# Is $\mathrm{EllipticPi}(n,\varphi, m)$ periodic for $n>1?$

The Wolfram function site lists the quasi-periodicity relation for the incomplete elliptic integral EllipticPi restricted to $-1\le n\le 1$ although actually it seems to work for $n<-1$ and $n \sin^2 \varphi < 1$ as well. $$\Pi(n,\varphi+k\pi,m)=2k\,\Pi(n,m) + \Pi(n,\varphi,m), \quad k \in \mathbb{Z},\; -1\le n \le 1$$ I could not find any Mathematica documention for a relation when $n \sin^2 \varphi > 1;$ and the values from other CAS differ from Mathematica

Pi(n,phi,m) for n=2, phi=1+k*pi, m=1/2
Mathematica (10.3.1 Linux/ARM)
k=0    0.704583746768798 - 1.813799364234218 I
k=8    0.704583746768798 - 1.813799364234218 I
k=-4   0.704583746768798 - 1.813799364234218 I
MPMath
k=0    0.704583746768798  -  1.813799364234217 I
k=8   -4.312131188674144  - 30.834589191981706 I
k=-4   3.212941214490268  + 12.696595549639525 I
Maxima
k=0    0.704583746768798
k=8   -4.312131188674136
k=-4   3.212941214490266


So actually the Mathematica results seems to be periodic for $n> 1.$

Is this periodicity a bug or a documented feature?

Edit: Here the exported Mathematica text

In[1]:= N[EllipticPi[2,1, 1/2],15]
Out[1]= 0.70458374676880-1.81379936423422 I
In[2]:= N[EllipticPi[2,1+8*Pi, 1/2],15]
Out[2]= 0.70458374676880-1.81379936423422 I
In[3]:= N[EllipticPi[2,1-4*Pi, 1/2],15]
Out[3]= 0.70458374676880-1.81379936423422 I

• It's not a bug.For n>1 integral of EllipicPi converages only if fi < ArcSin[1/Sqrt[n]].See 62:12:3 on page 661 in the book books.google.pl/… – Mariusz Iwaniuk May 22 '18 at 14:40
• @mariusz-iwaniuk: IMO 62:12:3 lists the range restriction for their program Equator but says nothing about periodicity. – gammatester May 22 '18 at 14:46
• Look at math.stackexchange.com/questions/17182/… to this end. The question is not related to Mathematica, but to math. – user64494 May 22 '18 at 16:49
• @user64494: No this is related to Mathematica (obviously because either it is a bug or feature). IMO the 'answers' in Math.StackExchange do not apply: The accepted answer ignores the fact, that the two integrals at Wikipedia are the same only if $|\varphi| \le \pi/2$ (and cannot explain the quasi-periodicity for $n<1$). The second 'answer' declares itself as a comment and is restricted to $0<m<1.$ And if it were correct, it would show that $\Pi(n,\varphi,m)$ is not periodic because $F(\varphi, m)$ is without doubt quasi-periodic. – gammatester May 22 '18 at 17:48
• The Maple command MmaTranslator:-FromMma("EllipticPi[a,z, b]") outputs EllipticPi(sin(z), a, sqrt(b)). There is no restrictions on the parameters a and b in Maple. These two statements implies the periodicity of EllipticPi in Maple. It should be noticed that the values of EllipticPi[a,z, b] in Mathematica and EllipticPi(sin(z), a, sqrt(b)) in Maple differ. I find nothing about the periodicity under consideration in Abramowitz&Stegun. – user64494 May 23 '18 at 4:04