Taylor expansion Definition and 174 Threads

How do you Taylor expand e^{i \vec{k} \cdot \vec{r}} the general formula is \phi(\vec{r}+\vec{a})=\sum_{n=0}^{\infty} \frac{1}{n!} (\vec{a} \cdot \nabla)^n \phi(\vec{a}) but \vec{k} \cdot \vec{r} isn't of the form \vec{r}+\vec{a} is it?

Homework Statement I'm having a hard time following a taylor expansion that contains vectors... http://img9.imageshack.us/img9/9656/blahz.png http://g.imageshack.us/img9/blahz.png/1/ Homework Equations The Attempt at a Solution Here's how I would expand it: -GMR/R^3 -...

Can someone please tell me how to expand \ln(x + \sqrt{1+x^2}) for small x? I'd like to retain terms at least up to order x^5. Thanks!

Homework Statement f(E) = \left(\frac{E_c}{E} \right)^{1/2} + \frac{E}{kT} Expand this as a Taylor function with the form... f \approx a_0 + a_1(E-E_0) + a_2(E-E_0) Hint being a_1 will be 0, because E_0 is a Gamow peak in this case, so slope will be 0. What I need to do is...

Homework Statement Develop the Taylor expansion of ln(1+z). Homework Equations Taylor Expansion: f(z) = sum (n=0 to infinity) (z-z0)n{f(n)(z0)}/{n!} Cauchy Integral Formula: f(z) = (1/(2*pi*i)) <<Closed Integral>> {dz' f(z')} / {z'-z} The Attempt at a Solution I have NO idea...

Homework Statement With n>1, show that (a) \frac{1}{n}-ln\frac{n}{n-1}<0 and (b) \frac{1}{n}-ln\frac{n+1}{n}>0 Use these inequalities to show that the Euler-Mascheron constant (eq. 5.28 - page330) is finite. Homework Equations This is in the chapter on infinite series, in the section...

Homework Statement What is the quadratic approximation to the potential function? Homework Equations U(x) = U0((a/x)+(x/a)) U0= 20 a=4 The Attempt at a Solution This is just the last part of a question on my engineering homework, I never learned Taylor expansions before even...

Hi There. Was working on these and I think I managed to get most of them but still have a few niggling parts. I've managed to do questions 2,3,3Part2 and I've shown my working out so I'd be greatful if you could verify whether they are correct. Please could you also guide me on Q1 & 4. Q1...

can anybody help me with this integration? Integral of e to the -2x times x squared dx it expands to 1/4, but I'm not sure how to start.

\biggl(-\frac{x^2}2 + \frac{x^4}{24} - \frac{x^6}{720} +\mathcal{O}(x^8)\biggr)-\frac12\biggl(-\frac{x^2}2+\frac{x^4}{24}+\mathcal{O}(x^6)\biggr)^2+\frac13\biggl(-\frac{x^2}2+\mathcal{O}(x^4)\biggr)^3 + \mathcal{O}(x^8)\\ & =-\frac{x^2}2 + \frac{x^4}{24}-\frac{x^6}{720} - \frac{x^4}8 +...

Homework Statement I need to find the bloch vector for the density matrix \frac{1}{N}\exp{-\frac{H}{-k_bT}} where the Hamiltonian is given by H=\hbar\omega\sigma_z. The Attempt at a Solution I can break the Taylor series of exp into odd and even terms because sigma z squared is the...

Homework Statement For g=Hf = sin (f), use a Taylor expansion to determine the range of input for which the operator is approximately linear within 10 % Homework Equations The taylor series from 0 to 1 , the linearization, is the most appropriate equation The Attempt at a Solution...

Homework Statement Linearize the system operator illustrated below by applying a Taylor series expansion. f(t) ----> e^f(t) -----> g(t) Homework Equations I only find the general form of a taylor series relevant. g(x)= sum (0,infinity) of [f^n*(a)*(x-a)^n]/n! The system is...

This is just part of a larger problem, but I have a basic equation r'=k-g*r, where k and a start out as constants, but then I need to treat everything as if it can vary slightly from the average. For this, I set r=r_ave+dr, g=g_ave+dg, and k=k_ave+dk. Now I need to work these into the first...

Hi I wonder if there is a simpler way to obtain the first three non-zero terms of Taylor Expansion for the function \frac{Ln(1+x)}{1-x} about x=0? I differentiated it directly, but it was such a nightmare to do:mad: . So I am wondering if there is a simpler way to do it?

Homework Statement I have the following question to answer: Show that (X^2/h^2)*((1/2*y1) - y2 + (1/2*y3)) + (X/h)*((-1/2 y1)+(1/2 y3))+y2 (sorry about the format) is equal to (taylor expansion): y = y2+(x(dy/dx)¦0 + (x^2/2*((d^2)y)/(dx^2))¦0 Homework Equations also given in...

Hey Everyone. I'm ALMOST finished this problem... To spare you the long story, I need to take the difference between an gravitational acceleration, and the same gravitational acceleration at a slightly larger height. The two functions are a(r) and a(r+d), where d is very small Now... VERY...

I'm having trouble determining the order of the pole of [exp(iz) - 1]/((z^2) + 4) at z=2i I know I can't just expand the exponential as 1 + iz + [(iz)^2]/2 ... because this formula only works near the origin. Can I still use Taylor's theorem to find the expansion at z=2i (i.e does...

the problem reads develop expansion of ln(1+z) of course I just tried throwing it into the formula for taylor expansions, however I do not know what F(a) is, the problem doesn't specify, so how can I use a taylor series?

Can anyone please give me a hint on any of the following Taylor expansions? That would be so helpful! for all three: Find the first non-zero term in the Taylor series about x = 0 problem 1 \frac{1} {sin^2x} - \frac{1} {x^2} everytime I differentiate the result is zero...so that...

I am supposed to find an approximation of this integral evaluated between the limits 0 and 1 using a taylor expansion for cos x: \int \frac{1 - cos x}{x}dx and given cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!}... i should get a simple series similar to this for...

Hi How do you expand (1-exp(-1))^-1 as Taylor series Callisto

V = 2\pi \sigma(\sqrt{R^2+a^2}-R) Show that for large R, V \approx \frac{\pi a^2 \sigma}{R} I figured if I could develop the MacLaurin serie with respect to an expression in R such that when R is very large, this expression is near zero, then the first 1 or 2 terms should be a fairly...

basic taylor expansion... Hi, could some one explain how i could use the taylor series to expand out: f(x)= 1/sqrt(1-x^2) Any help would be appreciated, thanks.