I asked an LLM to provide an expansion of the MacDonald function of arbitrary order (a modified Bessel function of the second kind with purely imaginary order and positive argument), K(I*y,r), as a weighted sum of MacDonald functions of integer order. It came back with

         K(I*y,z)=2*sinh(Pi*y)/Pi* [K(0,r)/2*y+sum( (-1)^n*y*BesselK(0,r)/(y^2+n^2),n=1..infinity)]

(see below for more readable text)

I evaluated the LHS and RHS using MAPLE 2026 for various choices of y and r and found numerical agreement using both "sum" and "Sum".  I was very pleased until I realized that the RHS isn't a convergent series!

Can anyone explain to me how MAPLE pulls this off! 

(I asked MAPLE Tech Support but they said it is above their pay grade... I suspect that MAPLE is using Borel summability to evaluate the RHS but I haven't been able to verify that)

I apologize, but I can't see how to attach a .mw file, so I've cut and pasted the code below

restart;

T := R(xi)*R(xi) + lambda;

u := A[0] + A[1]*R(xi) + B[1]/R(xi);

d[1] := A[1]*T - B[1]*T/R(xi)^2;

d[2] := 2*A[1]*R(xi)*T - 2*B[1]*T/R(xi) + 2*B[1]*(R(xi)^2 + lambda)*T/R(xi)^3;

expand(((-alpha^2*b^2 + a^2)*alpha^2)/(2*beta)*d[2] + (omega + alpha^2*(alpha^2*l^2 + k^2)/2 - a*C[1]/(-alpha^2*b^2 + a^2))*u[0]/(beta - 2*beta*a^2/(-alpha^2*b^2 + a^2)) + u[0]*u[0]*u[0]);

value(%);

simplify(%);

collect(%, R(xi));

      /      6  4    4      2\      3
 A[1] \-alpha  b  + a  alpha / R(xi) 
 ------------------------------------
             /     2  2    2\        
        beta \alpha  b  + a /        

                  /      6  4    4      2\         
      A[1] lambda \-alpha  b  + a  alpha / R(xi)   
    + ------------------------------------------ + 
                     /     2  2    2\              
                beta \alpha  b  + a /              

                         /                               /     
             1           |/                    B[1] \    |     
   --------------------- ||A[0] + A[1] R(xi) + -----|[0] |beta 
        /     2  2    2\ \\                    R(xi)/    \     
   beta \alpha  b  + a /                                       

                                                  2
   /     2  2    2\ /                    B[1] \    
   \alpha  b  + a / |A[0] + A[1] R(xi) + -----|[0] 
                    \                    R(xi)/    

      1  2  2      6   1 /  2  2    2  2\      4
    + - b  l  alpha  + - \-a  l  + b  k / alpha 
      2                2                        

                                                       \\
      /  1  2  2    2      \      2    2               ||
    + |- - a  k  + b  omega| alpha  - a  omega + a C[1]||
      \  2                 /                           //

                  /      6  4    4      2\
      B[1] lambda \-alpha  b  + a  alpha /
    + ------------------------------------
               /     2  2    2\           
          beta \alpha  b  + a / R(xi)     

            6  4       2         4      2       2     
      -alpha  b  lambda  B[1] + a  alpha  lambda  B[1]
    + ------------------------------------------------
                     /     2  2    2\      3          
                beta \alpha  b  + a / R(xi)           

WHen I open many worksheets at same time, say 10. The new UI do not stack them all (i.e. the tab at the top), forcing one to use the small arrow to navigate to each worksheet.

Is there a way to tell the UI to show all tabs (may be double rows and 3 rows as needed) to make it easier to jump from one worksheet to the other?

I do not know if this is new feature in the new ribbon UI or not. 

Here is screen show where I have 10 worksheets open

There is also a pull down menu, but it only shows 8 worksheets and one can have more open but they do not show. So have to scroll down looking for the rest. Even that does not work well. many times when I try to scroll down, the window closes. It will not give me time to move the mouse to the scroll bar to move it before it closes.

