Series expansion Definition and 183 Threads

Homework Statement The problem just states to find the Laplace Transform of cos(kt) from its power series expansion, instead of using the formula for the transform of a periodic function.Homework Equations Equation for Laplace transform of a function f(t) ->\int(e^{-st}f(t))dt Power Series...

When solving diff-eq's given initial values, e.g. y'' - 2y ' + y = 0 y (0) = 0 y ' (0) = 1 Can one assume immediately that y(0) = c0 and y ' (0) = c1 ? Since these are the first 2 terms in the series? Thanks!

what is the taylor's series expansion polynomial for the function f(x) = [(sinx)/x] p/s : i can't open the latex reference, sorry

Homework Statement [Directions to problem] Show that the function of x gives a power series expansion on some interval centered at the origin. Find the expansion and give its interval of validity. \int_0^x e^{-t^2} dt Homework Equations The Attempt at a Solution I have...

Homework Statement power series expansion of: ((cosh x)/(sinh x)) - (1/x) Homework Equations cosh x = (1/2)(ex + e-x) sinh x = (1/2)(ex - e-x) The Attempt at a Solution what i have so far: I simplified the first part of the eq to read : e2x-1 e2x-1 now I am stuck...

Homework Statement Test the convergence of the series for the surface charge density: \sum^{\infty}_{s=0}(-1)^s(4s+3)\frac{(2 s -1)!}{(2s)!} Homework Equations (2s-1)! = \frac{(2s)!}{2^s s!}; (2s)! = 2^s s! Stirling's asymptotic formula for the factorials: s! = \sqrt{2 \pi s}s^s...

Hi, I'm new to this forum, so if there is anything wrong in this post please forgive me, I'm not sure my post will be shown correctly, so I attached a doc file. The question is A lightly damped harmonic osillator, γ<<ω0, is driven at frequency ω. 1, Find the frequency of the...

Okay, so I am supposed to expand ln(cos(x)) basing my calculations on tan(x) = x + (1/3)x^3 + (2/15)x^5 + (17/315)x^7 + ... and expanding out to that fourth term is sufficient. I am having a major brain meltdown, I can't seem to find any equivalencies present, and I'm sure there are. Any help...

Homework Statement Need to calculate fractional uncertainty f, of M (mass of a star in this case), where f is much less than one. The hint i was given was all i need to know is M \alpha d3, and use a taylor expansion to the first order in f. M = mass of a star, d = distance to star...

Hi all, I am trying to work out a series expansion for ln ((x+1)/(x-1)). I have got the series expansion for ln(x+1) ie x- (x^2/2) + (x^3/3) - (x^4/4) ... and for ln(x-1) -x- (x^2/2) - (x^3/3) - (x^4/4) ... Can I tie these two together to get the series for ln...

Find the Fourier series of the 2Pi-periodic function f(x)= {0 , Abs(x) <= Pi/2 {Abs(x)-Pi/2, Pi/2 < Abs(x) <= Pi My attempt at a solution I have sketched the function... It equals zero between -Pi/2 and Pi/2 and it equals Pi/2 at -Pi and Pi. Then the...

Homework Statement find the first four nonzero terms in the power series expansion of tan(x) about a=0 Homework Equations \Sigma_{n=0}^{\infty} \frac{f^n (a)}{n!}(x-a)^n The Attempt at a Solution Well the series has a zero term at each even n (0,2,4 etc) for n=1 I got x, for...

Homework Statement expand the exponential term in the equation y=2[e^{x+(x²/2)}-1] as a power series Homework Equations on wikipedia I found this... http://img297.imageshack.us/img297/1088/15139862vw6.jpg The Attempt at a Solution Do I substitute x+(x²/2) as "x" in the above...

Homework Statement I want to show/prove that √(y)√(1+y) - ln[√(y)+√(1+y)] = 2y^(3/2)/3 when y<<1 by series expansion. Homework Equations √(1+y) = 1+y/2 - (y^2)/8 + ... and ln[√(y)+√(1+y)] = ln[1 + √(y) + y/2 -(y^2)/8 + ...] The Attempt at a Solution I'm thinking I sub in the...

May be simple, but I'm getting problem with doing Fourier series expansion of Sin(x) for -pi\leqx\leqpi

Anyone recognize this series expansion?? 1+3t+\frac{9t^2}{2!}+\frac{27t^3}{3!}+\frac{51t^4}{4!}+... I looks kind of like e^t but i am not sure how to deal with it. Can I factor something... I kind of suck at these. Someone give me a hint.

Hello Did an exercise and a small simulation to expand f(x)=x, defined on 0<x<3 in a Fourier-Bessel series using Bessel functions of order one that satisfy the boundary condition J_1(3\lambda)=0 and I have some questions: 1.- Is there a rule to use an specific Bessel function order to do...

