Re: LogIntegral^(-1)
- To: mathgroup at smc.vnet.net
- Subject: [mg48682] Re: LogIntegral^(-1)
- From: "Roger L. Bagula" <rlbtftn at netscape.net>
- Date: Thu, 10 Jun 2004 02:43:39 -0400 (EDT)
- References: <ca3gvi$rto$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
I really need to know if this is: Li(x)^(-1) or ArcLi(x) since it makes a great difference and Mathematica notation on this seems unclear. One approximation ( Euler's I think) of the distribution of primes is Pi(n)=Li(n)--> n/log(n): asymptotic If it is "arc" that is very different than 1): 1) s[t_]=C[2]*(-1+1/Li[w*(t-C[1])/C[2]]) 2) s[t_]=C[2]*(-1+ArcLi[w*(t-C[1])/C[2]]) 1/Li(n)-->0 as n->Infinity That is the signal dies out as time gets large. But for Li[n] ArcLi[x]=n which becomes infinite as time goes on. It is my conclusion that it should be the first: 1/Li[n]--> 1-ArcTanh(n): approximation that tends the same way But that gives: s[t_]= -C[2]*ArcTanh[w*(t-C[1])/C[2]] as the approximate signal function. I don't like the implications of that. Roger L. Bagula wrote: > I have an information theory problem: > Suppose that the noise has a constant expectation value with time > so that when you take the differential you get > DN/dt=0 > Mathematica for the Shannon capascity as rate of transmission is: > s1[t_]=D[s[t],t] > n[t_]=s[t]/(Exp[s1[t]/w]-1) > n1[t_]=D[n[t],t] > DSolve[FullSimplify[n1[t]]==0,s[t],t] > > I get an answer involving: > LogIntegral^(-1) > specifically: > > s[t_]=C[2]*(-1+LogIntegral^(-1)[w*(t-C[1])/C[2]]) > > I'm pretty sure that C[2] represents the average noise. > The problem is I can't get Mathematica to plot this function. > I replaced c2 and c1 with real constants and used a constant w > while trying to plot Re[s[t]]: > I get errors like: > Plot::"plnr": > "\!\(Re[\(s[t]\)]\) is not a machine-size real number at \!\(t\) = \ > \!\(-0.999999916666666699`\)." > Plot::"plnr": > \:f3b5\!\(Re[s[t]]\) is not a machine-size real number at t = \ > -0.918866016854168421`. > Plot::"plnr": > "\!\(Re[\(s[t]\)]\) is not a machine-size real number at \!\(t\) = \ > \!\(-0.830382400281252586`\)." > General::"stop": > "Further output of \!\(Plot :: \"plnr\"\) will be suppressed during > this \ > calculation." >