Both of these solutions are not good. Having to use the arrow key to look and navigate for a different worksheet is bad UI design.

How to see all tabs for all open worksheet in same UI?  If the tabs do not fit on one row, why not make second row? If two rows do not fit, make 3rd row. This should be an option for the user. But I did not see one so far. But will keep looking.

I find tabs where all worksheet show much better design that this UI design.   

I only use worksheet and not document mode. Windows 10.

To give you idea what I mean, These are examples found on the net of stacked tabs

Where in Maple, each tab above will have the name of the worksheet open. Font can be small, is OK.

Is it possible to have this in the new UI for open worksheets?

One option I might try to make my worksheets names much shorter. May be then they will fit all in same window.

The below problem has already occured several times to me. In all such instances Maple did not realise that extracting a factor from a square root is the key for further simplification. Doing this by hand is obvious and often easy when extracted factors are positive.  

Did I overlook something? Are there other ways avoid disassembling an expression with the op command?
Should simplify or other commands be improved to adress such problems?

Download Simplify_radical_02.mw

Help me rewrite the code to create visible Bar chart of different colors. I can't figure out why this code is not giving me a visible bar graph

Download Bargraph.mw

restart;
with(plottools);
with(plots);
a := 1;
b := 1;
c := 1;
k := 1;
l := 1;
omega := 1;
A[2] = 2;
alpha := 2;
beta := 1;
kappa := 0.5;
C[1] := 1;
lambda := -1;

omega := (-alpha^6*b^4*lambda + 2*alpha^6*b^2*l^2 - 2*a^2*alpha^4*l^2 + 2*alpha^4*b^2*k^2 + a^4*alpha^2*lambda - 2*a^2*alpha^2*k^2 + 4*a*C[1])/(-4*alpha^2*b^2 + 4*a^2);

a[0] := 0;

a[1] := sqrt(-(-alpha^2*b^2 + a^2)/(4*beta))*alpha;

b[1] := sqrt(-(alpha^2*b^2*lambda*sigma - a^2*lambda*sigma)/(4*beta))*alpha;

sigma := A[1]*A[1] - A[2]*A[2];

T := A[1]*sinh(xi*sqrt(-lambda)) + A[2]*cosh(xi*sqrt(-lambda)) + mu/lambda;

t := diff(T, xi);

S := t/T;

R := 1/T;

mu := 0;

A[1] := 0;

y := 0;

xi := k*x^kappa/kappa + l*y^kappa/kappa - omega*t^kappa/kappa;

  Error, recursive assignment

Dear all,

I'm reporting what seems to me as a bug in the SMTLIB library in maple. 

    |\^/|     Maple 2026 (X86 64 LINUX)
._|\|   |/|_. Copyright (c) Maplesoft, a division of Waterloo Maple Inc. 2026
 \  MAPLE  /  All rights reserved. Maple is a trademark of
 <____ ____>  Waterloo Maple Inc.
      |       Type ? for help.
> SMTLIB:-Satisfiable({x^2=2,y^2=2,x<y});
                                     true

> SMTLIB:-Satisfiable({x^2=2,y^2=2,y<x});
                                     false

> SMTLIB:-Satisfiable({x^2=2,a^2=2,a<x});
                                     true

The Satisfiable command do not output the correct decision on two formulas of equivalent realization by switching x<y (output SAT) to y<x (output UNSAT). I suspect this is because some alphabetical order depandance in the variables as for a<y we get SAT again.

I tried to feed Z3 with the code given by ToString on the problematic formula and I get two different outputs :

• on the Z3 version 4.8.12 from the ubuntu repository (apt install) I also get the wrong UNSAT output;
• one the Z3 version 4.17.0 build from the official github repository I finally get the correct SAT output.

Thus, I suspect a version problem in SMTLIB that do not take in account the last updates made in SMT solvers (Z3?).

