## 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.

## Taylor expansion of x^2*arctan((1-x)/(1+x))

I’m trying to compute the Taylor expansion of this function:

$$f(x) = x^2\arctan\bigg(\frac{1-x}{1+x}\bigg)$$

Is it necessary to use the Cauchy Product?

## 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.

## How would I apply partial fraction expansion to this expression?

$$\displaystyle\frac{1}{X\bigg(1-\dfrac{X}{Y}\bigg)\bigg(\dfrac{X}{Z}-1\bigg)}$$

I want it in the form $$\frac{A}{X} + \frac{B}{(1-\dfrac{X}{Y})}+\frac{C}{(\dfrac{X}{Z}-1)}$$where I am required to find the values for A, B and C. I have found $$B=C$$ but I am unsure if that is correct

## Spherical expansion of an exponential function?

We know that a normal planewave can be Rayleigh expanded by spherical harmonics as : \begin{align} e^{i\vec{k}\cdot\vec{r}} = 4\pi \sum_{l=0}^\infty \sum_{m=-l}^l i^l j_l(kr)Y_{lm}(\hat{\vec{k}})Y_{lm}^\star(\hat{\vec{r}}). \end{align} Does any body know how to expand an exponential function $$e^{gz}$$ by spherical harmonics? Thanks!

Linhui

## 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:

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?