Articles of complex analysis

roots of a polynomial inside a circle

I am asked to show that for $n$ larger or equal to $2,$ the roots of $1 + z + z^{n}$ lie inside the circle $\|z\| = 1 + \frac{1}{n-1}$ Attempt1: Induction for the case $n = 2,$ the roots of $1 + z + z^{2}$ lie inside the circle of radius $2.$ Now we […]

Analytic continuation of factorial function

We know that the factorial can be extended to the whole complex plane except at negative integers and $0$ . But are there any theorems that allow us to do so ? . I know we can use the Identity theorem in complex analysis to extend functions on open sets but how does that work […]

Inverse of the Joukowski map $\phi(z) = z + \frac{1}{z}$

We know the Joukowski map $$\phi(z) = z + \frac{1}{z}$$ which maps the upper semidisc of radius $1$ in the lower half plane, and the lower semidisc of radius $1$ in the upper half plane. What is the inverse of this function ? We obtain $z ^{2}-zy + 1 = 0$ and this equation has […]

Type and number of points at infinity on Riemann surface

Consider a polynomial $P(z) = z^4 \in \mathbb{C}[z]$. Set-theoretically $P(z)$ has one root equal to zero. From algebraic point of view it has four roots: root zero has multiplicity four. Also we can’t draw a curve in $\mathbb{C}$ around one of such roots but not around the others. Now consider a Riemann surface $X$ given […]

Residue of complex function

I know that if function is meromorphic then it will have resiue. My simple question is if function is not meromorphic then can it have a residue, because I am not getting any such statement in my book. If it is correct or incorrect then plz help me with suitable example.

Difficulty evaluating complex integral

The integral along the path $\gamma(t)=e^{2ti},\;t\in[0,2\pi]$ is $\begin{equation*} \int_{\gamma}\frac{1}{z^{2}-1}dz \end{equation*}$. I approached this like a real integral in the hopes things would work out, first by performing a u sub $\begin{equation*} u=e^{2ti}\Rightarrow du=2e^{2ti}dt\Rightarrow dt=\frac{du}{2e^{2ti}} \end{equation*}$ Which brought me to $\begin{align*} &\int_{t=0}^{t=2\pi}\frac{i}{u^{2}-1}du= i\int_{t=0}^{t=2\pi}\frac{1}{(u+1)(u-1)}du\\ &=i\frac{1}{2}\int_{t=0}^{t=2\pi}\frac{1}{(u+1)}du- i\frac{1}{2}\int_{t=0}^{t=2\pi}\frac{1}{(u-1)}du\end{align*}$ In the reals this would obviously $\frac{i}{2}\log(e^{2ti}-1)-\frac{i}{2}\log(e^{2ti}+1)|_{t=0}^{2\pi}$ but i am pretty […]

Why is the MacLaurin series proof for eulers formula $ e^{i\theta} = \cos(\theta) + i\sin(\theta) $ valid?

The proof for this $$ e^{i\theta} = \cos(\theta) + i\sin(\theta) $$ using the MacLaurin series is all right for a high school level, but I dont understand why the series that has been derived for the reals should hold for complex numbers too. Could someone give a sufficient reason why it is correct to use […]

How to show set of all bounded, analytic function forms a Banach space?

I am trying to prove that set of bounded, analytic functions $A(\mho)$, $u:\mho\to\mathbb{C}$ forms a Banach space. It seems quite clear using Morera’s theorem that if we have a cauchy sequence of holomorph functions converge uniformly to holomorph function. Now i am a bit confused what norm would be suitable in order to make it […]

Proving that $\phi_a(z) = (z-a)/(1-\overline{a}z)$ maps $B(0,1)$ onto itself.

I want to prove that if $\phi_a: B(0,1) \to \Bbb C$ is given by $\phi_a(z) = (z-a)/(1-\overline{a}z)$ with $|a| < 1$, then $|\phi_a(z)| < 1$. Resist the itch on your finger urging you to close the question: I already took a look at this question and this one. I’m supposed to prove things in the […]

identity of $(I-z^nT^n)^{-1} =\frac{1}{n}$

I am trying to understand the identity $$(I-z^nT^n)^{-1} =\frac{1}{n}[(I-zT)^{-1}+(I-wzT)^{-1}+…+(I-w^{n-1}zT)^{-1}] \quad (*),$$ where $T \in \mathbb{C}^{n\times n},z\in \mathbb{C}$ and the spectral radius of $T$, $\rho(T)=\max\{|\lambda|: \exists v, Tv=\lambda v\}\leq 1$ and $|z|<1$ and $w$ is a primitive $n$th root of 1,i.e. $w =e^{i2\pi/n}$. I have tried using the identity $$[I-A]^{-1} = I+A+A^2+A^3+… $$ which holds when […]