limit( EllipticF(x*a,r), x=infinity) = ?

Posted on 2008-09-06 17:27 By Axel Vogt ( Maple Rating 4

994)

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How do I ...? with Maple

What is limit( EllipticF(x*a,r), x=infinity), a and r complex?

The usual 'limit' returns unevaluated.

'MultiSeries:-limit' gives a result, but it seems to be wrong,
especially for a=I, r=1, but also for other values (no, it is
not a question of Digits):

  Tst := [alpha = (-2/(1+2*I*2^(1/2)))^(1/2), 
            rho = ((1+2*I*2^(1/2))/(1-2*I*2^(1/2)))^(1/2)];

  EllipticF(x*alpha,rho); 
  eval(%,x=2^100): # large value instead of limit 
  eval(%,Tst):
  evalf(%);
                  1.0010773804561 + 1.4157372084260 I


  EllipticF(x*alpha,rho);
  MultiSeries:-limit(%, x=infinity);
  eval(%,Tst):  
  evalf(%);
                                               2     1/2
                           1               (rho  - 1)
             -2 EllipticK(---) + EllipticK(-------------) I
                          rho                   rho
           - ----------------------------------------------
                                  rho

                -0.34551659067555 - 4.0028945009987 I


Converting to integrals gives another result, which alos can
not be correct.

What is the correct limit? I am missing a proper reference ...
Some of the countless papers of Temme?

Branch cuts

Comment on 2008-09-06 17:47 from JacquesC ( Maple Rating 4

1699)

The correct answer to your question is inevitably a piecewise-defined function, as in most cases that limit crosses EllipticF's branch cuts. Maple does not seem to know these explicitly, but it at least knows the branch points:

FunctionAdvisor(branch_points, EllipticF);
  [EllipticF(z, k), z  in  [-1, 1, - 1/k, 1/k, infinity + infinity I]]

So the answer is going to vary depending on where exactly a and r are in the complex plane. I am not even sure that the limit will exist for all complex a and r.

limit( EllipticF(x*a,r), x=infinity) = ?

Branch cuts

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