oscillatory integrals

*To*: mathgroup at smc.vnet.net*Subject*: [mg69451] oscillatory integrals*From*: dimmechan at yahoo.com*Date*: Wed, 13 Sep 2006 04:01:17 -0400 (EDT)

Hello to all. ***Let me consider the integral of the function q(x,r) (see below) over the range {x,0,Infinity}, for various values of r. In[17]:=$Version Out[17]=5.2 for Microsoft Windows (June 20, 2005) In[18]:=Clear["Global`*"] ***The function q(x,r) is defined as follows: In[19]:=q[r_,x_]:=(f[x]/g[x])*BesselJ[0,r*x] ***where In[20]:=f[x_]:=x*Sqrt[x^2+1/3]*(2*x^2*Exp[(-1/5)*Sqrt[x^2+1]]-(2x^2+1)Exp[(-1/5)*Sqrt[(x^2+1/3)]]) In[21]:=g[x_]:=(2x^2+1)^2-4x^2*Sqrt[x^2+1/3]Sqrt[x^2+1] ***The case r=2 was first considered by Longman on a well celebrated paper (Longman 1956). I will consider first this case also. ***From the following plot one can see that the integrand is an oscillatory function convergent to zero for large arguments. In[22]:=Plot[q[2,x],{x,0,20},PlotPoints\[Rule]1000] ***With its default settings NIntegrate fails to give any result In[23]:=NIntegrate[q[2,x],{x,0,Infinity}] NIntegrate::singd: NIntegrate's singularity handling has failed at point {x}={4.33675*10^14} for the specified precision goal. Try using larger values for any of $MaxExtraPrecision or the options WorkingPrecision, or SingularityDepth and MaxRecursion. More... NIntegrate::inum : (***I drop the rest of the message***) Out[23]=NIntegrate[q[2,x],{x,0,8}] ***However increasing the SingularityDepth (the default is 4) gives a result In[24]:=NIntegrate[q[2,x],{x,0,Infinity},SingularityDepth\[Rule]6] NIntegrate::ncvb: NIntegrate failed to converge to prescribed accuracy after \ 7 recursive bisections in x near x = 50.2`. More... Out[24]=-0.0267271 ***Although from the following comands it is clear that in latter case NIntegrate samples more points, I do not understand why with the default value of the SingularityDepth NIntegrate fails to give an answer. Note that for the first message above (NIntegrate::singd:) there is still no notes in the Online Documentation. In[26]:=Block[{Message},ListPlot[Reap[NIntegrate[q[2,x],{x,0,Infinity}, SingularityDepth\[Rule]6,EvaluationMonitor\[RuleDelayed]Sow[x]]][[2,1]]]] In[27]:=Block[{Message},ListPlot[Reap[NIntegrate[q[2,x],{x,0,Infinity}, SingularityDepth\[Rule]6,EvaluationMonitor\[RuleDelayed]Sow[x]]][[2,1]]]] ***Increasing also the maximum number of recursive subdivisions gives a very reliable result. In[24]:=NIntegrate[q[2,x],{x,0,Infinity},SingularityDepth\[Rule]20,MaxRecursion\[Rule]20,\ WorkingPrecision\[Rule]22]//Timing Out[24]={1.406 Second,-0.0266089981279} ***while the following commands dealing with the sampled points In[29]:= Length[Reap[NIntegrate[ q[2,x],{x,0,Infinity}, SingularityDepth\[Rule]20,MaxRecursion\[Rule]20,WorkingPrecision\ \[Rule]22,EvaluationMonitor\[RuleDelayed]Sow[x]]][[2,1]]] Take[Sort[Reap[NIntegrate[q[2,x],{x,0,Infinity},SingularityDepth\[Rule]20,\ MaxRecursion\[Rule]20,WorkingPrecision\[Rule]22, EvaluationMonitor\[RuleDelayed]Sow[x]]][[2,1]]],-10] Out[29]=2865 Out[30]= {1979.89661682140809934,2514.12574728867068961,3744.0426990391850147,3787.\ 78762150909090052,5029.2514945773413792,7489.085398078370029,10059.\ 5029891546827584,14979.170796156740059,29959.341592313480118,59919.\ 68318462696024} ***Next the option Method\[Rule]Oscillatory will be employed In[32]:=NIntegrate[q[2,x],{x,0,8},Method\[Rule]Oscillatory,WorkingPrecision\[Rule]22]//Timing 8::indet: Indeterminate expression 0\8 encountered. More... 8::indet: Indeterminate expression 0\8 encountered. More... Out[32]={1.187 Second,-0.026608998128} ***I do not understand why exist here the warning messages (8::indet:). Note that despite the presence of the message, the result is very accurate. ***Now I want to plot the function NIntegrate[q[r,x],{x,0,Infinity}] in the range {r,0,3}. What is the more reliable method to follow to get what I want? I simply executed Plot[NIntegrate[q[r,x],{x,0,8},Method\[Rule]Oscillatory, WorkingPrecision\[Rule]30],{r,0,3}] but although I got a plot, I need considerable time and there were a lot of warning messages so I believe this is not the case here. ***Next consider the function h[r,x] which is defined as follows: In[34]:=h[r_,x_]:=(f[x]/g[x])*BesselJ[1,r*x] ***For this function, we have e.g. In[34]:=NIntegrate[h[2,x],{x,0,Infinity},SingularityDepth\[Rule]20,MaxRecursion\[Rule]20,\ WorkingPrecision\[Rule]22]//Timing Out[34]={1.172 Second,-0.147430035385} ***and In[35]:=NIntegrate[h[2, x], {x, 0, 8}, Method ->Oscillatory, WorkingPrecision -> 22] // Timing 8::indet: Indeterminate expression 0\8 encountered. More... 8::indet: Indeterminate expression 0\8 encountered. More... Out[35]={1.078 Second, -0.147430035385} ***which clearly shows the reliability of Method ->Oscillatory. ***I want also here the plot of NIntegrate[h[r,x],{x,0,Infinity}] in the range {r,0,3}. Because of BesselJ[1,0]=0, I am a little worry how I will treat the point r=0. ***Any suggestions? ***Thanks in advance for any assistance. Dimitrios Anagnostou NTUA