Re: Re: Series expansion of ArcSin around 1

*To*: mathgroup at smc.vnet.net*Subject*: [mg21665] Re: [mg21634] Re: [mg21598] Series expansion of ArcSin around 1*From*: BobHanlon at aol.com*Date*: Fri, 21 Jan 2000 04:00:16 -0500 (EST)*Sender*: owner-wri-mathgroup at wolfram.com

I forgot to mention that since ArcSin[-x] == -ArcSin[x] True We can add to the definition of f f[x_?Negative] := -f[-x]; This eliminates most of the divergence and the only appreciable error is then for small magnitudes of x Bob Hanlon In a message dated 1/18/2000 4:48:15 AM, BobHanlon at aol.com writes: >series1 = Normal[Series[ArcSin[x], {x, 1, 4}]] > >The result is real for x < 1, for example > >series1 /. x -> .9 > >2.0218230504180923 > >The complex factors appear because (x-1) is negative and the powers are > >radicals. If you wish, you can eliminate the appearance of complex factors >by >a substitution: > >series2 = series1 /. (x - 1)^a_ :> ((1 - x)^a*(-1)^a) > >However, as seen on the Plot below, the result is in the wrong quadrant. > >Plot[{ArcSin[x], series2}, {x, -1, 1.}, > PlotStyle -> {RGBColor[0, 0, 1], RGBColor[1, 0, 0]}]; > >Both x and Pi-x have the same Sin > >Plot[Sin[x], {x, 0, 2Pi}]; > >Consequently, we need to modify the approximation as follows > >f[x_] := Evaluate[ > Pi - Normal[Series[ArcSin[x], {x, 1, 4}]] /. (x - 1)^ > a_ :> ((1 - x)^a*(-1)^a)] > >Plot[{ArcSin[x], f[x]}, {x, -1, 1.}, > PlotStyle -> {RGBColor[0, 0, 1], RGBColor[1, 0, 0]}]; > >As expected, the approximation diverges away from the expansion point. > > >Bob Hanlon > >In a message dated 1/17/2000 12:12:02 AM, pliszka at fuw.edu.pl writes: > >>I have the following problem. My x is close to 1 but sligthly >>smaller. I want to expand ArcSin[x] around 1 but this is what I get: >> >>In[53]:= Series[ArcSin[x],{x,1,4}] >> >> I 3/2 3 I 5/2 >> - (-1 + x) --- (-1 + x) >> Pi 6 80 >>Out[53]= -- - I Sqrt[2] Sqrt[-1 + x] + ------------- - --------------- >>+ >> 2 Sqrt[2] Sqrt[2] >> >> 5 I 7/2 >> --- (-1 + x) >> 448 9/2 >>> --------------- + O[-1 + x] >> Sqrt[2] >> >>How to tell Mathematica that my x is real and smaller than 1 >>so it will not return all this complex numbers? >