Formal Power Series - Maple Help
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Home : Support : Online Help : System : Information : Updates : Maple 2022 : Formal Power Series

Formal Power Series

The convert/FormalPowerSeries functionality was completely rewritten for Maple 2022. It offers a number of advantages over previous versions:

 

• 

Closed-form solutions can be found in a number of cases where previous versions failed.

• 

Solutions in terms of m-fold hypergeometric sequences for arbitrary positive integers m are now supported in more cases than before.

• 

Notwithstanding the name, formal Laurent and Puiseux series (i.e., with negative or fractional exponents) can be computed as well, now in more cases than before.

• 

convert/FormalPowerSeries will automatically attempt to return the series coefficients in purely real form, making the previous option makereal obsolete.

• 

In a number of cases, the new code returns more compact answers than previous versions.

• 

If a closed form expression for the power series coefficients cannot be found, and a recurrence relation of degree 1 or 2 exists, it will be returned instead. Previously, only linear recurrences could be computed, and would only be returned if option recurrence was specified.

• 

When a recurrence relation is returned, now the initial conditions are given as well.

• 

Additional options give more control over the underlying algorithm(s) used and the form of the output.

 

Maple 2021

Maple 2022

More closed-form solutions, notably, for sums of several terms and Puiseux solutions.

convert−z2 + z36arctanz,FormalPowerSeries

−12⁢z+16⁢z3⁢arctan⁡z

(1)

mapconvert,expand−z2 + z36arctanz,FormalPowerSeries 

∑k=0∞⁡−−1k⁢z2⁢k+24⁢k+2+∑k=0∞⁡−1k⁢z2⁢k+412⁢k+6

(2)

convert−z2 + z36arctanz,FormalPowerSeries

−z26+59+∑n=0∞⁡4⁢n−5⁢−1n⁢z2⁢n3⁢2⁢n−1⁢2⁢n−3

(3)

convertarctanz+arcsinz,FormalPowerSeries

arctan⁡z+arcsin⁡z

(4)

mapconvert,arctanz+arcsinz,FormalPowerSeries

∑k=0∞⁡−1k⁢z2⁢k+12⁢k+1+∑k=0∞⁡2⁢k!⁢4−k⁢z2⁢k+1k!2⁢2⁢k+1

(5)

convert⁡arctanz+arcsinz,FormalPowerSeries

∑n=0∞⁡−1n⁢n!2+2⁢n!⁢4−n⁢z2⁢n+12⁢n+1⁢n!2

(6)

convert⁡arctanz+arcsinz,FormalPowerSeries,output=expanded

∑n=0∞⁡−1n⁢z2⁢n+12⁢n+1+∑n=0∞⁡2⁢n!⁢4−n⁢z2⁢n+12⁢n+1⁢n!2

(7)

convert1−1−4 z2z24 1−4 z, FormalPowerSeries

1−1−4⁢z2⁢z24⁢1−4⁢z

(8)

convert1−1−4 z2z24 1−4 z, FormalPowerSeries

∑n=0∞⁡2⁢n+2!⁢n+2⁢n+1⁢zn+4n+2!2

(9)

convert8 z3+1−1,FormalPowerSeries

8⁢z3+1−1

(10)

convert8 z3+1−1,FormalPowerSeries

∑n=0∞⁡−1n⁢2−n+1⁢2⁢n+1⁢4⁢n!⁢z3⁢n+322⁢n+1!2

(11)

Solutions in purely real form by default.

convertsinz+z cosz,FormalPowerSeries

∑k=0∞⁡−I⁢Ik⁢k+12⁢k!+I⁢−Ik⁢k+12⁢k!⁢zk

(12)

convertsinz+z cosz,FormalPowerSeries,makereal

∑k=0∞⁡sin⁡k⁢π2⁢k+1⁢zkk!

(13)

convertsinz+z cosz,FormalPowerSeries

∑n=0∞⁡2⁢−1n⁢n+1⁢z2⁢n+12⁢n+1!

