Expansion Definition and 1000 Threads

In this section, they derive the Sommerfeld formula. In the first step it seems like they have expanded ##\frac{1}{(1+e^x))^2}##. I'm not sure why does the series taylor expand as ##e^{-nx}##? Also how did they get from the 2nd to the 3rd step? Simply by comparing terms we see they are...

Any boolean function on n variables can be thought of as a function f : \mathbb{Z}_2^n \rightarrow \mathbb{Z}_2 which can be written as f(x) = \sum_{s \in \mathbb{Z}_2^n} \hat{f}(s) \prod_{i : x_i = 1} (-1)^{x_i} where \hat{f}(s) = \mathbb{E}_t \left[ f(t) \prod_{i : s_i = 1}...

Homework Statement Find the n+1 and n-1 order expansion of \stackrel{df}{dy}Homework Equations (n+1)Pn+1 + nPn-1 = (2n+1)xPn ƒn = \sum CnPn(x) Cn = \int f(x)*Pn(u)The Attempt at a Solution I know you can use the recursion relation for Legendre Polynomials once you combine Cn with the...

Are there any theories or thoughts that view spacetime as 'having' a coefficient of thermal expansion... analogous to the CTE of water? An inflection with density in regards to temperature?

Show that for small positive $x$, $$\left( \sin x \right)^{\cos x} = x -\left( 3 \log x + 1\right) \frac{x^{3}}{3!} + \Big( 15 \log^{2} x + 15 \log x + 11 \Big) \frac{x^{5}}{5!} + \mathcal{O}(x^{7})$$

I have the following expression: $$\frac{1}{p} \ln\left(1+\frac{p^1}{1!n}\sum_{i=1}^n x_i + \frac{p^2}{2!n} \sum_{i=1}^n x_i^2 + \frac{p^3}{3!n} \sum_{i=1}^n x_i^3 + \frac{p^4}{4!n} \sum_{i=1}^n x_i^4 + \cdots \right)$$ Now let $$Y = \frac{p^1}{1!n}\sum_{i=1}^n x_i + \frac{p^2}{2!n}...

I was reading an article about the recent type Ia supernova in the big dipper when the question popped into my head that I've forgotten about until now. Since we use the Ia supernova as our candles to measure the expansion of the universe, how do we know that everything is accelerating...

in order to explain the big bang theory and the expansion of space itself, people often draw upon the analogy of blowing air into a balloon and the 2 dimensional surface of the balloon expanding. isn't the 2-D balloon surface expanding in the third dimension, since the volume of the balloon is...

Consider the following charge distribution:A positive charge of magnitude Q is at the origin and there is a charge -Q on each of the x,y and z axes a distance d from the origin. I want to expand the potential of this charge distribution using spherical coordinates.Here's how I did it...

Is there a simple way to series expand a function of the form $$ \frac{1}{\sum_{n=0}^{\infty} a_n x^n} $$ about the point ##x=0##, such that it can be expressed as another sum ##\sum_n c_n x^n##? I tried doing it by taylor expansion but I end up with a sum of sums of products of sums :)...

Hey fellas please consider helping me with this.. There is a bar of some material that is heated from state 1 to 2 to 3. If l1 is the length at 1 then we have, l2 - l1 = l1(1 + a(t2-t1)) l3 - l1 = l1(1 + a(t3-t1)) If, t3-t2 = t2-t1, Then l3-l2 = l2-l1 But if it is written l3 - l2 =...

I want to create a mechanical vacuum (semi not necessarily complete), I've heard the easiest way to create one is by expansion (basically expanded a pump). But is this also the easiest way physically, as in takes the least amount of energy? Or is there any other 'mechanical' way possible for...

You probably know that for two commutative quantities x and y,we have: (x+y)^n=\sum_{r=0}^n \left( \begin{array}{c} n \\ r \end{array} \right) x^{n-r} y^r Now I want to know is there a similar formula for the case when x and y don't commute and we have [x,y]=c and [x,c]=[y,c]=0 ? Thanks

Hello, this is probably one of those shoot yourself in the foot type questions. I am going through Landau & Lifshits CM for fun. On page 7 I do not understand this step: L' = L(v'^2) = L(v^2 + 2 \vec{v} \cdot \vec{\epsilon} + \epsilon^2) where v' = v + \epsilon . He then expands the...

Homework Statement 1 mole of an ideal gas initially at 100° C and 10 atm is expanded adiabatically against a constant pressure of 5 atm until equilibrium is re-established. Given that the temperature dependence of the heat capacity is CV = 18.83 + 0.0209T calculate deltaU, deltaH and deltaS...

