

sseriessinx&comma;x&equals;0&comma;4

s:=x112x5&sol;2&plus;Ox9&sol;2

(13)

types&comma;&apos;series&apos;

false

(14)

whattypes

`+`

(15)

Compatibility

• 

The series command was updated in Maple 2016; see Advanced Math.

See Also

coeftayl

convert

convert/FormalPowerSeries

convert/polynom

convert/Sum

envvar

initialfunctions

Order

powseries

series/leadterm

taylor

type/laurent

type/series

type/taylor

 


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