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
−12z+16z3arctanz
mapconvert,expand−z2 + z36arctanz,FormalPowerSeries
∑k=0∞−−1kz2k+24k+2+∑k=0∞−1kz2k+412k+6
−z26+59+∑n=0∞4n−5−1nz2n32n−12n−3
convertarctanz+arcsinz,FormalPowerSeries
arctanz+arcsinz
mapconvert,arctanz+arcsinz,FormalPowerSeries
∑k=0∞−1kz2k+12k+1+∑k=0∞2k!4−kz2k+1k!22k+1
convertarctanz+arcsinz,FormalPowerSeries
∑n=0∞−1nn!2+2n!4−nz2n+12n+1n!2
convertarctanz+arcsinz,FormalPowerSeries,output=expanded
∑n=0∞−1nz2n+12n+1+∑n=0∞2n!4−nz2n+12n+1n!2
convert1−1−4 z2z24 1−4 z, FormalPowerSeries
1−1−4z2z241−4z
∑n=0∞2n+2!n+2n+1zn+4n+2!2
convert8 z3+1−1,FormalPowerSeries
8z3+1−1
∑n=0∞−1n2−n+12n+14n!z3n+322n+1!2
Solutions in purely real form by default.
convertsinz+z cosz,FormalPowerSeries
∑k=0∞−IIkk+12k!+I−Ikk+12k!zk
convertsinz+z cosz,FormalPowerSeries,makereal
∑k=0∞sinkπ2k+1zkk!
∑n=0∞2−1nn+1z2n+12n+1!
convertln1+z+z2+z3,FormalPowerSeries
∑k=0∞−−1k+1k+1−Ik+1k+1−−Ik+1k+1zk+1
convertln1+z+z2+z3,FormalPowerSeries,makereal
∑k=0∞−1k+2sinkπ2zk+1k+1
∑n=0∞−1nzn+1n+1+∑n=0∞−1nz2n+2n+1
convert−z6+3 z2−3 z4+1,FormalPowerSeries
z23+∑k=0∞23−I3334k−3343kI3334k−−I3334k−3343k3343k+−I3334kI3334k3343k−−3343kI3334k3343kzk9−I3334k−3343kI3334k3343k
convert−z6+3 z2−3 z4+1,FormalPowerSeries,makereal
z23+∑k=0∞−2zk3k4+122coskπ2−−1k−19
z23+∑n=0∞83n−1z4n+2
More compact answers.
convert1q__1−z2⋅q__2−z3,FormalPowerSeries,z
∑k=0∞−2−123q__213kq__1k−q__1kq__213k−123q__13+2−−113q__213kq__1k−q__1kq__213kq__243−123q__1−−123q__213k−−113q__213kq__1kq__213k−123q__132q__2+−123q__213k−−113q__213k−q__1kq__213k−123q__132q__2+2−123q__213kq__1k−q__1kq__213k−113q__13+−123q__213k−−113q__213kq__1kq__213k−113q__132q__2−−123q__213k−−113q__213k−q__1kq__213k−113q__132q__2−2−−113q__213kq__1k−q__1kq__213kq__223q__12+2−123q__213k−−113q__213kq__1k−q__1kq__243q__1−2−123q__213kq__1k−q__1kq__213kq__243q__1−2−123q__213k−−113q__213kq__1kq__213kq__22−2−123q__213k−−113q__213k−q__1kq__213kq__22−2−123q__213kq__1k−q__1kq__213kq__243−123q__1+4−123q__213kq__1k−q__1kq__213kq__223−123q__12+2−123q__213k−−113q__213kq__1k−q__1kq__223−113q__12−2−123q__213kq__1k−q__1kq__213kq__223−113q__12+2−123q__213kq__1k−q__1kq__213kq__223q__12−2−123q__213k−−113q__213kq__1k−q__1kq__223−123q__12−2−−113q__213kq__1k−q__1kq__213kq__223−123q__12+−123q__213k−−113q__213kq__1kq__213kq__22−123+−123q__213k−−113q__213k−q__1kq__213kq__22−123−−123q__213k−−113q__213kq__1kq__213kq__22−113−−123q__213k−−113q__213k−q__1kq__213kq__22−113+2−123q__213k−−113q__213kq__1kq__213kq__132q__2−2−123q__213k−−113q__213k−q__1kq__213kq__132q__2+2−123q__213k−−113q__213kq__1k−q__1kq__13+2−−113q__213kq__1k−q__1kq__213kq__13zk2q__213k−q__1kq__1+q__213q__1kq__1−q__1+q__213−−113q__213k−q__1+−113q__213q__1+−113q__213−123q__213k−123q__213+q__1−123q__213−q__1−113+12−113−1q__2
∑n=0∞−q__12−n2+−1nq__12−n2+2q__2−13−n3q__152−2q__1−n2+12q__2−−1nq__1−1−n2q__22+2q__223−n3q__1−q__1−1−n2q__22zn2q__13−q__22q__132+q__2+∑n=0∞q__1q__2−n−1z3nq__243+q__223q__1+q__12+∑n=0∞−q__2−n−23z3n+1q__243+q__223q__1+q__12
Recurrence relations returned automatically if no closed form can be found, with initial conditions. Non-linear (degree 2) recurrences can be computed.
