Announcements
- Coach is at MathFest 2024 till next week.
- I'll host office hours both Wed and Fri in room 247 (a couple doors down from Coach's).
Brain Gains
- In section 5.5's homework, we showed the first few terms of the Laurent series of $\frac{1}{\sin{z}}$ for $0<|z|<\pi$ is $$\frac{1}{z}+\frac{z}{6}+\frac{7}{360}z^3 +\cdots.$$ Compute each of the following integrals (the path around $|z|=1$ is traversed counter clockwise):
- $\ds\int_{|z|=1}\frac{dz}{\sin(z)}$, $\ds\int_{|z|=1}\frac{dz}{z\sin(z)}$, $\ds\int_{|z|=1}\frac{dz}{z^2\sin(z)}$, $\ds\int_{|z|=1}\frac{dz}{z^3\sin(z)}$, $\ds\int_{|z|=1}\frac{dz}{z^4\sin(z)}$
- Explain why $\ds\left|\frac{1}{1+z^4}\right|\leq \frac{1}{R^4-1}$ for $|z|=R>1$.
- Let $\gamma$ the the portion of the circle $|z|=R$ with non-negative imaginary part (so a parametrization is $z(t)=Re^{i\theta}$ for $0\leq \theta\leq \pi$. Show that $$\lim_{R\to\infty}\int_\gamma\frac{1}{1+z^4}dz=0.$$
Group problems
- Compute $\ds\int_{-\infty}^{\infty}\frac{x^2}{(x^2+9)^2}dx$
|
Sun |
Mon |
Tue |
Wed |
Thu |
Fri |
Sat |