# What does this happen to my series expansion?

1. Sep 4, 2009

### rsq_a

I came across this really strange error when doing series expansions in Mathematica. Suppose I were to let,

$$F(z) = \frac{(z+2)^2}{(z+1)^2} - 0.4$$

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

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

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}]];
Sqrt[%]

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?