Many thanks for considering my problem!

restart;
solve({-alpha^4*b^2*lambda*mu*b[1] + a^2*alpha^2*lambda*mu*b[1] + 6*beta*lambda^2*sigma*a[0]*a[1]^2 + 6*beta*mu^2*a[0]*a[1]^2 - 6*beta*lambda*a[0]*b[1]^2 = 0, -alpha^4*b^2*lambda^2*mu*sigma - alpha^4*b^2*mu^3 + a^2*alpha^2*lambda^2*mu*sigma + a^2*alpha^2*mu^3 - 4*beta*lambda^2*sigma*a[0]*b[1] - 4*beta*lambda*mu*b[1]^2 - 4*beta*mu^2*a[0]*b[1] = 0, -alpha^4*b^2*lambda^2*sigma - alpha^4*b^2*mu^2 + a^2*alpha^2*lambda^2*sigma + a^2*alpha^2*mu^2 + beta*lambda^2*sigma*a[1]^2 + beta*mu^2*a[1]^2 - 3*beta*lambda*b[1]^2 = 0, -alpha^4*b^2*lambda^2*sigma - alpha^4*b^2*mu^2 + a^2*alpha^2*lambda^2*sigma + a^2*alpha^2*mu^2 + 3*beta*lambda^2*sigma*a[1]^2 + 3*beta*mu^2*a[1]^2 - beta*lambda*b[1]^2 = 0, -alpha^6*b^4*lambda^2*mu*b[1] + alpha^6*b^2*l^2*lambda^2*sigma*a[0] + alpha^6*b^2*l^2*mu^2*a[0] - a^2*alpha^4*l^2*lambda^2*sigma*a[0] + alpha^4*b^2*k^2*lambda^2*sigma*a[0] - a^2*alpha^4*l^2*mu^2*a[0] + alpha^4*b^2*k^2*mu^2*a[0] + 2*alpha^2*b^2*beta*lambda^2*sigma*a[0]^3 + a^4*alpha^2*lambda^2*mu*b[1] - a^2*alpha^2*k^2*lambda^2*sigma*a[0] - 6*alpha^2*b^2*beta*lambda^2*a[0]*b[1]^2 + 2*alpha^2*b^2*beta*mu^2*a[0]^3 - a^2*alpha^2*k^2*mu^2*a[0] + 2*a^2*beta*lambda^2*sigma*a[0]^3 + 2*alpha^2*b^2*lambda^2*omega*sigma*a[0] - 6*a^2*beta*lambda^2*a[0]*b[1]^2 + 2*a^2*beta*mu^2*a[0]^3 + 2*alpha^2*b^2*mu^2*omega*a[0] - 2*a^2*lambda^2*omega*sigma*a[0] - 2*a^2*mu^2*omega*a[0] + 2*a*lambda^2*sigma*C[1]*a[0] + 2*a*mu^2*C[1]*a[0] = 0, -2*alpha^6*b^4*lambda^3*sigma - 2*alpha^6*b^4*lambda*mu^2 + alpha^6*b^2*l^2*lambda^2*sigma + alpha^6*b^2*l^2*mu^2 - a^2*alpha^4*l^2*lambda^2*sigma + alpha^4*b^2*k^2*lambda^2*sigma + 2*a^4*alpha^2*lambda^3*sigma - a^2*alpha^4*l^2*mu^2 + alpha^4*b^2*k^2*mu^2 + 6*alpha^2*b^2*beta*lambda^2*sigma*a[0]^2 + 2*a^4*alpha^2*lambda*mu^2 - a^2*alpha^2*k^2*lambda^2*sigma - 6*alpha^2*b^2*beta*lambda^2*b[1]^2 + 6*alpha^2*b^2*beta*mu^2*a[0]^2 - a^2*alpha^2*k^2*mu^2 + 6*a^2*beta*lambda^2*sigma*a[0]^2 + 2*alpha^2*b^2*lambda^2*omega*sigma - 6*a^2*beta*lambda^2*b[1]^2 + 6*a^2*beta*mu^2*a[0]^2 + 2*alpha^2*b^2*mu^2*omega - 2*a^2*lambda^2*omega*sigma - 2*a^2*mu^2*omega + 2*a*lambda^2*sigma*C[1] + 2*a*mu^2*C[1] = 0, -alpha^6*b^4*lambda^3*sigma + alpha^6*b^4*lambda*mu^2 + alpha^6*b^2*l^2*lambda^2*sigma + alpha^6*b^2*l^2*mu^2 - a^2*alpha^4*l^2*lambda^2*sigma + alpha^4*b^2*k^2*lambda^2*sigma + a^4*alpha^2*lambda^3*sigma - a^2*alpha^4*l^2*mu^2 + alpha^4*b^2*k^2*mu^2 + 6*alpha^2*b^2*beta*lambda^2*sigma*a[0]^2 - a^4*alpha^2*lambda*mu^2 - a^2*alpha^2*k^2*lambda^2*sigma - 2*alpha^2*b^2*beta*lambda^2*b[1]^2 + 12*alpha^2*b^2*beta*lambda*mu*a[0]*b[1] + 6*alpha^2*b^2*beta*mu^2*a[0]^2 - a^2*alpha^2*k^2*mu^2 + 6*a^2*beta*lambda^2*sigma*a[0]^2 + 2*alpha^2*b^2*lambda^2*omega*sigma - 2*a^2*beta*lambda^2*b[1]^2 + 12*a^2*beta*lambda*mu*a[0]*b[1] + 6*a^2*beta*mu^2*a[0]^2 + 2*alpha^2*b^2*mu^2*omega - 2*a^2*lambda^2*omega*sigma - 2*a^2*mu^2*omega + 2*a*lambda^2*sigma*C[1] + 2*a*mu^2*C[1] = 0}, {omega, a[0], a[1], b[1]});
 

