NIntegrate with singular endpoints

*To*: mathgroup at smc.vnet.net*Subject*: [mg45390] NIntegrate with singular endpoints*From*: Goyder Dr HGD <h.g.d.goyder at cranfield.ac.uk>*Date*: Tue, 6 Jan 2004 04:16:46 -0500 (EST)*Sender*: owner-wri-mathgroup at wolfram.com

I wish to integrate numerically a function which is integrable but has singularities at the endpoints. Help states that "NIntegrate tests for singularities at the end points of the integration range" so I am optimistic that Mathematica can deal with my integral. However, the following happens. I start by defining my integrand where I have Rationalized all my numbers so that everything is defined exactly In[1]:= a = {48397463/62999302, 73182719/200515778, 20053837/72775587, \ 70325974/50617151, 72672481/359982590}; In[2]:= b = {26714619/25510582, 26714619/12755291, 245850922/78256779, \ 122925461/26085593, 491701844/78256779}; In[3]:= integrand = Product[(1 - z/E^(I*b[[n]]))^(a[[n]] - 1), {n, Length[a]}]; In[4]:= (* I try integrating between the first and second singularity *) NIntegrate[integrand, {z, E^(I*b[[1]]), E^(I*b[[2]])}] Out[4]= -2.28889+1.16263 \[ImaginaryI] In[5]:= (* This works so I am pleased. I move on to a second integral *) NIntegrate[integrand, {z, E^(I*b[[1]]), E^(I*b[[3]])}] \!\(Power::"infy" \(\(:\)\(\ \)\) " Infinite expression \!\(1\/\((\(\(\[LeftSkeleton] 28 \[RightSkeleton]\)\) \ + \(\(\(\(\[LeftSkeleton] 29 \[RightSkeleton]\)\)\\ \ \[ImaginaryI]\)\))\)\^\(52721750/72775587\)\) encountered."\) \!\(NIntegrate::"rnderr" \(\(:\)\(\ \)\) "Numerical approximation has caused \!\({ z}\) to take the value \!\({\(\(-1.`\)\) + \(\(7.817936619907544`*^-17\\ \ \[ImaginaryI]\)\)}\) where the integrand is singular."\) Out[5]= \!\(NIntegrate[integrand, {z, \[ExponentialE]\^\(\[ImaginaryI]\ b?1?\), \ \[ExponentialE]\^\(\[ImaginaryI]\ b?3?\)}]\) In[6]:= (* This does not work. So I try making the integrand a numerical value *) NIntegrate[N[integrand], {z, E^(I*b[[1]]), E^(I*b[[3]])}] \!\(Power::"infy" \(\(:\)\(\ \)\) " Infinite expression \!\(1\/\((\(\(\[LeftSkeleton] 28 \[RightSkeleton]\)\) \ + \(\(\(\(\[LeftSkeleton] 29 \[RightSkeleton]\)\)\\ \ \[ImaginaryI]\)\))\)\^\(52721750/72775587\)\) encountered."\) \!\(NIntegrate::"rnderr" \(\(:\)\(\ \)\) "Numerical approximation has caused \!\({ z}\) to take the value \!\({\(\(-1.`\)\) + \(\(7.817936619907544`*^-17\\ \ \[ImaginaryI]\)\)}\) where the integrand is singular."\) Out[6]= \!\(NIntegrate[N[integrand], { z, \[ExponentialE]\^\(\[ImaginaryI]\ b?1?\), \[ExponentialE]\^\(\[ImaginaryI]\ b?3?\)}]\) In[7]:= (* This does not work. So I try making the end points numerical *) NIntegrate[integrand, {z, N[E^(I*b[[1]])], N[E^(I*b[[3]])]}] \!\(NIntegrate::"ncvb" \(\(:\)\(\ \)\) "NIntegrate failed to converge to prescribed accuracy after \!\(7\) recursive bisections in \!\(z\) near \ \!\(z\) = \!\(\(\(-1.`\)\) + \(\(1.2250406712299894`*^-16\\ \[ImaginaryI]\)\)\ \)."\) Out[7]= -2.46651-2.71438 \[ImaginaryI] In[8]:= (* This begins to show promise so I increase the number of bisections *) NIntegrate[integrand, {z, N[E^(I*b[[1]])], N[E^(I*b[[3]])]},MaxRecursion -> 10 ] Out[8]= -2.46651-2.71438 \[ImaginaryI] In[9]:= (* This works, I am pleased, so now I go back to my first example and check out my method *) NIntegrate[integrand, {z, N[E^(I*b[[1]])], N[E^(I*b[[2]])]},MaxRecursion -> \ 10] \!\(Power::"infy" \(\(:\)\(\ \)\) " Infinite expression \!\(1\/\((\(\(0.` \[InvisibleSpace]\)\) + \(\(0.`\\ \ \[ImaginaryI]\)\))\)\^\(14601839/62999302\)\) encountered."\) NIntegrate::rnderr: Numerical approximation has caused {z} to take the value \ {0.5\[InvisibleSpace]+0.866025 \[ImaginaryI]} where the integrand is singular. Out[9]= \!\(NIntegrate[integrand, {z, N[\[ExponentialE]\^\(\[ImaginaryI]\ b?1?\)], N[\[ExponentialE]\^\(\ \[ImaginaryI]\ b?2?\)]}, MaxRecursion \[Rule] 10]\) In[10]:= (* This fails. It worked without the refinements and now it fails with \ refinements. I attempt a third integral *) NIntegrate[integrand, {z, N[E^(I*b[[1]])], N[E^(I*b[[ 5]])]},MaxRecursion -> 10] NIntegrate::slwcon: Numerical integration converging too slowly; suspect one \ of the following: singularity, value of the integration being 0, oscillatory \ integrand, or insufficient WorkingPrecision. If your integrand is oscillatory \ try using the option Method->Oscillatory in NIntegrate. Out[10]= 1.08293\[InvisibleSpace]-2.77243 \[ImaginaryI] (* This may have worked but I have a new problem. I am clearly just tinkering \ around and need help. *) How do you do a numerical integration with singularities at the end ponts? Thanks for any help Hugh Goyder -- This message has been scanned for viruses and dangerous content by the Cranfield MailScanner, and is believed to be clean.