How do you go about deriving the series expansion of ln(x)? 0 < x I got the representation at math.com but i'd still like to know how they got it. It's been a while since i did calc. iii. Thanks. John

Homework Statement z is a complex number. find the taylor series expansion for g(z)=1/(z^3) about z0= 2.in what domain does the taylor series of g converge. z0 is z subscript 0 Homework Equations The Attempt at a Solution I wrote g(z)=1/(z^3) = 1/(2+(z^3)-2) = (1/2)*1/(1+(z^3...

Hi there guys. My first post here. I heard this forum was really helpful so I've signed up lol.I'm trying to get to grips with using Taylor's/McLaurin's formula for series expansions...My main problem lies with expansions of Logarithmic functions.. I want to work out how to expand Logs when...

Homework Statement Expand cos z into a Taylor series about the point z_0 = (pi)/2 With the aid of the identity cos(z) = -sin(z - pi/2) Homework Equations Taylor series expansion for sin sinu = \sum^{infty}_{n=0} (-1)^n * \frac{u^{2n+1}}{(2n+1)!} and the identity as given...

Homework Statement I am ashamed to ask this, but in my quantum final, there was a little mathematically-oriented subquestion that asked to show that the function V(r)=-\frac{V_0}{1+e^{(r-R)/a}} (r in [0,infty)) can be written for r>R as V_0\sum_{n=1}^{\infty}(-1)^ne^{-n(r-R)/a}The Attempt at...

I am expanding the function f(t) = e^{i \omega t} from (-π,π) as a complex Fourier series where w is not an integer. I am stuck figuring out how the series expands with n. c_n = \frac{1}{2 \pi} \int_{-\pi}^{\pi} e^{i \omega t} e^{-int} dt Join exponentials c_n = \frac{1}{2 \pi}...

Let,s suppose we have a function f(x) which is not on L^{2} space but that we choose a basis of orthononormal functions so the coefficients: c_{n}=\int_{0}^{\infty}dxf(x)\phi_{n}(x) are finite. would be valid to expand the series into this basis in the form...

Represent the function 5ln(7-x) as a power series, i.e., Maclaurin series, C_0= C_1= C_2= C_3= C_4= i got C_0 = 5 ln (7-0) and i think C_1 = 5/(7-1) but its wrong the textbook says that C_1 will be the derivative of C_0 anyway... please give me some hint

Anyone knows what function correspond to this series expansion? \begin{align} f(x)=1+x+x^2+x^3+... \end{align}

y=kcosh(x/k) = k(e^(x/w) + e^(-x/k)) y ~ k[1 + x/k + (1/2!)(x^2/k^2) + (1/3!)(x^3/k^3)] +...+ k[1 - x/k +(1/2!)(x^2/k^2) - (1/3!)(x^3/k^3) +...] All odd terms except 1 cancel out. So we are left with y = k [2 + (2/2!)(x^2/k^2) + (2/4!)(x^4/k^4) + (2/6!)(x^6/k^6) +...] I've been...

Hi, I have a question about Taylor series: I know that for a function f(x), you can expand it about a point x=a, which is given by: f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + ... but I would like to do it for f(x+a) instead of f(x), and expand it about the very same point...

the function f(x) = \frac{10}{1+100*x^2} is represented as a power series f(x) = \sum_{n=0}^{\infty} C_nX^n Find the first few coefficients in the power series: C_0 = ____ C_1 = ____ C_2 = ____ C_3 = ____ C_4 = ____ well f(x) = \frac{10}{1+100*x^2} can be written as...

A couple of days ago one of my teachers mentioned (when discussing the completeness of spherical harmonics) that {1,x,x^2,x^3,...} forms an overcomplete basis for (a certain class of) functions. This implies that a power series expansion of a function is not unique. And you can for instance...

I was wondering how to create a power series expansion for the function x^2 / (1+x^2)... I've tried using the geometric series, but somehow i got stuck. thanks.

I have been doing some questions on Binomial Series expansion and have been stuck on this particular question for a long time and desperately need some guidance. Q) Expand (1/(sqrt(1-b^2(sin^2)x)))), where b = sin(1/2(theta)) as a binomial series. Here is what I have done so far... Let...

[SOLVED] power series expansion for Laplace transform We are to find the Taylor series about 0 of e^t, take the tranform of each term and sum if possible. So I know the expansion of e^t is 1+x/1!+x^2/2!... x^n/n! then taking the tranform, 1/s + (1/1!)(1!/s^2) +(1/2!)(2!/s3)... and so on then...