Any explanation why this happens? notice, I did not supply the x and y ranges, let Maple decide.

Download bug_in_contourplot.mw

nowday i am very intrested in this method which really i something intresting  i want to generate all the type of function not just thus in this paper  i will update the other  layers too but need sometime, i try to apply the exact layer  as author did in this paper but i did something mistake in one part about finding the parameters  i didnt fix that part but i did the second  part of the paper and i got answer, but for first one i need some help, also if possible  it will be amazing if someone can construct the all layer which can be construct , this is just two cases i provide here and in one of them i am stuck and i can't find the parameter but other i did it , target  of finding in apply the formula  for more  hiden layer  becuase in other papers i saw a lot of the  hiden layer i want to apply all of them if possible not jsut tw or three of them i want to know  and see how many hiden layer exist by constructing generate function, and the graph i am not sure maple can do that or not like fig 1 and fig 2 which if do that it will  be more amazing 

thank you for any help 

t1 have problem about  first case of layer and t2 dont have problem

I asked Maple AI what a glyph is. Then I prompted this

A kernel lost message was returned and the AI pannel became irresponsive.

Maple is still running well in exsisting and new tabs. 

Can the AI service be restarted from the user interface?

(Is that crash reproducible?)

Edit:

For exercises involving Pick's Theorem, I need grid points within a Cartesian coordinate system. How can "all" grid points - at least within the first quadrant - be generated without the tedious manual entry of integer coordinates? Is it possible to draw grid polygons as closed polylines simply by clicking on the grid points? (BTW: At the moment, this works well in the good old "Cabri.")
My search within the "Help" section (using terms such as plot, grid, mesh, lattice, etc.) proved unsuccessful.

The HTML characters in the attached document cause problems here on MaplePrimes. You have to open the worksheet

Download HTML_characters_in_math_mode.mw

05-2-2.mws

Can you help me with this code?

Download 05-2-2.mws

Can anyone share additional information about the Maple conference to be held in 2026? I want to submit a talk and then submit a paper to the Maple Transactions journal based on the same.