Expansion Definition and 1000 Threads

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...

Homework Statement Find e^{2-x} using taylor/mclaurin expansionHomework Equations e^1 = \sum_{n=0}^\inf \frac{1}{n!} e^x = \sum_{n=0}^\inf \frac{x^n}{n!} The Attempt at a Solution Can I just do: e^{1+1-x} [\sum_{n=0}^\inf \frac{1}{n!}*\sum_{n=0}^\inf \frac{1}{n!} *...

Hi MHB, I've come across this problem and I think I've observed a pattern when I tried to solve it by using the method of comparison with some lower values of the exponents, but then I just couldn't deduce the answer to the problem because the pattern suggests that I can't. Here is the...

Hello everyone, I have a question about metric expansion of space. According to Wikipedia (ok probably is not the best source, but I have only a qualitative understanding of physics) the expansion occurs only at scales larger than galaxy clusters...

Homework Statement Show that \frac{dp}{p} =\frac{\gamma}{\gamma-1}\frac{dT}{T} if the decrease in pressure is due to an adiabatic expansion.Homework Equations Poisson equations: Pv^{\gamma} Tv^{\gamma - 1} Ideal Gas Law: Pv=R_{d}T, where R_{d} is the dry air gas constant. Hydrostatic...

Can anyone help me with this calculation? I'm trying to figure out how much compressed air (generated from screw compressor) will expand by a ratio of what when fed into a wood furnace. 1) Wood Furnace Fuel Input: 30kg dry wood fuel/hr 2) Compressed Air Pressure: 8 bar 3) Compressed Air Flow...

Solve inhomogeneous differential equation \[ y'' + k^2y = \phi(x) \] with homogeneous boundary conditions \(y(\ell) = 0\) and \(y(0) = 0\) by expanding \(y(x)\) and \(\phi(x)\) \begin{align*} y(x) &= \sum_na_nu_n(x)\\ \phi(x) &= \sum_nb_nu_n(x) \end{align*}...

Homework Statement In the VDW eqn of state, ##(P + a/v^2)(v-b)=RT## write Pv as a function of {P,T} and by expanding the result in powers of P or otherwise show that the first terms of a Virial expansion in powers of P for a VDW gas are given by $$Pv = RT + \left(b - \frac{a}{RT}\right)P +...

I read on 27 chapter from Blundell's Thermal Physics 2nd edition. For ideal gases, Joule expansion doesn't cool gases. But for real gases, Joule expansion makes cooling effects. And this book(page 314) says that when gases expand, their potential energy of molecular interactions...

Solve without expanding the determinant having elements which are listed as follows row-wise = {1^2,2^2,3^2,2^2,3^2,4^2,3^2,4^2,5^2} where , the determinant is of the order 3 I have tried some combinations of operations performed row or column wise that could produce 2 zeros either in one of...

I'm at the end of a very long Poisson Processes question, involving inhomogeneous Poisson Processes. I just need to be able to expand the following expression to be able to complete the question. exp[{(sin ∏h)/∏} -h] Would anyone please be able to provide some help, with steps please!

Homework Statement Show that: Ʃ(-1)n/(n^2+a^2) (from n=0 to ∞) = pi/[asinh(pi*a)], a\neq in, n\in Z. Homework Equations f(z) = f(0) + Ʃbn(1/(z-an)+1/an) (from n=1 to ∞) , where bn is the residue of f(z) at an. The Attempt at a Solution The main problem is I don't how to pick the...

Hi, Could someone please identify the following series expansion for me, if possible: $$f(x) = 1/h\int_x^{x+h}f(t)dt + A_{1}Δf(x) + A_{2}hΔf'(x) +...+ A_{m-1}h^{m-2}Δf^{(m-2)}(x) +r$$ where $$A_{m}$$ are, as far as I know, plain constants and $$Δ = [x, x+h]$$. I think this result was...

How would one expand f+df about x? I'm messing something up in the process and can't seem to resolve it lol

Homework Statement Show that 1- x/2 + x^2/3 - x^3/4 + x^4/5... (-1)^n (x^n)/(n+1) = ln(1+x)/x with |x| < 1 Homework Equations The Attempt at a Solution finding derivative of the function multiplied by x d/dx(xS(x)) = 1 -x + x^2 - x^3 + x^4 - x^5 +... absolute value of this...

Homework Statement Find the coefficient of x8 in the expansion of (x2-3)7 Homework Equations This one: (n n-j) an-jxj The Attempt at a Solution Hi! Well, I know that x= x2 a= -3 n= 7 and apparently, j=8. This is what confuses me. n-j is a negative number.. how would...

Homework Statement A brass bar and an aluminium bar are each attached opposite each other to immovable walls. (There is a diagram, but I think the idea is clear enough). There is a gap between the two bars of 1.3x10-3m at 28°C. At what temperature will the gap be closed? coefficient of...

Calculation of $57^{th}$ digit in the decimal expansion of $\displaystyle \frac{1}{57}$

Hello, according to my textbook, the Taylor expansion of first order of a scalar function f(t) having continuous 2nd order derivative is supposed be: f(t) = f(0) + f'(0)t + \frac{1}{2}f''(t^*)t^2 for some t^* such that 0\leq t^* \leq 1 Quite frankly, I have never seen such a formulation...

Homework Statement Find the Laurent expansion for \frac{1}{z^2-1} in the annulus 1 < |z-2| < 3 The Attempt at a Solution I've gotten to the last parts but getting stuck there. First I expanded the denominator and did a partial fraction decomposition and arrived at...

From my own(simplistic)perspective,dark energy is expanding the universe by creating further spacetime at a velocity faster than the speed of light,if gravitational waves propagate at the speed of light how is it possible that M31 and the milky way are still bound by gravity when the fabric of...