residue - Maple Programming Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : Algebra : Expression Manipulation : residue

residue

compute the algebraic residue of an expression

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

residue(f, x=a)

residue(f, x=a, n)

Parameters

f

-

arbitrary algebraic expression

x

-

variable

a

-

algebraic value at which the residue is evaluated

n

-

(optional) positive integer

Description

• 

Computes the algebraic residue of the expression f for the variable x around the point a. The residue is defined as the coefficient of 1xa in the Laurent series expansion of f.

• 

Maple compute the residue by successively performing series expansions at x=a, by default of orders 2,3,6,11,18,27, and then extracts the coefficient of 1xa. If this is unsuccessful, for example, because f does not have a Laurent expansion, or because order 27 is still too small, the residue command returns unevaluated.

• 

The maximal order of series expansions being tried can be raised by providing a positive integer n as third argument.

Examples

residueZetas,s=1

1

(1)

residueΨxGAMMAxx,x=0

π212+γ22

(2)

residueⅇx,x=1

0

(3)

In the following example, there is no Laurent expansion.

residuearcsinx,x=1

residuearcsinx,x=1

(4)

seriesarcsinx,x=1,3

π2I2csgnIx1x1+I2csgnIx1x132123I2csgnIx1x152160+Ox172

(5)

In the next example, the default order is too small.

residue1z271z3,z=0

378

(6)

residue1z281z3,z=0

residue1z281z3,z=0

(7)

residue1z281z3,z=0,27

residue1z281z3,z=0,27

(8)

The reason is that the series expansion of order 27 does not have enough terms:

series1z281z3,z=0,27

z−28+3z−27+6z−26+10z−25+15z−24+21z−23+28z−22+36z−21+45z−20+55z−19+66z−18+78z−17+91z−16+105z−15+120z−14+136z−13+153z−12+171z−11+190z−10+210z−9+231z−8+253z−7+276z−6+300z−5+325z−4+351z−3+378z−2+Oz−1

(9)

Raising the order to 28 helps:

residue1z281z3,z=0,28

406

(10)

Compatibility

• 

The residue command was updated in Maple 2019.

• 

The n parameter was introduced in Maple 2019.

• 

For more information on Maple 2019 changes, see Updates in Maple 2019.

See Also

numapprox[laurent]

series

singular