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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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