Re: Re: LogIntegral^(-1)

*To*: mathgroup at smc.vnet.net*Subject*: [mg48724] Re: [mg48682] Re: LogIntegral^(-1)*From*: "Roger L. Bagula" <rlbtftn at netscape.net>*Date*: Fri, 11 Jun 2004 03:53:14 -0400 (EDT)*References*: <ca3gvi$rto$1@smc.vnet.net> <200406100643.CAA29482@smc.vnet.net> <40C89DA6.1010403@wolfram.com>*Sender*: owner-wri-mathgroup at wolfram.com

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 > > -- Respectfully, Roger L. Bagula tftn at earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 : URL : http://home.earthlink.net/~tftn URL : http://victorian.fortunecity.com/carmelita/435/

**References**:**Re: LogIntegral^(-1)***From:*"Roger L. Bagula" <rlbtftn@netscape.net>