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 ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}(a_n+b_n)z^n}$?

Holomorphic Functions

4. Use the definition of the derivative to differentiate each of the following functions.
   1. $f(z)=z^2+z$
   2. $f(z)={\displaystyle \frac{1}{z}}$
5. Define $f:\mathbb{C}\to \mathbb{C}$ by $f(z)=|z{|}^2$.
   1. Verify that $f$ is differentiable at the origin.
   2. Is $f$ holomorphic on any domain?

Cauchy-Riemann Equations

6. If $h:{\mathbb{R}}^2\to \mathbb{R}$ is defined by $h(x,y)=2xy$ use the definitions of the partial derivatives to calculate ${\partial }_{1}h$ and ${\partial }_{2}h$.
7. Verify that the real and imaginary parts of $f(z)=1/z$ together satisfy the Cauchy-Riemann equations.
8. Use the Cauchy-Riemann equations to determine whether there are any $z\in \mathbb{C}$ where $f(z)=|z|$ is differentiable.
9. Verify that the functions $$u(x,y)=x^3-3x{y}^{2}v(x,y)=3x^{2}y-y^3$$ satisfy the Cauchy-Riemann equations for all $(x,y)$. Show that $u,v$ are the real and imaginary parts of a holomorphic function $f:\mathbb{C}\to \mathbb{C}$.
10. Verify that the functions $$u(x,y)=\frac{x^4-6x^2{y}^2+y^4}{(x^2+y^2{)}^4}v(x,y)=\frac{4x{y}^3-4x^{3}y}{(x^2+y^2{)}^4}$$ satisfy the Cauchy-Riemann equations for all $(x,y)\ne (0,0)$. Show that $u,v$ are the real and imaginary parts of a holomorphic function $f:\mathbb{C}\setminus \{0\}\to \mathbb{C}$.
11. Suppose that $f(z)=u(x,y)+iv(x,y)$ is holomorphic. Use the Cauchy-Riemann equations to show that both $u$ and $v$ satisfy Laplace's equation. That is, verify that $${\partial }_1({\partial }_{1}u)+{\partial }_2({\partial }_{2}u)=0{\partial }_1({\partial }_{1}v)+{\partial }_2({\partial }_{2}v)=0$$ both hold. (The Laplacian of $g:{\mathbb{R}}^2\to \mathbb{R}$ is $\mathrm{△}g={\partial }_1({\partial }_{1}g)+{\partial }_2({\partial }_{2}g)$. Functions satisfying the Laplace equation $\mathrm{△}g=0$ are called \define{harmonic}.)
12. Let $f(z)=z^3$. Determine real-valued functions $u,v$ so that $f(x+iy)=u(x,y)+iv(x,y)$. Verify that both $u$ and $v$ satisfy the Laplace equation.
13. Suppose $f(x+iy)=u(x,y)+iv(x,y)$ is holomorphic on $\mathbb{C}$. Suppose we know that $u(x,y)=x^5-10x^3{y}^2+5x{y}^4$. Use the Cauchy-Riemann equations to find all the possible forms of $v(x,y)$. (The Cauchy Riemann equations have the following remarkable implication: suppose $f(z)=u(x,y)+iv(x,y)$ is holomorphic and we know a formula for $u$. Then we can recover $v$ up to a constant; similarly, if we know $v$ then we can recover $u$ up to a constant. Hence for holomorphic functions, the real part of a function determines the imaginary part up to constants, and vice versa.)
14. Suppose that $$u(x,y)=x^3-kx{y}^2+12xy-12x$$ for some constant $k\in \mathbb{C}$. Find all values of $k$ for which $u$ is the real part of a holomorphic function $f:\mathbb{C}\to \mathbb{C}$.
15. Show that if $f:\mathbb{C}\to \mathbb{C}$ is holomorphic and $f$ has a constant real part then $f$ is constant.
16. Show that the only holomorphic function $f:\mathbb{C}\to \mathbb{C}$ of the form $f(x+iy)=u(x)+iv(y)$ is given by $f(z)=\lambda z+a$ for some $\lambda \in \mathbb{R}$ and $a\in \mathbb{C}$.
17. Suppose that $f(x+iy)=u(x,y)+iv(x,y)$. is a holomorphic function and that $$2u(x,y)+v(x,y)=5$$ for all $x+iy\in \mathbb{C}$. Prove that $f$ is constant.

A Cauchy-Riemann Converse

18. Let $f(z)=\sqrt{|xy|}$ where $z=x+iy$.
    1. Show from the definition that $f$ is not differentiable at the origin.
    2. Show however that the Cauchy-Riemann equations are satisfied at the origin. Why does this not contradict our converse to the Cauchy-Riemann equations?