Is this a correct Laurent series expansion for the given annulus?

Expand the function $ $ f(z) = \frac{1}{(z + 1)(z + 3)}$ $ in a Laurent series valid for $ 1 < |z| < 3$


My attempt:

$ $ \frac{1}{(z + 1)(z + 3)}=\frac{1}{4}.\frac{1}{1+z}-\frac{1}{4}.\frac{1}{3+z}$ $

$ $ =\frac{1}{4}.\frac{1}{1-(-z)}-\frac{1}{4}.\frac{1}{3-(-z)}$ $

$ $ =\frac{1}{4}.\frac{1}{z}\frac{1}{\frac{1}{z}-(-1)}-\frac{1}{4}.\frac{1}{3-(-z)}$ $

$ $ =\frac{1}{4}.\frac{1}{z}\frac{1}{1-(-\frac{1}{z})}-\frac{1}{4}.\frac{1}{3}.\frac{1}{1-(-\frac{z}{3})}$ $

and now both fraction can be expanded using the geometric series

$ $ =\frac{1}{4}.\frac{1}{z}(1-\frac{1}{z}+\frac{1}{z^2}…)-\frac{1}{12}.(1-\frac{z}{3}+\frac{z^2}{3^2}…)$ $


Is the expansion correct?

NOTE: What is written above is the entire question.

Expansion of a product of two q-pochhammer terms

For $ n$ an integer and $ k$ an integer > 0, with $ |q| < 1$ a real number, consider the product $ A B$ where $ A = (q^n; q^2)_k$ and $ B = (-q^{n+1}; q^2)_k$ . That is,

$ $ AB = (q^n; q^2)_k (-q^{n+1}; q^2)_k = (q^n, -q^{n+1}; q^2)_k \tag{1}\label{eq1}. $ $

Here $ (a; q)_n = \prod_{k=0}^{n – 1} (1 – a q^{k})$ is the q-pochhammer symbol.

Is there a nice formula to expand $ \eqref{eq1}$ beyond simply applying the q-binomial theorem, $ (a; q)_n = \sum_{k=0}^n (-1)^k q^\binom{k}{2} a^k {n \brack k}_q $ , to either one of the terms $ A$ or $ B$ ?

Note that I’m trying to prove an expansion formula for $ A B $ with a particular value of $ |q| < 1$ in mind. So far I’ve made no headway using the q-binomial theorem alone (even when attempting some further elementary transformations using identities from the appendix of Gasper & Rahman).

Any reference suggestions would also be helpful.

Creating a series expansion from two functions multiplied together

First I have this function:

$ $ g_0(1,\eta)=\frac{\frac{3\eta}{\eta_c-\eta}+\sum_{k=1}^4kA_k\left(\frac{\eta}{\eta_c}\right)^k}{4\eta}$ $

I need to multiply this function by another function:

$ $ C_{01}r^0\left(\frac{\eta}{\eta_c}\right)+C_{11}r^1\left(\frac{\eta}{\eta_c}\right)+C_{21}r^2\left(\frac{\eta}{\eta_c}\right)+C_{31}r^3\left(\frac{\eta}{\eta_c}\right)+C_{41}r^4\left(\frac{\eta}{\eta_c}\right)+C_{51}r^5\left(\frac{\eta}{\eta_c}\right)+C_{61}r^6\left(\frac{\eta}{\eta_c}\right)$ $

Then I need to accumulate all terms in a series expansion about $ \frac{\eta}{\eta_c}=0$ . I need to collect the coefficients in front of each order polynomial. Can anyone help with this?

Recurrence relation for the asymptotic expansion of an ODE

I want to solve for the asymptotic solution of the following differential equation

$ $ \left(y^2+1\right) R”(y)+y\left(2-p \left(b_{0} \sqrt{y^2+1}\right)^{-p}\right) R'(y)-l (l+1) R(y)=0$ $

as $ y\rightarrow \infty$ , where $ p>0$ . I did the standard way by obtaining a series solution by the Frobenius method prescription in the form

$ $ R(y)=\sum_{n=0}^\infty \frac{a_{n}}{y^{n+k}}$ $ where $ k=l+1$ is the indicial exponent. I had difficulty finding for a recurrence relation for the coefficients $ a_n$ for arbitrary value of the parameter $ p$ . Right now, I am just doing the brute force method of solving individual $ a_n$ for every value of $ p$ . But I am just wondering whether the recurrence relation is possible to solve. Any help is appreciated.

