Week 4 Worksheet

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Differentiation of Power Series

1. 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 $n\in \mathbb{N}$.
2. 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. $h(z)={\displaystyle \frac{2z}{(1-z^2{)}^2}}$

The Exponential Function and Trigonometric Functions

3. Prove that $\{\exp (z):z\in \mathbb{C}\}=\mathbb{C}\setminus \{0\}$.
4. 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?
5. Determine the real and imaginary parts of $\exp$ and verify the Cauchy-Riemann equations are satisfied everywhere.
6. For which complex numbers $z$ is $\exp (z)$ real? Complex?
7. Find the zeroes of $f(z)=1+\exp (z)$ and $g(z)=1+i-\exp (z)$.
8. Evaluate $\cos (i)$ and $\sin (i)$.
9. Can the ratio test be applied to the power series defining $\cos$ and $\sin$?
10. 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 $z,w\in \mathbb{C}$.
11. Verify that $\exp (z)=\cosh (z)+\sinh (z)$ for all $z\in \mathbb{C}$.
12. Verify that $(\cosh (z){)}^2-(\sinh (z){)}^2=1$ for all $z\in \mathbb{C}$.

Contours

13. Compute the length of the smooth path $\gamma (t)=t^3+it$ on $[3,5]$.
14. Let $R$ be the rectangle with vertices $0$, $2$, $2+3i$ and $3i$. Give formulas for smooth paths ${\gamma }_1$, ${\gamma }_2$, ${\gamma }_3$, ${\gamma }_4$ such that the contour $\mathrm{\Gamma }=({\gamma }_1,{\gamma }_2,{\gamma }_3,{\gamma }_4)$ traverses the perimeter of $R$ once anti-clockwise.
15. The \define{reverse} of a path $\gamma :[a,b]\to \mathbb{C}$ is the path $\tilde{\gamma }:[a,b]\to \mathbb{C}$ given by $\tilde{\gamma }(t)=\gamma (a+b-t)$.
    1. Draw the reverse of $\gamma (t)=it+2(1-t)$ on $[0,1]$.
    2. How would you define the reverse of a contour $\mathrm{\Gamma }=({\gamma }_1,{\gamma }_2,{\gamma }_3)$?