Re: Integrating "If"
- To: mathgroup at smc.vnet.net
- Subject: [mg84571] Re: Integrating "If"
- From: dflatin at rcn.com
- Date: Fri, 4 Jan 2008 05:40:03 -0500 (EST)
- References: <flc7vh$809$1@smc.vnet.net> <flfubb$jrm$1@smc.vnet.net>
On Jan 3, 8:34=A0pm, "David W.Cantrell" <DWCantr... at sigmaxi.net> wrote: > BTW, in case you're not aware of it, the last three terms of your second > case can be simplified somewhat. Namely, for the values of d under > consideration, > > Pi + 2*(-1+d^2)*ArcCot[Sqrt[-1+d^2]] - 2*(1+d^2)*ArcTan[Sqrt[-1+d^2]] > > is the same as > > d^2 (Pi - 4 ArcTan[Sqrt[d^2 - 1]]) > > but I know of no direct way to demonstrate that fact using Mathematica. > (Mathematica can show that the derivative of the difference of the two > expressions is 0, etc. But that's hardly a direct method.) For what it is worth: \[Pi]+2(d^2-1)*ArcCot[Sqrt[d^2-1]]-2*(d^2+1)*ArcTan[Sqrt[d^2-1]]==d^2(\ [Pi]-4ArcTan[Sqrt[d^2-1]]) d^2->\[Theta]^2+1 Sqrt[\[Theta]^2]->\[Theta] (*assuming \[Theta] > 0*) %%%/.%%/.% Collect[%,\[Theta]] (#-\[Pi]+4 ArcTan[\[Theta]])&/@% (#/\[Theta]^2)&/@% (#+4ArcTan[\[Theta]])/2&/@%//Expand Cot[#]&/@% TrigExpand[#]&/@%