Motivation behind the definition of order-$k$ (edge) expansion?

I’m trying to understand the motivation behind the idea of order-$ k$ (edge) expansion for partitions of a graph, defined below:

For simplicity, let’s focus on $ d$ -regular graphs. The definitions I’m working with are:

The edge expansion of a subset of vertices $ S$ is $ $ \phi(S) = \frac{E(S,V \setminus S)}{d \cdot |S|},$ $ where $ E(A,B)$ counts the number of edges with one endpoint in $ A$ and the other in $ B$ .

Let $ S_1, \ldots, S_k$ collection of disjoint vertices, then their order-$ k$ expansion is $ $ \phi_k(S_1, \ldots, S_k) = \max_{i=1,\ldots, k} \phi(S_i).$ $ The order-$ k$ expansion of a graph $ G$ is $ $ \phi_k(G) = \min_{S_1, \ldots, S_k \text{ disjoint} } \phi(S_1, \ldots, S_k).$ $

My question is: why do we consider the $ \max$ in the definition of $ \phi_k(S_1, \ldots, S_k)$ ? If $ S_j$ is the subset of vertices for which $ \phi(S_i)$ is a maximum, this means there are “a lot” of edges from $ S_j$ to $ V \setminus S_j$ , relative to $ d|S_j|$ . Isn’t the $ \min$ more interesting here? Doesn’t the $ \min$ correspond to $ S_k$ that can easily be removed from the graph (few edges need to be cut), and yet the subgraph being removed is relatively dense?

Expansion in directional derivate Riemannian metric.

Let $ M$ be a spherically symmetric $ C^2$ manifold which has the property that at any point in $ M$ , denoted by $ O$ there exists a small ball $ B_R$ centered at $ O$ and polar coordinates $ (r,\theta)$ such that the Riemannian metric in $ B_R\setminus O$ is given by $ ds^2=dr^2+\psi^2(r)d\theta^2$ , where $ d\theta^2$ is the metric on $ \mathbb{S}^{n-1}$ , $ \psi\in C^2$ .

Let us now introduce polar coordinates in a small ball $ B_R$ centered at $ x$ . In this case the geodesic distance $ s_{xy}$ between the points x and y is $ s_{xy}=p$ . And let $ u\in C^2$ . Show that, $ $ u(p,\theta)=u(x)+u_{p}(p.\theta)p+\frac{1}{2}u_{pp}(p,\theta)p^2 \qquad(*)$ $

My approach: This is an expansion in directional derivate of the function $ u$ , like answer of Muphrid. But the terms in (*) isn’t correct for me. And I don’t see a rigourous way of prove this with the riemannian metric. Any hint will be appreciated.

Let $E$ be the set of all $x \in [0,1]$ whose decimal expansion contains only the digits 4 and 7. Is $E$ compact? – Explanation

Let $ E$ be the set of all $ x \in [0,1]$ whose decimal expansion contains only the digits 4 and 7. Is $ E$ compact?

This question comes from Baby Rudin Ch.2 – 2.17.

I have a solution I found for the question and this is what my questions revolve around. I believe I have the correct understanding of what was set out to be accomplished in order to prove the claim (I will type my “understanding” below). But I am a little lost in how the bounds that were stated were derived.

As such here is the solution:

enter image description here

My understanding: In order to show that $ E$ is compact we have to show closed and boundedness of the set. Boundedness is obvious, to show it is closed we show that $ E^{c} = [0,1] -E $ is open. This is accomplished by showing that any point $ y \in E$ will be at the minimum a fixed distance away from every point of $ E^{c}$ which we can then use to choose $ \epsilon$ to always be less than this distance. which will show that $ E^{c}$ is open (this is a new technique to show openess that I’ve seen). But after concluding $ E^{c}$ is open then we can conclude $ E$ is closed.

Questions:

i) How are these bounds constructed? i.e how is $ \epsilon \leq \frac{7}{9} 10^{-m}$ found? And as such how do the other two bounds follow? i.e: $ \frac{2}{9 \cdot 10^{m}}$

ii) As mentioned above is this another wy to show a point is an interior point? i.e: by showing no point from outside the set could be a fixed distance away?