complex analysis - Simple case of analytic continuation

I have yet to formally study complex analysis, yet the topic of analytic continuation, specifically with respect to the Riemann Zeta function, is fascinating.

My question is, what are some relatively simple cases of analytic continuation?

I ask because I am looking for a basic example to study and in the case of Riemann Zeta, the derivation seems rather complicated.

Is the continuation of $f(z)=\sqrt{z}$ a relatively simple case of analytic continuation? Based on this Wolfram MathWorld page, I would think so; however, the article goes on to say that the continued power series of the function can become a multivalued function as shown in the following graphic. Must an analytic continuation necessarily yield a multivalued function?

Thank you in advance.

Answer

The analytic function given by the sum of the series
$$

f(z):=\sum_{n=0}^{+\infty}z^n
$$
and the function
$$
g(z)=\frac1{1-z}
$$
agree on the open disk $\{|z|<1\}$ (which is the domain of $f$), but the latter is analytic on $\Bbb C\setminus\{1\}$, thus $g$ extend the function $f$ on this bigger domain.

elementary set theory - Examples of surjective functions $mathbb{Nto Z}$ and $mathbb{Nto Ntimes N}$

I'm asked to give examples of surjective functions $\mathbb{N} \rightarrow \mathbb{Z}$ and $\mathbb{N} \rightarrow \mathbb{N} \times \mathbb{N}$. Could a function $\mathbb{N} \rightarrow \mathbb{Z}$, just be $\mbox{floor}(x)$, and a function $\mathbb{N} \rightarrow \mathbb{N} \times \mathbb{N}$ be $x^2$? In both cases, every element in the codomain would be mapped to. Or is it meant to be a function like: $x \mapsto (x,y)$?

How can I write this differential equation system in MATLAB?

I am newbie MATLAB user. How to write this equation system in MATLAB?

$$\frac{dN_{ik}}{dt}=\sum_{j=i}^{n}\sum_{l=1}^{n}\beta_{jl}b_{ikjl}S_{jl}N_{jl} - S_{ik}N_{ik}$$

Here, I want to know if I take $n=3$ or any constant number what kind of equation system will I have?

How can i run this loop without entering $\beta$, $b$ and $S$ functions. I just want to see matrix form of this system. Sorry for my bad English, thanks for help.

calculus - To prove a sequence is Cauchy

I have a sequence:
$ a_{n}=\sqrt{3+ \sqrt{3 + ... \sqrt { 3} } } $ , it repeats $n$-times.

and i have to prove that it is a Cauchy's sequence.
So i did this:
As one theorem says that every convergent sequence is also Cauchy, so i proved that it's bounded between $ \sqrt{3}$ and $ 3 $ (with this one i am not sure, please check if i am right with this one.)And also i proved tat this sequence is monotonic. (with induction i proved this: $ a_{n} \leq a_{n+1} $
so if it's bounded and monotonic, therefore it is convergent and Cauchy.
I am just wondering if this already proved it or not? And also if the upper boundary - supremum if you wish - is chosen correctly.
I appreciate all the help i get.

Answer

${ a }_{ n+1 }=\sqrt { a_{ n }+3 } $ $\Rightarrow \quad { a^{ 2 } }_{ n+1 }=a_{ n }+3$ as $n\rightarrow \infty $ $\Rightarrow \quad { a^{ 2 } }_{ n+1 }=a_{ n }+3$ $\quad x^{ 2 }=x+3\quad \Rightarrow $ $x^{ 2 }-x-3=0 $ $and\quad it\quad$ convergents to the $x=\frac { 1+\sqrt { 13 } }{ 2 } $

number theory - Generate unique integer from $n$ integers and solve to get the integers from result

What could be the best way to generate a unique integer from $n$ integers in order $(n_1,n_2,\ldots)$?

Further, from $n$, we should be able to get back each $n_1, n_2,\ldots $ etc.
For example, from $n_1=120, n_2=135, n_3=789, n_4=980$, we need a number $n$. And from $n$, we should be able to get back numbers $n_1=120, n_2=135, n_3=789, n_4=980$. For the sake of computation effort, it would be better if we could generate as much small number as possible.

Thank you.

calculus - Find $lim_{x to 0}frac{cos 2x-1}{cos x-1}$ without L'Hopital's rule.

$$\lim_{x \to 0}\frac{\cos 2x-1}{\cos x-1}$$

I have found the above limit using L'Hopital's rule but since this rule is not given in the book so I'm supposed to do it without using this rule.

I know $$\lim_{x \to 0}\frac{1-\cos x}{x}=0$$

I tried to get something of the form of the above limit but I failed to do so.

Kindly help me solve this problem without using L'Hopital's rule.

Answer

Recall $\cos(2x)=2\cos^2(x)-1$ so we may rewrite as

$$\lim\limits_{x\to 0} 2\frac{\cos^2(x)-1}{\cos(x)-1}=\lim\limits_{x\to 0} 2\frac{(\cos(x)-1)(\cos(x)+1)}{\cos(x)-1}=2(\cos(0)+1)=4$$

linear algebra - Can we prove $BA=E$ from $AB=E$?

I was wondering if $AB=E$ ($E$ is identity) is enough to claim $A^{-1} = B$ or if we also need $BA=E$. All my textbooks define the inverse $B$ of $A$ such that $AB=BA=E$. But I can't see why $AB=E$ isn't enough. I can't come up with an example for which $AB = E$ holds but $BA\ne E$.
I tried some stuff but I can only proof that $BA = (BA)^2$.

Edit: For $A,B \in \mathbb{R}^{n \times n}$ and $n \in \mathbb{N}$.

Answer

If $AB = E$, then (the linear application associated to) $A$ has a right inverse, so it's surjective, and as the dimension is finite, surjectivity and injectivity are equivalent, so $A$ is bijective, and has an inverse. And the inverse is also a right inverse, so it's $B$