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