Week 3 Worksheet

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Power Series

1. Determine the radius of convergence of each of these power series.
   1. ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}\frac{2^n}{n}{z}^n}$
   2. ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}n!z^n}$
   3. ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}{n}^p{z}^n}$ for a fixed $p\in \mathbb{N}$.
2. The zeroth Bessel function $J_0(z)$ is defined by $$J_0(z)=\sum _{n=0}^{\mathrm{\infty }}(-1{)}^n\frac{1}{(n!{)}^2}\frac{z^n}{2n}$$ but what is its radius of convergence?
3. The power series $$\sum _{n=0}^{\mathrm{\infty }}{a}_n{z}^n$$ and $$\sum _{n=0}^{\mathrm{\infty }}{b}_n{z}^n$$ have radii of convergence $R>0$ and $S>0$ respectively. What is the radius of convergence of the series ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}(a_n+b_n)z^n}$?

Differentiation of Power Series

4. Fix $A$ and $B$ distinct complex numbers. Prove that $${\displaystyle \frac{A^n-B^n}{A-B}}=A^{n-1}{B}^0+A^{n-2}{B}^1+\cdots +A^1{B}^{n-2}+A^0{B}^{n-1}$$ for all $n\in \mathbb{N}$.
5. Starting from the geometric series $${\displaystyle \frac{1}{1-z}}=1+z+z^2+z^3+\cdots$$ on $\mathsf{B}(0,1)$ find a power-series representation for each of the following functions on $\mathsf{B}(0,1)$.
   1. $f(z)={\displaystyle \frac{1}{(1-z{)}^2}}$
   2. $g(z)={\displaystyle \frac{1}{1-z^2}}$
   3. $h(z)={\displaystyle \frac{2z}{(1-z^2{)}^2}}$

Special Functions

6. Prove that $\{\exp (z):z\in \mathbb{C}\}=\mathbb{C}\setminus \{0\}$.
7. A number $\tau$ is a \define{period} of a function $f:\mathbb{C}\to \mathbb{C}$ if $f(z+\tau )=f(z)$ for all $z\in \mathbb{C}$.
   1. Verify that $2\pi in$ is a period of $\exp$ for every $n\in \mathbb{Z}$.
   2. Does $\exp$ have any other periods?
8. Determine the real and imaginary parts of $\exp$ and verify the Cauchy-Riemann equations are satisfied everywhere.
9. For which complex numbers $z$ is $\exp (z)$ real? Complex?
10. Find the zeroes of $f(z)=1+\exp (z)$ and $g(z)=1+i-\exp (z)$.
11. Evaluate $\cos (i)$ and $\sin (i)$.
12. Can the ratio test be applied to the power series defining $\cos$ and $\sin$?
13. Verify the addition formulae $$\begin{array}{rl}\cos (z+w)& =\cos (z)\cos (w)-\sin (z)\sin (w)\\ \sin (z+w)& =\sin (z)\cos (w)+\sin (w)\cos (z)\end{array}$$ for all $z,w\in \mathbb{C}$.
14. Verify that $\exp (z)=\cosh (z)+\sinh (z)$ for all $z\in \mathbb{C}$.
15. Verify that $(\cosh (z){)}^2-(\sinh (z){)}^2=1$ for all $z\in \mathbb{C}$.