Series expansion Definition and 187 Threads

Homework Statement Using the series expansion of e^KX and the fact that a^X=e^Xlna, evaluate 2^-3.4 accurate to 3 dp? Homework Equations The Attempt at a Solution So a^X=e^Xlna. Basically we need to expand e^-3.4ln(2). e^x=1+X+(X^2)/2!+(X^3)/3!... e^-3.4ln(2)=...

Homework Statement y'' + y' + y = 1 + x + x2 Homework Equations y = Ʃ CN*xN N starts at 0 y' = Ʃ N*CN*x(N-1) N starts at 1 y'' = Ʃ N*(N-1)*CN*x(N-2) N starts at 2 [b]3. The Attempt at a Solution [/] I know how solve the equations...

Homework Statement f(t) defined by f(t) = |t| for (-pi,pi) and f(t+2pi)=f(t) the graph is just ^^^ where w=2pi/T = 1 Homework Equations Periodic function using Trigonometric from Even Function f(t) = (1/2)anot + (the sum from n=1 to inf) (an)*COS(nwt), where an = 4/T Integrated from 0 to...

Homework Statement Determine the coefficients c_n of the Laurent series expansion \frac{1}{(z-1)^2} = \sum_{n = -\infty}^{\infty} c_n z^n that is valid for |z| > 1. Homework Equations none The Attempt at a Solution I found expansions valid for |z|>1 and |z|<1: \sum_{n =...

Homework Statement Find the Taylor series expansion of f(x) = (x-1)/(1+(x-1)^2) about x=1 and use this to compute f(9)(1) and f(10)(1) Homework Equations The sum from n=0 to infinity of f(k)(c)/(k!) (x-c)k The Attempt at a Solution I'm not sure how to approach this...

Hello, I recently learned about the Fourier Series and how it can be used decompose a periodic signal into a sum of sinusoids. I can calculate all the coefficients by hand, but I wanted Mathematica to do that for me. I attempted to write a code, and it does give the desired output. I...

I am just curious to know if it is possible to derive a series expansion for the following sigmoid equation: base + max/(1+exp(H-x/rate)) where the coefficients are base, max, H, and rate. Would appreciate any insight.

Homework Statement f(t)= -1 if -∏ < t ≤ 0 1 if 0 < t ≤ ∏ f(t+2∏) = f(t) question asks to compute first 3 non-zero terms in Fourier series expansion of f(t) Homework Equations The Attempt at a Solution since this is an odd function i used the Fourier sine series...

Homework Statement Find the Taylor series expansion for f(x)=x*e^(-x^2) about x = -1 Homework Equations The Attempt at a Solution I have tried replacing x with (x-1) and f(x-1) = (x-1)*e^(-(x-1)^2). Consider the power series for e^(-(x-1)^2) about x = 0, f(x-1) =...

Homework Statement Find the Taylor series expansions for f(z) = −1/z^2 about z = i + 1. Homework Equations The Attempt at a Solution I'm just not sure what format I'm supposed to leave it in. Is it meant too look like this: f(z)=f(i+1)+f'(i+1)(x-i-1)... or this Ʃ\frac{1}{n!}f^{(n)}(1+i) *...

Homework Statement Does the principal branch square root of z have a Laurent series expansion in the domain C-{0}? The Attempt at a Solution Well I'm not really sure what a principal branch is? I believe that there is a Laurent series expansion for z^(1/2) in C-{0} because originally our...

Hi Every body! I wan to compute the power series expansion of dedekind eta function. Specifically, I want to know the power series expansion of η(τ)/η(3τ)? How could I expand this function? I would be happy if you could help me as I am stuck at this state when I am computing the modular...

basically i have to check if xln\frac{(x+1)}{x} → 1 as x→∞ the first term is 0 as x→∞ in the answers they say they used maclaurin series and got x(\frac{1}{x} + O\frac{1}{1^{2}}) but don't show how they did it. would the first term in the series be a(ln(\frac{a+1}{a}))...

I want to show that e^x e^x = e^{2x} using a power series expansion. So I start with \sum_{n=0}^\infty \frac{x^n}{n!} \sum_{m=0}^\infty \frac{x^m}{m!} \sum_{n=0}^\infty \sum_{m=0}^\infty \frac{x^n}{n!} \frac{x^m}{m!} \sum_{n=0}^\infty \sum_{m=0}^\infty \frac{x^{m+n}}{m!n!} I am...

I need to find the solution to the geometric series expansion of the form... \sumn^2*x^n , for n=0,1,2,... most resources I've found only have answers for n*x^n or n*x^(n-1). I have no idea how to calculate this, so I was wondering if there's a book out there that has massive lists of...

