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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

k1xaϵ<coeffi<k2xaϵ

  

for some constants k1 and k2, for any 0<ϵ 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=1x and then call series with a=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, or if it is a series at ±.

• 

series implements a caching mechanism (see also option remember), so that when the same series is requested a second time but with a lower order, instead of recomputing it the terms from the cache are used.

Examples

seriesx1xx2&comma;x&equals;0

x+x2+2x3+3x4+5x5+Ox6

(1)

If the second argument is x then the equation x=0 is assumed.

seriesx1xx2&comma;x

x+x2+2x3+3x4+5x5+Ox6

(2)

convert&comma;polynom

5x5+3x4+2x3+x2+x

(3)

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

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

2+x12+Ox13

(4)

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

1+x+12x2+16x3+124x4+1120x5+1720x6+15040x7+Ox8

(5)

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

x−1+1+12x+16x2+124x3+1120x4+1720x5+15040x6+Ox7

(6)

seriesGAMMAx&comma;x&equals;0&comma;2

x−1γ+π212+γ22x+Ox2

(7)

The truncation order may be lower than expected if cancellation happens.

series1&ExponentialE;x1x&comma;x&comma;6

2x−223x−1+118+1270x+Ox2

(8)

series1&ExponentialE;x1x&comma;x&comma;10

2x−223x−1+118+1270x13240x2113608x312041200x4+1874800x5+Ox6

(9)

The caching mechanism is used when series is called again with the original truncation order, leading to a more accurate result than before.

series1&ExponentialE;x1x&comma;x&comma;6

2x−223x−1+118+1270x13240x2113608x312041200x4+1874800x5+Ox6

(10)

The cache can be cleared by means of the command forget.

forgetseries&colon;

series1&ExponentialE;x1x&comma;x&comma;6

2x−223x−1+118+1270x+Ox2

(11)

&int;&ExponentialE;x3&DifferentialD;x

−1232x−113π33Γ23x313x−113Γ13&comma;x3x3133

(12)

series&comma;x&equals;0

x+14x4+Ox7

(13)

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

p1+lnxx+12lnx2x2+Ox3

(14)

typep&comma;&apos;series&apos;

true

(15)

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

sxx5212+Ox92

(16)

types&comma;&apos;series&apos;

false

(17)

whattypes

`+`

(18)

The same holds for expansions at ±.

tseriesx3x4&plus;4x5&comma;x&equals;&infin;

t1x4x4+5x5+O1x7

(19)

whattypet

`+`

(20)

Compatibility

• 

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

See Also

asympt

coeftayl

convert

convert/FormalPowerSeries

convert/polynom

convert/Sum

envvar

initialfunctions

Order

powseries

series/leadterm

taylor

type/laurent

type/series

type/taylor