# Why cant this limit be evaluated?

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### Mark “Essa King” Sukaiti

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Feb 19, 2024, 3:25:51 PMFeb 19
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D(r)=-1/2*((3*r^2 - 1)*elliptic_ec((4*r/(r^2 + 2*r + 1)))^2 + (r^2 - 2*r + 1)*elliptic_ec((4*r/(r^2 + 2*r + 1)))*elliptic_kc((4*r/(r^2 + 2*r + 1))))/(pi^2*r^8 - 2*pi^2*r^7 - pi^2*r^6 + 4*pi^2*r^5 - pi^2*r^4 - 2*pi^2*r^3 + pi^2*r^2)

D(r).limit(r=0)

The limit should be -0.125 (or -1/8) but it seems maxima doesnt know the limit of elliptic_ec(x) for x->0
Sympy also fails giving -5/8.

The other algorithms cant evaluate it either.

### Emmanuel Charpentier

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Feb 20, 2024, 2:40:13 AMFeb 20
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Indeed. That’s probably an oversight :

sage: elliptic_kc(x).limit(x=0) 1/2*pi sage: elliptic_ec(x).limit(x=0) limit(elliptic_ec(x), x, 0)

Curioisly :

sage: elliptic_ec(0) 1/2*pi

FWIW :

sage: D(r)._mathematica_().Limit(mathematica.Rule(r, 0)) -1/8

HTH,

### dim...@gmail.com

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Feb 20, 2024, 7:29:12 AMFeb 20
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On Mon, Feb 19, 2024 at 11:40:13PM -0800, Emmanuel Charpentier wrote:
>
> Indeed. That’s probably an oversight :
> sage: elliptic_kc(x).limit(x=0) 1/2*pi sage: elliptic_ec(x).limit(x=0)
> limit(elliptic_ec(x), x, 0)
>

this looks like a maxima bug:

(%i17) elliptic_ec(0);
%pi
(%o17) ---
2
(%i18) limit(elliptic_ec(x),x,0);
(%o18) limit elliptic_ec(x)
x -> 0
(%i19) elliptic_kc(0);
%pi
(%o19) ---
2
(%i20) limit(elliptic_kc(x),x,0);
%pi
(%o20) ---
2

No idea why limit(elliptic_ec(x),x,0) is a problem for Maxima.

> Curioisly :
> sage: elliptic_ec(0) 1/2*pi
>
> FWIW :
> sage: D(r)._mathematica_().Limit(mathematica.Rule(r, 0)) -1/8
>
> HTH,
> ​
> Le lundi 19 février 2024 à 21:25:51 UTC+1, Mark “Essa King” Sukaiti a
> écrit :
>
> > D(r)=-1/2*((3*r^2 - 1)*elliptic_ec((4*r/(r^2 + 2*r + 1)))^2 + (r^2 - 2*r +
> > 1)*elliptic_ec((4*r/(r^2 + 2*r + 1)))*elliptic_kc((4*r/(r^2 + 2*r +
> > 1))))/(pi^2*r^8 - 2*pi^2*r^7 - pi^2*r^6 + 4*pi^2*r^5 - pi^2*r^4 -
> > 2*pi^2*r^3 + pi^2*r^2)
> >
> > D(r).limit(r=0)
> >
> > The limit should be -0.125 (or -1/8) but it seems maxima doesnt know the
> > limit of elliptic_ec(x) for x->0
> > [image: 2024-02-20_00-22.png]
> > Sympy also fails giving -5/8.
> >
> > The other algorithms cant evaluate it either.
> >
>
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### Oscar Benjamin

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Feb 24, 2024, 6:28:17 AMFeb 24
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On Mon, 19 Feb 2024 at 20:25, Mark “Essa King” Sukaiti <xzark....@gmail.com> wrote:
D(r)=-1/2*((3*r^2 - 1)*elliptic_ec((4*r/(r^2 + 2*r + 1)))^2 + (r^2 - 2*r + 1)*elliptic_ec((4*r/(r^2 + 2*r + 1)))*elliptic_kc((4*r/(r^2 + 2*r + 1))))/(pi^2*r^8 - 2*pi^2*r^7 - pi^2*r^6 + 4*pi^2*r^5 - pi^2*r^4 - 2*pi^2*r^3 + pi^2*r^2)

D(r).limit(r=0)

The limit should be -0.125 (or -1/8) but it seems maxima doesnt know the limit of elliptic_ec(x) for x->0

Sympy also fails giving -5/8.

To follow up on this the SymPy bug is now fixed in master:

In [1]: e = ((1 - 3*x**2)*elliptic_e(4*x/(x**2 + 2*x + 1))**2/2 - (x**2 - 2*x + 1)*elliptic_e(4*x/(x*
...: *2 + 2*x + 1))*elliptic_k(4*x/(x**2 + 2*x + 1))/2)/(pi**2*x**8 - 2*pi**2*x**7 - pi**2*x**6 +
...: 4*pi**2*x**5 - pi**2*x**4 - 2*pi**2*x**3 + pi**2*x**2)

In [2]: e.limit(x, 0)
Out[2]: -1/8

Oscar
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