Homework Statement A question asks me to find the first three non-zero terms of ln(cosx) Homework Equations The Attempt at a Solution I wrote cos x as 1+(1-cos x), used the power series of ln function, expanded cosx and simplified, here is my answer: 1/2x^2 - 1/6x^4 + 1/16x^6...

Homework Statement Find first three non zero terms in series expansion where the argument of funstion is small ln(5+p) Homework Equations The Attempt at a Solution The only way I could think how to do this is by saying ln(5+p) = ln(1+(4+p)) and expanding to (4+p)-...

So I'm computing a second order Taylor series expansion on a function that has multiple variables. So far I have this I(x,y,t)=dI/dx(change in x)+dI/dy(change in y)+dI/dt(change in t)+2nd order terms Would it still be a better approximation than just he first order if I included some...

Hi there. I have some trouble with this problem, it asks me to find the Fourier expansion series for the function f(t)=0 if -pi<t<0, f(t)=t^2 if 0<t<pi So I've found the coefficients a_0=\displaystyle\frac{1}{\pi}\displaystyle\int_{0}^{\pi}t^2dt=\displaystyle\frac{\pi^2}{3}...

I consider an array of lattice points and construct a vector at each lattice points. How to convert this discrete system into a continuum one by using the Taylor series expansion by considering the lattice distance say \lambda? thanks in well advance?

Homework Statement I am doing this multiplication with power series and I am just stuck at this one and other questions that similar to this one. http://img5.imageshack.us/img5/9526/img1261r.jpg Homework Equations The Attempt at a Solution It seems that I suppose to add n-k...

Homework Statement Ok so I have to expand in a power series of ({\alpha} Z)^{2}, the equation E_{nj}=mc^{2}\left\{ \left[1+\left(\frac{Z{\alpha}}{n-(j+1/2)+\sqrt{(j+1/2)^{2}-\alpha^{2}Z^{2}}}\right)^{2}\right]^{-\frac{1}{2}}-1\right\} Homework Equations I know that a series expansion of...

What is the difference in expanding a solution of a complex differential equation in terms of Laurent and Taylor series? Thanks in well advance.

Given that the sum of the geometric series is: 1+x+x^(2)+x^(3)+x^(4)...=1/1-x for -1<x<1 Find power series for 1/1+x Not to sure where to start, any help would be great

Homework Statement In attachment The Attempt at a Solution I break into partial fractions, then get stuck. Please help me in layman terms f(z) = -1/3[3(z+1)] + 4/3[z+4] Now I am stuck.

So this is a REALLY elementary question but I can't seem to find the answer on the net, or maybe I did but just keep skipping over it some how. (by the way, this is with respect to complex numbers z \in C which is used in Complex Analysis, thus why I chose this forum). I know what it means...

Homework Statement This is the how the question begins. 1. Bessel's equation is z^{2}\frac{d^{2}y}{dz^{2}} + z\frac{dy}{dz} + \left(z^{2}- p^{2}\right)y = 0. For the case p^{2} = \frac{1}{4}, the equation has two series solutions which (unusually) may be expressed in terms of elementary...

Given a function g(t), define the function f(r) as follows f(r) = \int_r^\infty g(t) dt I want to find the series expansion of f(r) around the point r = 0, without actually doing the integral. Is this possible? Basically, can i use any particular series expansion of g to find the...

Homework Statement Find the power series expansion of Log z about the point z = i-2. Show that the radius of convergence of the series is R = \sqrt{5}. Homework Equations None The Attempt at a Solution I know that Log z = (z-1) - (1/2)(z-1)^2 + (1/3)(z-1)^3 -... So wouldn't this...

Homework Statement This is actually not a problem, it's something in my notes. The function I am supposed to be approximating is V(x) = V0(1 - ex/a)2 - V0 V0 and a are constants. Homework Equations The Attempt at a Solution It says that the function given is not a parabola. But it can be...

Homework Statement 1) Find the Laurent series for (z^2)*cos(1/3z) in the region \left|z\right| 2) Find the Laurent series expansion of (z^2 - 1)^(-2) valid in the following region a) 0 < \left|z - 1\right| < 2 b) \left|z + 1\right| > 2 Homework Equations The Attempt at a Solution I...

Hi everyone, once I again I turn to all of your expertise in complex analysis. Homework Statement Evaluate \int\frac{(ln(x))^{2}}{1+x^{2}}dx from 0 to +infinity by appropriate series expansion of the integrand to obtain 4\sum(-1)^{n}(2n+1)^{-3} where the sum goes from n=0 to...

Homework Statement Expand the function into Fourier series f(x) = coshx, |x|\leq \pi Homework Equations Fourier series will be C_{n}=\frac{1}{2\pi}\int_{-\pi}^{\pi}(\frac{e^{x}+e^{-x}}{2}})e^{-inx}}dx \frac{1}{4\pi}\int_{-\pi}^{\pi}({e^{x}e^{-inx})dx+...

