Try to understand what Sum does, if the upper bound is not an integer. That is what you see.

h:= n -> add(p, p=1 .. n);
plot('h'(n), n=1 .. 6.5);

Try to understand what Sum does, if the upper bound is not an integer. That is what you see.

h:= n -> add(p, p=1 .. n);
plot('h'(n), n=1 .. 6.5);

Why is it broken or weird?

What do you expect for Sum( f(p), p = 1 .. m), if m is set to some non-integer,
like it happens if it is plotted?

Why is it broken or weird?

What do you expect for Sum( f(p), p = 1 .. m), if m is set to some non-integer,
like it happens if it is plotted?

I hesitated first, since the question was about Argument(Fct), but up to jumping
from + to - for Fct values close to the negative axis that is ok.

And with 7 terms (using 15 Digits) I get enough desired relative precision:

 'JacobiTheta1(z,q) = 
    Sum(2*q^((_k1+1/2)^2)*sin(z*(2*_k1+1))*(-1)^_k1,_k1 = 0 .. infinity)';
  subs(infinity=6, %); #subs(Sum = add, %);
  subs(q=nome, %);
  subs(a=4, b=0, %);

  tst:=evalf(%);
  abs(1 - rhs(%)/lhs(%));


                             29                       30
        -0.858032541978655 10   + 0.715689869090575 10   I =
                             29                       30
        -0.857653930247745 10   + 0.715705552048839 10   I

                        0.0000568533871177845

  map(argument, tst);
  abs(1 - rhs(%)/lhs(%));

                 1.69011570515771 = 1.69006096404286

                          0.000032388974721

I hesitated first, since the question was about Argument(Fct), but up to jumping
from + to - for Fct values close to the negative axis that is ok.

And with 7 terms (using 15 Digits) I get enough desired relative precision:

 'JacobiTheta1(z,q) = 
    Sum(2*q^((_k1+1/2)^2)*sin(z*(2*_k1+1))*(-1)^_k1,_k1 = 0 .. infinity)';
  subs(infinity=6, %); #subs(Sum = add, %);
  subs(q=nome, %);
  subs(a=4, b=0, %);

  tst:=evalf(%);
  abs(1 - rhs(%)/lhs(%));


                             29                       30
        -0.858032541978655 10   + 0.715689869090575 10   I =
                             29                       30
        -0.857653930247745 10   + 0.715705552048839 10   I

                        0.0000568533871177845

  map(argument, tst);
  abs(1 - rhs(%)/lhs(%));

                 1.69011570515771 = 1.69006096404286

                          0.000032388974721

A nice estimate!

For the search one can use the function isqrt (integer square root, having lower costs
and gives a proof, in case that sqrt(n) is an integer, but Maple have missed it [I felt
unsure, if it checks a square root reminder]):

  a:=4^27+4^1016;
  for j from 1 to 2032 do
    J:= a+4^j;
    S:=isqrt(a+4^j);
    if S*S = J then print(j) end if;
  end do: j:='j':

                                 522
                                 2004

A nice estimate!

For the search one can use the function isqrt (integer square root, having lower costs
and gives a proof, in case that sqrt(n) is an integer, but Maple have missed it [I felt
unsure, if it checks a square root reminder]):

  a:=4^27+4^1016;
  for j from 1 to 2032 do
    J:= a+4^j;
    S:=isqrt(a+4^j);
    if S*S = J then print(j) end if;
  end do: j:='j':

                                 522
                                 2004

Alejandro,

Thank you looking into details for that issue.

BTW: using 'simplify' before evaluation let it work, it brings it to the desired form (valid in the range of integration).

@Kitonum  Thx :-)

That makes me guessing, that Maple only considers the additive group
structure of Z / (n) and ignores, that it is a ring.

Also that may be an explanation, why it does not know Little Fermat
(or generalizations).

@Kitonum  Thx :-)

That makes me guessing, that Maple only considers the additive group
structure of Z / (n) and ignores, that it is a ring.

Also that may be an explanation, why it does not know Little Fermat
(or generalizations).

Strange: it can not even do it for n=1, to show that 4*m^2+4*m = 4*m*(m+1) is zero modulo 8

Strange: it can not even do it for n=1, to show that 4*m^2+4*m = 4*m*(m+1) is zero modulo 8

Nice obfuscation (I will not even try to understand the code)!

Though I was educated as Catholic I never understood the 12 days: from Christmas Night
(today: Monday, 24 Dec) to "Heilig-Drei-König" (Sunday 06 Jan) we have 2 weeks, which
is 14 days and even starting on 25 Dec these are 13 days :-)

I vaguely remember sayings, that Xmas once was 06 Jan and was fixed as 24/25 Dec, due
to imprecise calendars - and mainly because of pagan/celtic feasts for Solstice.

Anyway: ich wünsche ein frohes Fest, einen guten Rutsch und ein glückliches neues Jahr!

Usually a hard disk will not make a different machine to work, so
I do not understand your question (it only makes some sense as
additional data storage).

You may want to ask Maple's support for your special needs.
And also want to study the License agreement for your version.