convertarcsinz3,FormalPowerSeries
arcsinz3
convertarcsinz3,FormalPowerSeries,recurrence
k4ak−2k+1k+2k2+2k+2ak+2+k+1k+2k+3k+4ak+4=0
∑n=0∞Anzn+1,RESoln4+4n3+6n2+4n+1An+−2n4−18n3−62n2−98n−60An+2+n4+14n3+71n2+154n+120An+4=0,An,A0=0,A1=0,A2=1,A3=0,INFO
convertzⅇz−1,FormalPowerSeries
zⅇz−1
convertzⅇz−1,FormalPowerSeries,recurrence
∑n=0∞Anzn,RESolAn+3+An+2+∑_k=1n+2A_kAn+3−_kn+4,An,A0=1,A1=−12,A2=112,INFO
convertLambertWz,FormalPowerSeries
LambertWz
convertLambertWz,FormalPowerSeries,recurrence
∑n=0∞Anzn,RESolAn+4+An+3+∑_k=1n+2_k+1A_k+1An+3−_kn+3,An,A0=0,A1=1,A2=−1,A3=32,INFO
New method option (by default, all three methods are tried in sequence).
convertarcsinz2,FormalPowerSeries
∑n=0∞2n!24nz2n+22n+2!
convertarcsinz2,FormalPowerSeries,method=hypergeometric
convertarcsinz2,FormalPowerSeries,method=holonomic
∑n=0∞Anzn+1,RESol−n2−2n−1An+n2+5n+6An+2=0,An,A0=0,A1=1,INFO
convertarcsinz2,FormalPowerSeries,method=quadratic
∑n=0∞Anzn,RESolAn+4−4n+2An+2+∑_k=2n_k+1A_k+1n+3−_kAn+3−_k+∑_k=2n+2−_k+1A_k+1n+5−_kAn+5−_k4n+3,An,A0=0,A1=0,A2=1,A3=0,INFO
converttanz,FormalPowerSeries
∑n=0∞Anzn,RESolAn+3+−2An+1+∑_k=1n−2_k+1A_k+1An+1−_kn+2n+3,An,A0=0,A1=1,A2=0,INFO
converttanz,FormalPowerSeries,method=hypergeometric
tanz
converttanz,FormalPowerSeries,method=holonomic
converttanz,FormalPowerSeries,method=quadratic
New output option (default: combined).
convertsinz+cosz3,FormalPowerSeries
∑n=0∞−−1n9n−3z2n22n!+∑n=0∞3−1n9n+1z2n+122n+1!
convertsinz+cosz3,FormalPowerSeries,output=combined
convertsinz+cosz3,FormalPowerSeries,output=expanded
∑n=0∞−−1n9nz2n22n!+∑n=0∞3−1nz2n22n!+∑n=0∞3−1n9nz2n+122n+1!+∑n=0∞3−1nz2n+122n+1!
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