(14)

convertln1+z+z2+z3,FormalPowerSeries

∑k=0∞⁡−−1k+1k+1−Ik+1k+1−−Ik+1k+1⁢zk+1

(15)

convertln1+z+z2+z3,FormalPowerSeries,makereal

∑k=0∞⁡−1k+2⁢sin⁡k⁢π2⁢zk+1k+1

(16)

convertln1+z+z2+z3,FormalPowerSeries

∑n=0∞⁡−1n⁢zn+1n+1+∑n=0∞⁡−1n⁢z2⁢n+2n+1

(17)

convert−z6+3 z2−3 z4+1,FormalPowerSeries 

z23+∑k=0∞⁡2⁢3⁢−I3⁢334k⁢−3343k⁢I3⁢334k−−I3⁢334k⁢−3343k⁢3343k+−I3⁢334k⁢I3⁢334k⁢3343k−−3343k⁢I3⁢334k⁢3343k⁢zk9⁢−I3⁢334k⁢−3343k⁢I3⁢334k⁢3343k

(18)

convert−z6+3 z2−3 z4+1,FormalPowerSeries,makereal

z23+∑k=0∞⁡−2⁢zk⁢3k4+12⁢2⁢cos⁡k⁢π2−−1k−19

(19)

convert−z6+3 z2−3 z4+1,FormalPowerSeries

z23+∑n=0∞⁡8⁢3n−1⁢z4⁢n+2

(20)

More compact answers.

convert1q__1−z2⋅q__2−z3,FormalPowerSeries,z 

∑k=0∞⁡−2⁢−123⁢q__213k⁢q__1k⁢−q__1k⁢q__213k⁢−123⁢q__13+2⁢−−113⁢q__213k⁢q__1k⁢−q__1k⁢q__213k⁢q__243⁢−123⁢q__1−−123⁢q__213k⁢−−113⁢q__213k⁢q__1k⁢q__213k⁢−123⁢q__132⁢q__2+−123⁢q__213k⁢−−113⁢q__213k⁢−q__1k⁢q__213k⁢−123⁢q__132⁢q__2+2⁢−123⁢q__213k⁢q__1k⁢−q__1k⁢q__213k⁢−113⁢q__13+−123⁢q__213k⁢−−113⁢q__213k⁢q__1k⁢q__213k⁢−113⁢q__132⁢q__2−−123⁢q__213k⁢−−113⁢q__213k⁢−q__1k⁢q__213k⁢−113⁢q__132⁢q__2−2⁢−−113⁢q__213k⁢q__1k⁢−q__1k⁢q__213k⁢q__223⁢q__12+2⁢−123⁢q__213k⁢−−113⁢q__213k⁢q__1k⁢−q__1k⁢q__243⁢q__1−2⁢−123⁢q__213k⁢q__1k⁢−q__1k⁢q__213k⁢q__243⁢q__1−2⁢−123⁢q__213k⁢−−113⁢q__213k⁢q__1k⁢q__213k⁢q__22−2⁢−123⁢q__213k⁢−−113⁢q__213k⁢−q__1k⁢q__213k⁢q__22−2⁢−123⁢q__213k⁢q__1k⁢−q__1k⁢q__213k⁢q__243⁢−123⁢q__1+4⁢−123⁢q__213k⁢q__1k⁢−q__1k⁢q__213k⁢q__223⁢−123⁢q__12+2⁢−123⁢q__213k⁢−−113⁢q__213k⁢q__1k⁢−q__1k⁢q__223⁢−113⁢q__12−2⁢−123⁢q__213k⁢q__1k⁢−q__1k⁢q__213k⁢q__223⁢−113⁢q__12+2⁢−123⁢q__213k⁢q__1k⁢−q__1k⁢q__213k⁢q__223⁢q__12−2⁢−123⁢q__213k⁢−−113⁢q__213k⁢q__1k⁢−q__1k⁢q__223⁢−123⁢q__12−2⁢−−113⁢q__213k⁢q__1k⁢−q__1k⁢q__213k⁢q__223⁢−123⁢q__12+−123⁢q__213k⁢−−113⁢q__213k⁢q__1k⁢q__213k⁢q__22⁢−123+−123⁢q__213k⁢−−113⁢q__213k⁢−q__1k⁢q__213k⁢q__22⁢−123−−123⁢q__213k⁢−−113⁢q__213k⁢q__1k⁢q__213k⁢q__22⁢−113−−123⁢q__213k⁢−−113⁢q__213k⁢−q__1k⁢q__213k⁢q__22⁢−113+2⁢−123⁢q__213k⁢−−113⁢q__213k⁢q__1k⁢q__213k⁢q__132⁢q__2−2⁢−123⁢q__213k⁢−−113⁢q__213k⁢−q__1k⁢q__213k⁢q__132⁢q__2+2⁢−123⁢q__213k⁢−−113⁢q__213k⁢q__1k⁢−q__1k⁢q__13+2⁢−−113⁢q__213k⁢q__1k⁢−q__1k⁢q__213k⁢q__13⁢zk2⁢q__213k⁢−q__1k⁢q__1+q__213⁢q__1k⁢q__1⁢−q__1+q__213⁢−−113⁢q__213k⁢−q__1+−113⁢q__213⁢q__1+−113⁢q__213⁢−123⁢q__213k⁢−123⁢q__213+q__1⁢−123⁢q__213−q__1⁢−113+12⁢−113−1⁢q__2

