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I am trying to work on this old qual exam.

Here is the question:

Find the number of roots (counting multiplicities) of the function $$f(z)=\cos(z)-1 + \frac{z^2}{2}$$ inside the domain $\vert z \vert <1$.

My work: I first thought of Rouché's theorem. But then I figured that $f(z)=z^4\left(\frac{1}{4!}-\frac{z^2}{6!}+\cdots\right)$. So $f(z)=z^4 g(z)$ for some analytic function $g(z)$ such that $g(0)\neq 0$. And then I used the argument principle to conclude that the number of zeroes is $4$. Is this correct?

Also, how do I know for sure that there are no other zeroes of $g$ inside the unit disk centered at $0$. Any hints? Thanks

Antonio Vargas
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Cousin
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1 Answers1

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Since $f$ is analytic in the domain $D:=\{z:|z|<1\}$, the Argument Principle says that the number of zeros of $f$ in $D$ is given by $${1\over 2\pi i}\int_D {f'(z)\over f(z)}\,dz,$$ which we will compute via the Residue Theorem.

First, expanding about $z=0$, $${f'(z)\over f(z)}=\frac{4}{z}-\frac{z}{15}+\frac{z^3}{6300}+\frac{z^5}{189000}+\cdots,$$ so by the Residue Theorem, $${1\over 2\pi i}\int_D {f'(z)\over f(z)}\,dz=\text{Res}(f'(z)/f(z),0)=4.$$

Hence, $f(z)$ has 4 zeros (counting multiplicities) in $D$.

JohnD
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    I still don't see how the last integral is equal to $8\pi i$. Can you please explain a little more?. Thanks. – Cousin Jan 09 '13 at 23:37
  • The earlier integral I computed with Mathematica without much thought and got $4$. Perhaps the revision is clearer. – JohnD Jan 10 '13 at 04:36
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    Yes, it makes sense now. Plus I was actually able to do this with Rouche's theorem as well. It never hurts to know two methods. Like one of my Professors once said, if you can prove it in two ways, it must be really really true. :).Thanks!! – Cousin Jan 10 '13 at 07:28
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    Why do you only take the residue at $0$ to calculate the integral? Don't all of the zeroes of $f$ need to be considered? – Bartek Feb 07 '13 at 00:48
  • @user54755 How did you solve this using Rouche's theorem? – Arpan Dutta Aug 11 '13 at 05:24
  • This computation only shows that $f(z)$ has a zero of multiplicity $4$ at $z=0$. By only calculating the residue at $z=0$ you have implicitly taken your contour to be an arbitrarily small circuit about the origin. You should have taken it to be the circle $|z| = 1$. – Antonio Vargas Oct 13 '13 at 18:51