Re: LogIntegral^(-1)
- To: mathgroup at smc.vnet.net
- Subject: [mg48740] Re: LogIntegral^(-1)
- From: "Roger L. Bagula" <rlbtftn at netscape.net>
- Date: Fri, 11 Jun 2004 23:59:14 -0400 (EDT)
- References: <ca3gvi$rto$1@smc.vnet.net> <200406100643.CAA29482@smc.vnet.net> <40C89DA6.1010403@wolfram.com> <cabqd9$orm$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
I tried your approach. Works about the same as mine. what works is setting the solution equal to y and solving for t which reinverts the ArcLogIntegral... The soltion looks like a square root function or the other way a parabolic function. Anyway a function like: s[t_]=y=a+t^s for s in the range of 1/2 or 1/3 about. Roger L. Bagula wrote: > Dear Daniel Lichtblau, > As a signal goes it doesn't look good as an answer, but thanks for your > reply. > > >>The InverseFunction wrapper means, not surprisingly, that it is an >>inverse function. Not a reciprocal (which would be written in >>OutputForm as LogIntegral[...]^(-1) rather than LogIntegral^(-1)[...]). > > > This is not a real clear distinction > and I've never seen it in any of > my Mathematica books or software, so thans for giving me that! > I frankly would rather it was ^(-1) instead of an inverse function. > > It's not the kind of result I had hoped for. > > But I'm glad I did get a reply. > > Daniel Lichtblau wrote: > > >>Roger L. Bagula wrote: >> >> >>>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. >> >> >>It's very clear. >> >>n[t] = s[t]/(Exp[D[s[t],t]/w]-1); >>n1[t_] = FullSimplify[D[n[t],t]]; >> >>In[10]:= InputForm[soln = DSolve[n1[t]==0, s[t], t]] >> >>Solve::ifun: Inverse functions are being used by Solve, so some >>solutions may >> not be found; use Reduce for complete solution information. >> >>InverseFunction::ifun: >> Inverse functions are being used. Values may be lost for multivalued >> inverses. >> >>Out[10]//InputForm= >>{{s[t] -> (-1 + InverseFunction[LogIntegral, 1, 1][ >> E^(C[1]/w)*w*(t + C[2])])/E^(C[1]/w)}} >> >>The InverseFunction wrapper means, not surprisingly, that it is an >>inverse function. Not a reciprocal (which would be written in >>OutputForm as LogIntegral[...]^(-1) rather than LogIntegral^(-1)[...]). >> >> >> >>>One approximation ( Euler's I think) of the distribution of primes is >>>Pi(n)=Li(n)--> n/log(n): asymptotic >> >> >>This is a bit misleading. Pi(n) is approximated by Li(n) but they are >>not equal. >> >> >> >>>[...] >> >> >> >>Daniel Lichtblau >>Wolfram Research >> >> > >
- References:
- Re: LogIntegral^(-1)
- From: "Roger L. Bagula" <rlbtftn@netscape.net>
- Re: LogIntegral^(-1)