RE: Re: Evaluation of AiryAi on the complex plane
- To: mathgroup at smc.vnet.net
- Subject: [mg44986] RE: [mg44959] Re: Evaluation of AiryAi on the complex plane
- From: "Wolf, Hartmut" <Hartmut.Wolf at t-systems.com>
- Date: Fri, 12 Dec 2003 04:41:24 -0500 (EST)
- Sender: owner-wri-mathgroup at wolfram.com
>-----Original Message----- >From: Jens-Peer Kuska [mailto:kuska at informatik.uni-leipzig.de] To: mathgroup at smc.vnet.net >Sent: Thursday, December 11, 2003 11:28 AM >To: mathgroup at smc.vnet.net >Subject: [mg44986] [mg44959] Re: Evaluation of AiryAi on the complex plane > > >Hi, > >you know what you are doing ?? >The Airy function decay exponential with >Exp[- 2 z^(3/2)/3] for Re[z]>0 > >for 11.18 this gives a factor of Exp[-25] >and you will need hig precision arithmetic >to evaluate that. > >Regards > Jens > Yes, of course. I might add to Mauricio: observe x0 = SetPrecision[11.18, 20] Plot[Evaluate[Through[{Re, Im}[AiryAi[x0 + x \[ImaginaryI]]]]], {x, -10^-1, 10^-1}, PlotStyle -> {Hue[2/3], Hue[0]}, Evaluate[opt]] and you'll see the problems of precision (entering through machine precision x values). You may overcome these $MaxExtraPrecision = 100. Plot[{Re@AiryAi[x0 + SetPrecision[x, 20] \[ImaginaryI]], Im@AiryAi[x0 + SetPrecision[x, 20] \[ImaginaryI]]}, {x, -10^-1, 10^-1}, Compiled -> False, PlotStyle -> {Hue[2/3], Hue[0]}] For your examples: Plot[Abs[AiryAi[x0 + SetPrecision[x, 20]]], {x, 0, 10^-100}, PlotStyle -> Hue[0], PlotRange -> {All, {0, 10^-11}}, Compiled -> False] Plot[Abs[AiryAi[x0 + SetPrecision[x, 20] I]], {x, 0, 10^-100}, PlotStyle -> Hue[0], PlotRange -> {All, {0, 10^-11}}, Compiled -> False] (Also, don't Compile the expression.) -- Hartmut Wolf >Mauricio Bedoya wrote: >> >> Hello: >> >> I'm in trouble evaluating the AiryAi function. I need to evaluate it >> around the point (11.18,0) or (11.18 + I 0.0). When I sweep the real >> axis there is no problem [1], it has a value of 2 10^-12 on all the >> range. >> >> But when I choose a range on a complex line [2], the >function throws a >> 0 value on all the range. >> >> when I evaluate the function [3] I obtain 0 in the complex plane >> around 11.18 and 2.31151*10^-12 in 11.18, no matter how close I am >> from 11.18 + 0.0 I. >> >> This doesn't happen around the points 10.03 + 0.0 I and >23.52 + 0.0 I. >> >> I work in mathemathica 4.1.2.0. >> >> I have tried to evaluate the Ai function in the version 3 and the >> result is 0 in the complex plane, but when I plot the function with >> complex values I obtain values different from 0 and also the absolute >> value of the function is continuous. I think that there is a problem >> with the evaluation of the function around 11.18 because I >plotted the >> function in mathematica 3 with no problems but I couldn't >evaluate it. >> In the version 4 I couldn't plot or evaluate it. >> >> I would be grateful if someone could help me with this problem. >> >> Mauricio Bedoya >> >> List of commands >> [1] Plot[{AiryAi[11.18 + x]}, {x, 0, 10^-100}, PlotRange -> {All, {0, >> 10^-11}}] >> [2] Plot[{AiryAi[11.18 + x I]}, {x, 0, 10^-100}, PlotRange -> {All, >> {0, 10^-11}}] >> [3] Abs[ai[11.18]] >> Abs[ai[11.18 + 0 10^-11 I]] >