Re: InverseFunction: how to manage?

*To*: mathgroup at smc.vnet.net*Subject*: [mg124298] Re: InverseFunction: how to manage?*From*: Murray Eisenberg <murray at math.umass.edu>*Date*: Mon, 16 Jan 2012 17:09:07 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <201201142217.RAA13006@smc.vnet.net>*Reply-to*: murray at math.umass.edu

First, Mathematica indicates that you need to impose a condition in order that your paper-and-pencil integral gives the result you claim: Integrate[1/Sqrt[Sin[u]], {u, 0, f}] ConditionalExpression[ -2*EllipticF[(-2*f + Pi)/4, 2] + Sqrt[2]*EllipticK[1/2], Sin[f] >= 0] Thus take: t[f_] = Integrate[1/Sqrt[Sin[u]], {u, 0, f}, Assumptions -> 0 <= f <= Pi] -2*EllipticF[(-2*f + Pi)/4, 2] + Sqrt[2]*EllipticK[1/2] Is the following of any use? f /. First@Solve[t[f] == tau, f] (Pi - 4*JacobiAmplitude[(-tau + Sqrt[2]*EllipticK[1/2])/2, 2])/2 (You'll get a warning that inverse functions are being used by Solve and so that some solutions may not be found.) On 1/14/12 5:17 PM, Dr. Wolfgang Hintze wrote: > I apologize for asking this very elementary question but how do I > manage InverseFunction? > > Here is an example > > When I solve the equation of motion for a pendulum > > f''[t] == Cos[f[t]], f[0]== 0, f'[0] == 0 > > I get (with paper and pencil) the time t as a function of the angle f > thus > > t[f_] = Integrate[1/Sqrt[Sin[u]], {u, 0, f}] > Out[22]= > 2*(EllipticF[f/2 - Pi/4, 2] + EllipticF[Pi/4, 2]) > > Now I want the the angle as a function of time (f[t]) like this > > "f[t_] = InverseFunction[t[f]]" > > But this does not work. I also tried to define t as a pure function > > t = 2*(EllipticF[#1/2 - Pi/4, 2] + EllipticF[Pi/4, 2])& > > but again, I have not seen a way to invert this, and for instance carry > out Plot[f,{t,0,2 Pi}]. > > Thanks in advance for any hints. > > Best regards, > Wolfgang > > > > -- Murray Eisenberg murray at math.umass.edu Mathematics & Statistics Dept. Lederle Graduate Research Tower phone 413 549-1020 (H) University of Massachusetts 413 545-2859 (W) 710 North Pleasant Street fax 413 545-1801 Amherst, MA 01003-9305

**References**:**InverseFunction: how to manage?***From:*"Dr. Wolfgang Hintze" <weh@snafu.de>