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What does this happen to my series expansion?

  1. Sep 4, 2009 #1
    I came across this really strange error when doing series expansions in Mathematica. Suppose I were to let,

    [tex]F(z) = \frac{(z+2)^2}{(z+1)^2} - 0.4[/tex]

    Now [tex]F(z) = 0[/tex] gives [tex]z \approx -3.72076, -1.61257[/tex]. Suppose we take the second value of [tex]z^* = -1.61257[/tex]. What is the series expansion of [tex]\sqrt{F(z)}[/tex] about this point?

    [tex]F(z) \sim 0 + F'(z^*)(z-z^*) + \ldots \approx 3.37088 (z-z^*) + \ldots[/tex]. Thus [tex]\sqrt{F(z)} \sim \sqrt{3.37088}\sqrt{z-z^*}[/tex].

    Now let's try doing this in Mathematica:

    Code (Text):

    moo = z /. Solve[disc[z] == 0, z]
    F[z_] = -0.4 + ((z+2)/(z+1))^2;
    Chop[Series[F[z], {z, moo[[2]], 4}]];
    This seems to give the right answer. The first two terms are:

    Code (Text):
    1.83599 Sqrt[z + 1.61257] + 3.43262 (z + 1.61257)^(3/2)
    Now here's the problem. I try:

    Code (Text):

    Series[Sqrt[F[z]], {z, moo[[2]], 4}]
    The output of the first three terms are,

    Code (Text):

    0. + 1.05367*10^-8 I) - (0. + 1.59959*10^8 I) (z + 1.61257) - (0. + 1.21417*10^24 I) (z + 1.61257)^2
    which is nonsensical.

    What is the problem? Does it have to do with the fact that we're using an approximation to a root?
  2. jcsd
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