Homework Statement Question is= Find all Laurent series expansion of f(z)=z^4/(3+z^2) around 1. I will be very very thankful if someone can help me to do this question. Homework Equations The Attempt at a Solution can I assume (z-1=u) here and change the function in terms of...

"Partial" Taylor Series Expansion It has been awhile since I have had to use a Taylor series expansion (from scratch). I looked it up on wiki and the rules are easy enough, I am just a little confused as to how I apply it to a multivariable function, but only expand it about one variable...

Taylor Series Expansion About the Point "i" Homework Statement Calculate the radius of convergence of the Taylor series for \frac{1}{z^2-2z+2} about the point i. The Attempt at a Solution I can find the radius of convergence if I can determine the expansion but I can't seem to...

This start out as homework but my question is not about helping me solving the problem but instead I get conflicting answers depend on what way I approach the problem and no way to resolve. I know the answer. I am not going to even present the original question, instead just the part that I have...

Homework Statement Find a power series expansion for log(1-z) about z = 0. Find the residue at 0 of 1/-log(1-z) by manipulation of series, residue theorem and L'Hopitals rule. Homework Equations The Attempt at a Solution Is this power series the same as the case for real numbers.

For finding series expansion solution of problems like f(x) = h(x) for 0<x<1 f(x) = 0 for 1<x<2 0<x<2 Where the Fourier series expansion only integrate from x=0 to x=1 only and totally ignor the portion of x=1 to x=2. This is also true for Fourier bessel series expansion...

[b]1. I am trying to calculate the laurent series expansion of the function 1/(1-z)² in the region 1<|z| [b]2. None [b]3. I can get an answer informally by doing the polynomial division like in high school, but I don't know if this is the right answer and in case it is I cannot prove...

Homework Statement I am trying to find the Tn(x) for sqrt[x] centered at a=1 Homework Equations The Attempt at a Solution right now i have f'(x)=1/2x^-1/2 f''(x)=-1/4x^-3/2 f'''(x)=3/8x^-5/2 f''''(x)=-15/16x^-7/2 f'(1)=1/2 f''(1)=-1/4 f'''(1)=3/8 f''''(1)=-15/16 how...

Homework Statement Using the technique of Taylor expansion, find an approximate expression for the relativistic factor γ for small v (i.e., expanded around v = 0) that is correct to order v2. Homework Equations γ=1/SQRT(1+ V2/C2). But in class, my professor just substituted X=V/C, so...

The problem: find the laurent series for 1/(z^2-1)^2 valid in 0 < |z-1| < 2 and |z + 1| > 2 we know that f(z) has poles of order 2 at 1 and -1... In the first region, there are no poles (since z=-1 isn't a part of it). We can write the equation as a product of 1/(z-1)^2 and 1/(z+1)^2...

Homework Statement Find the Laurent series expansion of f(z) = (e^z - 1) / (sinz)^3 at z = 0.The Attempt at a Solution Ok, so I'm confused on a number of fronts here. For e^z - 1, I assume you just use the standard power series expansion of e^z and then tack on a -1 at the end, which would...

I'm trying to work something on inverse Laplace transform. I need to express a transfer function F(s) to the form F(s)=\frac{s^{-1} (a_0 + a_1s^{-1} + a_2s^{-2}+ ... }{b_0 + b_1s^{-1} + b_2s^{-2}+ ... } I can easily do it for rational function e.g. \frac{s^3+2s^2+3s+1}{s+4}= \frac{s^{-1}...

Does f(t)=1 have Fourier series expansion or not?

By Weierstrass approximation theorem, it seems to be obvious that every continuous function has a power expansion on a closed interval, but I'm not 100% sure about this. Is this genuinely true or there're some counterexamples?

Homework Statement Find the power-series expansion about the given point for the function; find the largest disc in which the series is valid. f(z) = z^3 + 6z^2-4z-3 about z0=1. Homework Equations The Attempt at a Solution The series is fine. Since it's a polynomial, there are only three...

I have to solve the nonlinear DE y'=x²-y² by using an infinite series expansion y=\sum_{n=0}^{\infty} a_n x^n, but I've tried in vain. Maybe a change of variables would make it easier, but I don't know which one. Thanks

I came across this really strange error when doing series expansions in Mathematica. Suppose I were to let, F(z) = \frac{(z+2)^2}{(z+1)^2} - 0.4 Now F(z) = 0 gives z \approx -3.72076, -1.61257. Suppose we take the second value of z^* = -1.61257. What is the series expansion of \sqrt{F(z)}...