Homework Statement You are building a device for monitoring ultracold environments. Because the device will be used in environments where its temperature will change by 211°C in 2.99s, it must have the ability to withstand thermal shock (rapid temperature changes). The volume of the device is...

Homework Statement A clock based on a simple pendulum is situated outdoors in Anchorage, Alaska. The pendulum consists of a mass of 1.00kg that is hanging from a thin brass rod that is 2.000m long. The clock is calibrated perfectly during a summer day with an average temperature of 19.5°C...

Homework Statement Use Maclaurin’s theorem to derive the first five terms of the series expansion for ##(1+x)^{r}##, where -1<x<1. Assuming the series, obtained above, continues with the same pattern, sum the following infinite series ##1 + \frac{1}{6} - \frac{(1)(2)}{(6)(12)} +...

Homework Statement When the temperature of liquid mercury increases by one degree Celsius (or one kelvin), its volume increases by one part in 550,000. The fractional increase in volume per unit change in temperature (when the pressure is held fixed) is called the thermal expansion coefficient...

Homework Statement Given that ##f(x)=(1+x) ln (1+x)##. (a) Find the fifth derivative of f(x), (b) Hence, show that the series expansion of f(x) is given by ##x+\frac{x^{2}}{2} -\frac{x^{3}}{6} + \frac{x^{4}}{12} - \frac{x^{5}}{20}## (c) Find, in terms of r, an expression for the rth term...

Homework Statement Function f(x) = x^2/(x-1) should be expanded by Taylor method around point x=2 and 17th order derivative at that point should be calculated. Homework Equations Taylor formula: f(x)=f(x0)+f'(x0)*(x-x0)+f''(x0)*(x-x0)^2+... The Attempt at a Solution I...

Homework Statement For ##n>0##, the expansion of ##(1+mx)^{-n}## in ascending powers of ##x## is ##1+8x+48x^{2}+...## (a) Find the constants ##m## and ##n## (b) Show that the coefficient of ##x^{400}## is in the form of ##a(4)^{k}##, where ##a## and ##k## are real constants. Homework...

Homework Statement I'm taking the Laplace transform of F(s), and the first thing is to expand it by partial fraction or something so that I can match F(s) with a table of laplace transforms. Homework Equations The Attempt at a Solution Does partial fraction even work? I've got two...

Hello, I've been looking round trying to find a history of records, showing the speed of which the universe is expanding? Would this just be the hubbles constant? If so I'm looking for past Hubble constant values. Is there an archive for this? or is there a way to work past Hubble constant's to...

My question is regarding the early inflationary phase of the Big Bang. As I understand it, inflation is what gave rise to the expansion energy of the universe. Meaning, inflation gave the 'push' so to speak that set the everything moving apart. This makes sense because obviously the universe...

The Unruh Effect predicts that a uniformly accelerating observer in a vacuum field (full of perturbations) will observe an effective temperature. We know that space is expanding at an accelerating rate. My question is then, in all inertial reference frames would all 'observers' in the universe...

Homework Statement Which of the signals is not the result of Fourier series expansion? options : (a) 2cos(t) + 3 cos(3t) (b) 2cos(\pit) + 7cos(t) (c) cos(t) + 0.5 Homework Equations Dirichlet conditionsThe Attempt at a Solution From observation, I thought all are periodic and so must be...

Could you please clarify for me how much each of these contribute to the red shifting of light from distant objects? It seems to me that red shifting of light from near by objects i.e. within our galaxy would be affected more by the Doppler effect whereas for intergalactic objects the red...

Use taylor series to show that the infinite series from n=0 of $$\frac{s^n}{n!}\frac{d^n}{dq^n}(e^{-q^2})=e^{-(q+s)^2}$$

Here is the question: I have posted a link there to this topic so the OP can view my work.

Homework Statement Expand (1-2i)^10 without the Binomial Expansion Theorem I know I need to put this in polar form and then it's simple from there, however, I am simply having a difficult time finding the angle. Drawing the complex number as a vector in the complex plane I get a...

Is it possible to do a binomial expansion of (x+y)^{1/2}? I tried to compute it with the factorial expression for the binomial coefficients, but the second term already has n=1/2 and k=1, which makes the calculation for the binomial coefficient (n 1) weird, I think. Any advice?

Homework Statement 500 Cal are added to a gas inside a cylinder with a piston (containing one mole) by an external heating device. The volume of the gas doubles without any change in it's temperature of 300K. How much work is done on the piston? Homework Equations The Attempt at a...

Homework Statement There are 2 containers.Outside temperatue is -190 degree celcius.(boiling point of liquid oxygen is -180 degree celcius).One is filled with vacuum and can have a minimum height of 6000m and other is filled with liquid oxygen (1140 kg/m3 density) at -190 degree celcius...

