Announcements
- Coach is at MathFest 2024 till next week.
- Jen is ill. Coach asked me to hold class today in her place.
Brain Gains
- What does it mean to say that a function $f$ is meromorphic on a domain $D$?
Solution
We say $f$ is meromorphic on a domain $D$ if at every point of $D$ it is either analytic or has a pole.
- Give an example of a function $f$ that is meromorphic on a domain $D$. Then give an example of a function $f$ that is not meromorphic on a domain $D$.
Solution
- Any polynomial or rational function is meromorphic on the entire plane. If a function is analytic on $D$, then it is meromorphic on $D$.
- If a function has an essential singularity at point, then it is not meromorphic on a domain that includes that point. The function $\text{Log}$ is not meromorphic on a the unit circle.
Note: If a function is meromorphic inside a simple closed contour $C$, then we can show $f$ must have finitely many poles inside $C$. Suppose, by way of contradiction, that there were instead infinitely many poles. The fact that each pole is isolated, along with the Bolzano-Weierstrass theorem, allows us to produce a non-isolated singularity, a contradiction.
- Suppose $f$ is analytic in a neighborhood of $z_0$ and has a zero of order $m$ at $z_0$. We can then write $f(z)=(z-z_0)^m h(z)$. What do we know about $h(z)$?
- Suppose that $f$ is analytic and has a zero of order $m$ at $z_0$.
- Show that $g(z) = \frac{f'(z)}{f(z)}$ has a simple pole at $z_0$.
- What is $\text{Res}(g;z_0)$?
- Suppose that $f$ is analytic and has a pole of order $k$ at $z_p$.
- Show that $g(z) = \frac{f'(z)}{f(z)}$ has a simple pole at $z_p$.
- What is $\text{Res}(g;z_p)$?
- Let $\ds f(z) = \frac{ z^2(z-1)^3(z+4i) }{ (z+2i)^2(z^2+1)(z-8) }$ and compute $\ds\oint_{|z|=3}\frac{f'(z)}{f(z)}dz$.
Discussion
If $f$ is analytic and nonzero at each point of a simply closed positively oriented countour $C$ and is meromorphic inside $C$, then $$\frac{1}{2\pi i}\int_C\frac{f'(z)}{f(z)}dz = N_0(f) - N_p(f)$$ where $N_0(f)$ and $N_p(f)$ are, respectively, the number of zeros and poles of $f$ inside $C$, including multiplicities.
In other words, the integral above is the sum of the orders of the zeros minus the sum of the orders of the poles.
Group problems
- Let $P$ be a polynomial of degree $n$. Compute $$\ds \lim_{R\to \infty }\int_{|z|=R}\frac{P'(z)}{P(z)}dz.$$
- Create a function $f$ that is meromorphic inside $|z|=5$ such that $f$ has 2 zeros and 3 poles inside $|z|=5$ and $$\ds \frac{1}{2\pi i}\int_{|z|=5}\frac{f'(z)}{f(z)}dz = 4.$$
Discussion
Let's examine the following Mathematica notebooks.
When we view the argument principle as a discussion about how many times the image of $C$ wraps around the origin, we get a new theorem. The theorem allows us perturb a function $f$ by some amount $h$ without changing the sum of the orders of the zeros. We just need to make sure that for each $z$, the perturbation $h(z)$ is small enough to keep the sum from winding around the origin.
If $f$ and $h$ are each functions that are analytic inside and on a simple closed contour $C$ and if the strict inequality $$|h(z)|< |f(z)|$$ holds at each point on $C$, then $f$ and $g=f+h$ must have the same total number of zeros (counting multiplicities) inside $C$.
Practice
- Show that the zeros of $f(z) = z^5+10z+1$ must lie in the disk $|z|<3$.
- Prove that the equation $z^3+9z+27=0$ has no solution in the disk $|z|<2$.
- Find $R$ sufficiently large to guarantee the polynomial $p(z) = z^5+3z^4+7z^3-8z^2+10z+13$ has all 5 zeros inside $|z|=R$.
- Prove the fundamental theorem of algebra, using Rouché's theorem.
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