I came across this really strange error when doing series expansions in Mathematica. Suppose I were to let,(adsbygoogle = window.adsbygoogle || []).push({});

[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:

This seems to give the right answer. The first two terms are: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}]];

Sqrt[%]

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

The output of the first three terms are,Code (Text):

Series[Sqrt[F[z]], {z, moo[[2]], 4}]

which is nonsensical.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

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

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# What does this happen to my series expansion?

Can you offer guidance or do you also need help?

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