Homework Statement Measurements of XRay scattering from a metal are made. The bragg peaks are θ = 53° and 48° for temperatures of 300K and 1272K. What is the linear expansion coefficient? Homework Equations linear expansion coef is given by (1/L)(dL/dT) Bragg equation (differential...

Homework Statement Hey guys, So I need a bit of help with this question: Find three Laurent expansions around the origin, valid in three regions you should specify, for the function f(z)=\frac{30}{(1+z)(z-2)(3+z)} Homework Equations None that I know of...just binomial expansion...

Good day. For $k\geq0$ a continuous function on $\mathbb{R}_+$ and $\{W_t\}$ a standard Brownian motion, could you help me find the Taylor's expansion of the following exponential: $e^{-\int_0^t k(s) dW_s}.$ For the case where $e^{-k W_t}$ where $k>0$ is a constant, I was able to recompute...

Homework Statement Expand f(z)=\frac{1}{z-4} in a laurent series valid for (a) |z|<4 and (b) |z|>4 Homework Equations The formula for laurent expansion... \sum_{n=-∞}^{+∞}a_{n}(z-z_{0})^{n} where a_{n}=\frac{1}{2\pi i}\oint_c \frac{f(z)}{(z-z_{0})^{n+1}}dz The Attempt at a...

Homework Statement Consider two rigid cylinders, P and Q, with respective volumes VP and VQ with VP << VQ. They are connected by a capillary. Initially cylinder Q is empty and cylinder P contains an ideal monatomic gas at 300K and pressure 1 bar. A moveable piston in P is used to maintain the...

Homework Statement How would a circle on a sphere expand as a function of the sphere's radius as the sphere expands? Homework Equations none were provided The Attempt at a Solution \S=4\,\pi \,{R}^{2} A=\pi \,{r}^{2} {\it dA}=2\,\pi \,r{\it dr} {\it dS}=8\,\pi \,R{\it dR}...

Homework Statement A scuba diver is 17.4m below the surface of the lake, where the water temperature is 8.25∘C. The density of fresh water is 1000 kg/m3. The diver exhales a 23.6cm3 bubble.What's the bubble's volume as it reaches the surface, where the water temperature is 15.1∘C? Homework...

Orthogonality of spherical bessel functions Homework Statement Proof of orthogonality of spherical bessel functions The book gave \int_{0}^{a}\int_{0}^{2\pi}\int_{0}^{\pi} j_{n} (\lambda_{n,j}r) j_{n'} (\lambda_{n',j'}r) Y_{n,m}(\theta,\phi)\overline{Y}_{n',m'}(\theta,\phi) \sin\theta...

Homework Statement A 61.00m long steel measuring tape is calibrated for use at 20.0∘C. How long is this tape under the following conditions: a hot day with 38∘C? Express your answer using four decimal places and include the appropriate units. i've been losing my mind over this, i swear...

This is not a homework question, but a conceptual question that I am trying to understand. We all have experienced/heard that if you put a completely full water bottle with the cap on in the freezer, the bottle cracks as the water freezers. The explanation is that the water expand as it...

As I understand it distant objects are so far away from us, not because they are moving away from us through space but because the very space between them and us is expanding. Am I right in thinking that one bit of evidence for this is that some galaxies are 30 billion light years away? So...

Hi all, below is a video showing a compilation of many expansions of my Wilson cloud chamber. The tracks shown are mostly alpha and cosmic rays. The chamber uses a rubber bulb from a turkey baster as the expansion piston. The object in the chamber is a cork with a radium paint coated pin. I'm...

Hi guys, i need your help to go about his question, Question: $$\text{Show that the coefficient }C_n \text{in the Laurent expansion of }$$ $$f(z)=(z+\frac{1}{z}) \text{ about z=0 is given by}$$ $$C_n=\frac{1}{2\pi}\int^{2\pi}_0 \cos(2cos(\theta))cos(n\theta)\, d\theta ,n\in\mathbb{z}$$

In the harmonic approximation, why is the volume of a crystal not temperature dependent? Does it have something to do with the fact that the amplitude of a harmonic oscillator is independent of the frequency?

Homework Statement Show by direct expansion that: det (I + εA) = 1 + εTr(A) + O(ε2) Homework Equations f(x) = f(a) + (x-a)f'(a) + (1/2)(x-a)2f''(a) + ... The Attempt at a Solution Does the question mean Taylor expansion when they say 'direct expansion'? I'm kind of stuck on...

Hello, I am just wondering if it is possible to actually do the following question based on the information given Homework Statement An isothermal reversible expansion, for which pV=constant as shown by the curve joining State A and State B, takes a system from State A to State B along...