Re: how can I solve a function Erfc

*To*: mathgroup at smc.vnet.net*Subject*: [mg48447] Re: how can I solve a function Erfc*From*: Daniel Lichtblau <danl at wolfram.com>*Date*: Sun, 30 May 2004 06:12:08 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

David W. Cantrell wrote: > montgermont.aude at ec-lille.fr (aude) wrote: > >>I have to solve this function: >> >>Erfc[L/(4*(Dg*t)^(1/2))]= 0.9 >> >>Dg is constant. >> >>Dg=5*10^5 >> >>I want to plot t as a function of L. > > > If plotting t as a function of L is all you want to do, you can just use > ImplicitPlot. But trying to use Mathematica to solve an equation of the > form > > Erfc[L/(4*(Dg*t)^(1/2))] == k > > for t in terms of L, Dg, and k raises some interesting points, one of which > I suspect is a bug. > > First, note that Mathematica has a built-in function InverseErfc. With that > in mind, you might think that one of the following would work, but alas, > no. > > In[1]:= Solve[Erfc[L/(4*(Dg*t)^(1/2))] == k, t] > ... > Out[1]= {} > > In[2]:= Reduce[Erfc[L/(4*(Dg*t)^(1/2))] == k, t] > ... > Out[2]= Reduce[Erfc[L/(4*Sqrt[Dg*t])] == k, t] > > But it seems that we should be able, trivially, to assist Mathematica by > rewriting the equation using InverseErfc. (Note that I've now used your > specific k value, so that there is no doubt that InverseErfc is defined > there.) > > In[3]:= Solve[L/(4*Sqrt[Dg*t]) == InverseErfc[9/10], t] > Out[3]= {} > > Surely this is a bug. The solution for t should have been > > L^2/(16*Dg*InverseErf[Infinity, -(9/10)]^2) > > which, BTW, is obtainable with Reduce. I can't imagine why Solve didn't > work there! > > David Cantrell > The Solve verification process did not like that value (no doubt due to presence of the Infinity). That may indicate a bug, I'm not sure. In[6]:= InputForm[Solve[Erfc[L/(4*(Dg*t)^(1/2))] == k, t, VerifySolutions->False]] Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. Out[6]//InputForm= {{t -> L^2/(16*Dg*InverseErf[Infinity, -k]^2)}} Daniel Lichtblau Wolfram Research