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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>