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MathGroup Archive 1993

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thanks and more help...

  • To: mathgroup at yoda.physics.unc.edu
  • Subject: thanks and more help...
  • From: gray at cmgroup.engr.wisc.edu
  • Date: Sat, 23 Jan 93 23:45:40 CST

Thank you to those that responded to my quest for a test for
"evenness" and "oddness" of functions.


Hello Mathematica (and I hope residue integral) gurus...

I have encountered a problem and I can't tell if it is mine or
Mathematica's.  I need to integrate fI5 defined below from -Infty to
Infty with respect to the variable z.

I have chosen a rectangular contour that runs from -Infty to Infty along
the real axis, then from Infty to Infty + 2 Pi I/d, then back to
-Infty + 2 Pi I/d and finally back down to -Infty.  Omitting the
details, I can write the integral around that contour of fI5 as a
constant times the integral I am interested in

                      d x 2
          Cos[x] Cosh[---]
                       2
          ------------------
                           2
          (a + b Cosh[d x])


First define fI5 (a,b,d are real constants, a^2 > b^2, a > 0, b > 0) ...

In[1]:=
  fI5 = ( Exp[I z] Cosh[d z/2]^2 )/(( a + b Cosh[d z] )^2)

Out[1]=
    I z      d z 2
   E    Cosh[---]
              2
  ------------------
                   2
  (a + b Cosh[d z])


The poles of order 2 within the contour are shown to be zo1 and zo2 ...

In[2]:=
  zo1 = Log[ a/b - Sqrt[(a/b)^2 - 1] ]/d + Pi I/d

Out[2]=
                         2
                        a     a
         Log[-Sqrt[-1 + --] + -]
                         2    b
  I Pi                  b
  ---- + -----------------------
   d                d


In[3]:=
  zo2 = Log[ a/b + Sqrt[(a/b)^2 - 1] ]/d + Pi I/d

Out[3]=
                        2
                       a     a
         Log[Sqrt[-1 + --] + -]
                        2    b
  I Pi                 b
  ---- + ----------------------
   d               d


Now find the residue using the Residue function ...

In[4]:=
  i5Residue = Residue[fI5, {z, zo1}](* +
              Residue[fI5, {z, zo2}]*)

Out[4]=
  Series::esss: 
     Essential singularity encountered in 

                    Log[-<<1>> + <<1>>]
             I Pi
      Exp[I (---- + -------------------) + <<2>>].
              d              d
  Series::esss: 
     Essential singularity encountered in 

                    Log[-<<1>> + <<1>>]
             I Pi
          d (---- + -------------------)
              d              d
      Exp[------------------------------ + <<2>>].
                      2
  Series::esss: 
     Essential singularity encountered in 

                    Log[-<<1>> + <<1>>]
             I Pi
      Exp[d (---- + -------------------) + <<2>>].
              d              d
  General::stop: 
     Further output of Series::esss
       will be suppressed during this calculation.
                                                         2
                                                        a     a
            I z      d z 2               Log[-Sqrt[-1 + --] + -]
           E    Cosh[---]                                2    b
                      2           I Pi                  b
  Residue[------------------, {z, ---- + -----------------------}]
                           2       d                d
          (a + b Cosh[d z])


fI5 doesn't seem like it should have essential singularities.  Therefore
I try and use the definition for poles of order m (mine are order2) and
I get ...

In[5]:=
  i5zo1Residue = Limit[ D[ (z - zo1)^2 fI5, z ], z -> zo1 ]

Out[5]=
  0


In[6]:=
  i5zo2Residue = Limit[ D[ (z - zo2)^2 fI5, z ], z -> zo2 ]

Out[6]=
  0


This can't be correct either since the integral is surely nonzero from
the following numerical integration around the contour ...

In[7]:=

  fI5check1 = NIntegrate[ fI5 /. {a -> .62348, b -> .34753,
                                  d -> .283462},
                          {z, -2000, 2000, 2000 + 2 Pi I/.283462,
                          -2000 + 2 Pi I/.283462, -2000},
                          MinRecursion -> 3, MaxRecursion -> 14,
                          WorkingPrecision -> 20 ]

Out[7]=

  -0.0023531134895 + 0. I


In[8]:=
  sumResI5 = fI5check1/(2 Pi I)

Out[8]=
  0. + 0.0011765567447 I
  ----------------------
            Pi


The residue is therefore nonzero and the integral must be nonzero.

I also tried using the Series function to get the Laurent series.  This
also screamed at me about essential singularities.

Why can't Mathematica find the residues at the poles?  Am I missing
something?  Is it possible to find the residues at those poles?

By the way, I did look in *A LOT* of integral tables before undertaking
this task (including my favorite, "Gradshteyn and Ryzhik").

Any help, guidance, or suggestions would be enormously appreciated.

Best regards,



    _/_/_/   _/       _/_/_/  | Gary L. Gray
   _/    _/ _/       _/    _/ | Engineering Mechanics & Astronautics
  _/       _/       _/        | University of Wisconsin-Madison
 _/  _/_/ _/       _/  _/_/   | gray at cmgroup.engr.wisc.edu
_/    _/ _/       _/    _/    | AOL: GLGray
 _/_/_/ _/_/_/_/   _/_/_/     | (608) 262-0679





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