(21)

convert1q__1−z2⋅q__2−z3,FormalPowerSeries,z

∑n=0∞⁡−q__12−n2+−1n⁢q__12−n2+2⁢q__2−13−n3⁢q__152−2⁢q__1−n2+12⁢q__2−−1n⁢q__1−1−n2⁢q__22+2⁢q__223−n3⁢q__1−q__1−1−n2⁢q__22⁢zn2⁢q__13−q__22⁢q__132+q__2+∑n=0∞⁡q__1⁢q__2−n−1⁢z3⁢nq__243+q__223⁢q__1+q__12+∑n=0∞⁡−q__2−n−23⁢z3⁢n+1q__243+q__223⁢q__1+q__12

(22)

Recurrence relations returned automatically if no closed form can be found, with initial conditions.
Non-linear (degree 2) recurrences can be computed.

convertarcsinz3,FormalPowerSeries 

arcsin⁡z3

(23)

convertarcsinz3,FormalPowerSeries,recurrence  

k4⁢a⁡k−2⁢k+1⁢k+2⁢k2+2⁢k+2⁢a⁡k+2+k+1⁢k+2⁢k+3⁢k+4⁢a⁡k+4=0

(24)

convertarcsinz3,FormalPowerSeries

∑n=0∞⁡A⁡n⁢zn+1,RESol⁡n4+4⁢n3+6⁢n2+4⁢n+1⁢A⁡n+−2⁢n4−18⁢n3−62⁢n2−98⁢n−60⁢A⁡n+2+n4+14⁢n3+71⁢n2+154⁢n+120⁢A⁡n+4=0,A⁡n,A⁡0=0,A⁡1=0,A⁡2=1,A⁡3=0,INFO

(25)

convertzⅇz−1,FormalPowerSeries 

zⅇz−1

(26)

convertzⅇz−1,FormalPowerSeries,recurrence 

zⅇz−1

(27)

 

convertzⅇz−1,FormalPowerSeries

∑n=0∞⁡A⁡n⁢zn,RESol⁡A⁡n+3+A⁡n+2+∑_k=1n+2⁡A⁡_k⁢A⁡n+3−_kn+4,A⁡n,A⁡0=1,A⁡1=−12,A⁡2=112,INFO

(28)

convertLambertWz,FormalPowerSeries

LambertW⁡z

(29)

convertLambertWz,FormalPowerSeries,recurrence

LambertW⁡z

(30)

convertLambertWz,FormalPowerSeries

∑n=0∞⁡A⁡n⁢zn,RESol⁡A⁡n+4+A⁡n+3+∑_k=1n+2⁡_k+1⁢A⁡_k+1⁢A⁡n+3−_kn+3,A⁡n,A⁡0=0,A⁡1=1,A⁡2=−1,A⁡3=32,INFO

(31)

 

New method option (by default, all three methods are tried in sequence).

