Services & Resources / Wolfram Forums
-----
 /
MathGroup Archive
1999
*January
*February
*March
*April
*May
*June
*July
*August
*September
*October
*November
*December
*Archive Index
*Ask about this page
*Print this page
*Give us feedback
*Sign up for the Wolfram Insider

MathGroup Archive 1999

[Date Index] [Thread Index] [Author Index]

Search the Archive

Problems with EllipticE[p,1/k] - bug or property?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg20580] Problems with EllipticE[p,1/k] - bug or property?
  • From: Robert Prus <robert at fuw.edu.pl>
  • Date: Sat, 30 Oct 1999 14:54:54 -0400
  • Organization: Warsaw University, Physics Department, Institute of Theoretical Physics
  • Sender: owner-wri-mathgroup at wolfram.com

Hi,

I try to check the identity:

EllipticE[ArcSin[Sqrt[k]Sin[p]],1/k]-(EllipticE[p,k]-(1-k)EllipticF[p,k])/Sqrt[k]==0

(it is taken from one of the the books: Abramowitz, Stegun or Gradshteyn,
Ryzhik with some corrections due to Mathematica notation).

Here are the calculations (I use Mathematica 3.0 for SGI):

First of all I check that functions EllipticE[p,k] and EllipticF[p,k] have
period Pi in p (in the sense of identities:

EllipticE[p+Pi,k] == 2EllipticE[k]+EllipticE[p,k]
EllipticF[p+Pi,k] == 2EllipticK[k]+EllipticF[p,k] ):


Mathematica 3.0 for Silicon Graphics
Copyright 1988-97 Wolfram Research, Inc.
 -- Motif graphics initialized -- 

In[1]:= Table[EllipticE[p+Pi,k]-(2EllipticE[k]+EllipticE[p,k])/.{p->Random[],k->Random[]}//Chop,{10}]

Out[1]= {0, 0, 0, 0, 0, 0, 0, 0, 0, 0}

In[2]:= Table[EllipticE[p+Pi,k]-(2EllipticE[k]+EllipticE[p,k])/.{p->Random[Real,{-4Pi,4Pi}],k->Random[Real,{-10,10}]}//Chop,{10}]

Out[2]= {0, 0, 0, 0, 0, 0, 0, 0, 0, 0}

In[3]:= Table[EllipticF[p+Pi,k]-(2EllipticK[k]+EllipticF[p,k])/.{p->Random[],k->Random[]}//Chop,{10}]

Out[3]= {0, 0, 0, 0, 0, 0, 0, 0, 0, 0}

In[4]:= Table[EllipticF[p+Pi,k]-(2EllipticK[k]+EllipticF[p,k])/.{p->Random[Real,{-4Pi,4Pi}],k->Random[Real,{-10,10}]}//Chop,{10}]

Out[4]= {0, 0, 0, 0, 0, 0, 0, 0, 0, 0}


then I check the identity for EllipticE[p,1/k]:


In[5]:= eq=EllipticE[ArcSin[Sqrt[k]Sin[p]],1/k]-(EllipticE[p,k]-(1-k)EllipticF[p,k])/Sqrt[k]

                                          1
Out[5]= EllipticE[ArcSin[Sqrt[k] Sin[p]], -] - 
                                          k
 
     EllipticE[p, k] - (1 - k) EllipticF[p, k]
>    -----------------------------------------
                      Sqrt[k]

In[6]:= Table[eq/.{p->Random[Real,{-Pi/2,Pi/2}],k->Random[Real,{-10,10}]}//Chop,{10}]

Out[6]= {0, 0, 0, 0, 0, 0, 0, 0, 0, 0}

In[7]:= Table[eq/.{p->Random[Real,{0,Pi}],k->Random[Real,{-10,10}]}//Chop,{10}]

Out[7]= {0, 0.520936 I, 0, 0, -1.90617 I, 0.363113 I, 0.0053501 I, 
 
>    -1.00235 I, 0, 1.22196 I}

In[8]:= Table[eq/.{p->Random[Real,{-Pi,Pi}],k->Random[Real,{-10,10}]}//Chop,{10}]

Out[8]= {0, 0, -0.39191, 0, -0.0333185 I, 0, 0.364729 I, 0, 0, 0.0519677 I}


then I can plot the values of eq:


In[9]:= f[pp_,kk_]:=Chop[N[eq/.{p->pp,k->kk}]]

In[10]:= Plot3D[Re[f[p,k]],{p,-2Pi,2Pi},{k,-10,10}]

Out[10]= -SurfaceGraphics-

In[11]:= Plot3D[Im[f[p,k]],{p,-2Pi,2Pi},{k,-10,10}]

Out[11]= -SurfaceGraphics-


I don't understand why Out[7] and Out[8] have entries different than 0.

Maybe the identity I check is valid only for p in interval (-Pi/2,Pi/2)?

Any comments?

RP

--------------------
Robert Prus, robert at fuw.edu.pl
Institute of Theoretical Physics, Warsaw University
Hoza 69, 00-681 Warsaw, Poland


  • Prev by Date: Re: From Mathematica to EPS
  • Previous by thread: Re: Assumptions in Integrate