MaplePrimes - comments on Post, hypergeom convert standard functions

MaplePrimes - comments on Post, hypergeom convert standard functions http://www.mapleprimes.com/posts/120598-Hypergeom-Convert-Standard-Functions en-us 2026 Maplesoft, A Division of Waterloo Maple Inc. Maplesoft Document System Wed, 10 Jun 2026 13:38:52 GMT Wed, 10 Jun 2026 13:38:52 GMT The latest comments added to the Post, hypergeom convert standard functions http://www.mapleprimes.com/images/mapleprimeswhite.jpg MaplePrimes - comments on Post, hypergeom convert standard functions http://www.mapleprimes.com/posts/120598-Hypergeom-Convert-Standard-Functions Special cases http://www.mapleprimes.com/posts/120598-Hypergeom-Convert-Standard-Functions?ref=Feed:MaplePrimes:hypergeom convert standard functions:Comments#comment120602 Maple traditionally ignores special cases. One of the well-known examples is <pre>int(x^k,x); (k + 1) x -------- k + 1 </pre> which is wrong for k=-1. In this example with hypergeom, not only conversion to StandardFunctions is wrong. Conversion to elementary is wrong even with assumptions, <pre>convert(P,elementary) assuming k::posint; (-2 k - 1) 2 1/2 (2 k + 1) 2 (1 + (z ) ) ---------------------------------- 2 1/2 (z ) eval(%,k=1); 2 1/2 3 (1 + (z ) ) -------------- 2 1/2 8 (z ) </pre> while the correct answer is <pre>convert(eval(P,k=1),elementary); 2 z 3/4 + ---- 4 </pre> Without assumptions, series are also wrong for positive integers k, <pre>series(P,z,1); (-2 k - 1) -1 (-2 k - 1) 2 z + 2 (2 k + 1) + O(z) series(P,z,3) assuming k::posint; (-2 k) 2 2 (2 k + 1) + O(z ) </pre> Alec The latest comments added to the Post, hypergeom convert standard functions 120602 Sun, 29 May 2011 00:40:48 Z Alec Mihailovs Alec Mihailovs So I guess this hypergeometric is worse than http://www.mapleprimes.com/posts/120598-Hypergeom-Convert-Standard-Functions?ref=Feed:MaplePrimes:hypergeom convert standard functions:Comments#comment120604 So I guess this hypergeometric is worse than I thought.  1 is on the boundary of the circle of convergence, so the behavior there (z=0 in the worksheet) is from analytic continuation.  When k is not a positive integer there is a pole there, but when k is a positive integer it is a polynomial and has no pole there.  The residue for the pole does not go to zero as k approaches an integer.  --- G A Edgar The latest comments added to the Post, hypergeom convert standard functions 120604 Sun, 29 May 2011 01:29:19 Z edgar edgar Yes http://www.mapleprimes.com/posts/120598-Hypergeom-Convert-Standard-Functions?ref=Feed:MaplePrimes:hypergeom convert standard functions:Comments#comment120612 Yes, it's that weird case in which approaching [-1, -1/2, -2] by [a, b, c] along different curves (in the (a,b,c)-space) gives different limits for the hypergeom. In particular, <pre>evalf(Limit(hypergeom([-k,1/2-k],[-2*k],0.99),k=1)); 1.663750000 evalf(Limit(hypergeom([-k,1/2-k],[3*k-5],0.99),k=1)); 0.1450000000 </pre> while <pre>eval(3/4+z^2/4,z=0.1); 0.7525000000 </pre> The singularity at z=0 for non-integer k is not actually a pole in its usual sense - because the function is even. There is a pole on every branch though (there are 2 branches), with residues of the opposite sign, and a branch cut coming through them. The series given by Maple for P is correct only in the right half-plane (with Re(z)>0 or Re(z)=0 and Im(z)>0 - in other words, with csgn(z)=1) and should have the opposite sign at z-1 and other odd powers of z in the left half plane (with Re(z)<0 or Re(z)=0 and Im(z)<0 - i.e. with csgn(z)=-1). The residue defined through the contour integral (with a half of the contour on one branch and another half - on another branch, so it is not actually a contour) is 0 - the integral on the right hand side of it cancels with the integral on the left hand side, and the value given by Maple, <pre>residue(P,z=0); (-2 k - 1) 2 </pre> is just plain wrong - either residues of both poles should be listed - this one and one with the opposite sign, or their sum which is 0. Otherwise the total sum of residues will be wrong. By the way, the series for sqrt(z^2) and 1/sqrt(z^2) etc. are also wrong, <pre>series(sqrt(z^2),z); z </pre> Simplifying a function helps, <pre>series(simplify(sqrt(z^2)),z); csgn(z) z series(simplify(P),z,1); (-k) -1 0 1/2 4 csgn(z) z + O(z ) simplify(P); (-k) (2 k + 1) 4 csgn(z) (1 + csgn(z) z) 1/2 -------------------------------------- z </pre> (which is again wrong for positive integer k). But no help for the residue, it's actually getting even worse, including z in the answer, <pre>residue(simplify(P),z=0); csgn(z) 1/2 ------- k 4 </pre> Also, that gives a proof that 0=1 in Maple, <pre>f:=(sqrt(z^2)-z)/2: eval(series(f,z)=series(simplify(f),z),z=-1); 0 = 1 </pre> The following is interesting and puzzling a little bit, <pre>eval(simplify(P),k=1) assuming z>0: #limit on one branch eval(simplify(P),k=1) assuming z<0: #limit on another branch expand(%+%%); #the sum of limits on both branches 2 z 3/4 + ---- 4 </pre> Alec The latest comments added to the Post, hypergeom convert standard functions 120612 Sun, 29 May 2011 06:10:52 Z Alec Mihailovs Alec Mihailovs