 

convertarcsinz2,FormalPowerSeries

∑n=0∞⁡2⁢n!2⁢4n⁢z2⁢n+22⁢n+2!

(32)

convertarcsinz2,FormalPowerSeries,method=hypergeometric

∑n=0∞⁡2⁢n!2⁢4n⁢z2⁢n+22⁢n+2!

(33)

convertarcsinz2,FormalPowerSeries,method=holonomic

∑n=0∞⁡A⁡n⁢zn+1,RESol⁡−n2−2⁢n−1⁢A⁡n+n2+5⁢n+6⁢A⁡n+2=0,A⁡n,A⁡0=0,A⁡1=1,INFO

(34)

convertarcsinz2,FormalPowerSeries,method=quadratic

∑n=0∞⁡A⁡n⁢zn,RESol⁡A⁡n+4−4⁢n+2⁢A⁡n+2+∑_k=2n⁡_k+1⁢A⁡_k+1⁢n+3−_k⁢A⁡n+3−_k+∑_k=2n+2⁡−_k+1⁢A⁡_k+1⁢n+5−_k⁢A⁡n+5−_k4⁢n+3,A⁡n,A⁡0=0,A⁡1=0,A⁡2=1,A⁡3=0,INFO

(35)

 

converttanz,FormalPowerSeries

∑n=0∞⁡A⁡n⁢zn,RESol⁡A⁡n+3+−2⁢A⁡n+1+∑_k=1n⁡−2⁢_k+1⁢A⁡_k+1⁢A⁡n+1−_kn+2⁢n+3,A⁡n,A⁡0=0,A⁡1=1,A⁡2=0,INFO

(36)

converttanz,FormalPowerSeries,method=hypergeometric

tan⁡z

(37)

converttanz,FormalPowerSeries,method=holonomic

tan⁡z

(38)

converttanz,FormalPowerSeries,method=quadratic

∑n=0∞⁡A⁡n⁢zn,RESol⁡A⁡n+3+−2⁢A⁡n+1+∑_k=1n⁡−2⁢_k+1⁢A⁡_k+1⁢A⁡n+1−_kn+2⁢n+3,A⁡n,A⁡0=0,A⁡1=1,A⁡2=0,INFO

(39)

New output option (default: combined).

 

convertsinz+cosz3,FormalPowerSeries

∑n=0∞⁡−−1n⁢9n−3⁢z2⁢n2⁢2⁢n!+∑n=0∞⁡3⁢−1n⁢9n+1⁢z2⁢n+12⁢2⁢n+1!

(40)

convertsinz+cosz3,FormalPowerSeries,output=combined

∑n=0∞⁡−−1n⁢9n−3⁢z2⁢n2⁢2⁢n!+∑n=0∞⁡3⁢−1n⁢9n+1⁢z2⁢n+12⁢2⁢n+1!

(41)

convertsinz+cosz3,FormalPowerSeries,output=expanded

∑n=0∞⁡−−1n⁢9n⁢z2⁢n2⁢2⁢n!+∑n=0∞⁡3⁢−1n⁢z2⁢n2⁢2⁢n!+∑n=0∞⁡3⁢−1n⁢9n⁢z2⁢n+12⁢2⁢n+1!+∑n=0∞⁡3⁢−1n⁢z2⁢n+12⁢2⁢n+1!

(42)