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

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

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

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

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

Now find the residue using the Residue function ...

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

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

Out=
0

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

Out=
0

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

In:=

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=

-0.0023531134895 + 0. I

In:=
sumResI5 = fI5check1/(2 Pi I)

Out=
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

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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