Series expansion Definition and 187 Threads

In mathematics, a series expansion is a method for calculating a function that cannot be expressed by just elementary operators (addition, subtraction, multiplication and division).
The resulting so-called series often can be limited to a finite number of terms, thus yielding an approximation of the function. The fewer terms of the sequence are used, the simpler this approximation will be. Often, the resulting inaccuracy (i.e., the partial sum of the omitted terms) can be described by an equation involving Big O notation (see also asymptotic expansion). The series expansion on an open interval will also be an approximation for non-analytic functions.
There are several kinds of series expansions, such as:

Taylor series: A power series based on a function’s derivatives at a single point.
Maclaurin series: A special case of a Taylor series, centred at zero.
Laurent series: An extension of the Taylor series, allowing negative exponent values.
Dirichlet series: Used in number theory.
Fourier series: Describes periodical functions as a series of sine and cosine functions. In acoustics, e.g., the fundamental tone and the overtones together form an example of a Fourier series.
Newtonian series
Legendre polynomials: Used in physics to describe an arbitrary electrical field as a superposition of a dipole field, a quadrupole field, an octupole field, etc.
Zernike polynomials: Used in optics to calculate aberrations of optical systems. Each term in the series describes a particular type of aberration.
Stirling series: Used as an approximation for factorials.

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  1. S

    Using the series expansion of e^KX, evaluate 2^-3.4 accurate to 3 dp?

    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)=...
  2. B

    Differential Equations: Non-homogeneous Series Expansion

    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...
  3. E

    Help With Fourier Series Expansion of a Periodic Function

    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...
  4. K

    Laurent Series Expansion coefficient for f(z) = 1/(z-1)^2

    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 =...
  5. T

    Taylor Series Expansion to Compute Derivatives

    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...
  6. L

    Fourier Series Expansion using Mathematica

    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...
  7. H

    Series Expansion for a Sigmoid Equation

    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.
  8. K

    Computing Fourier Series for Odd Functions

    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...
  9. H

    A question about Taylor series expansion

    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) =...
  10. A

    Taylor Series Expansion for f(z) = −1/z^2 about z = i + 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) *...
  11. R

    "Principal Branch Square Root of z in Domain C-{0}

    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...
  12. T

    Power Series Expansion of Dedekind Eta Function: How to Expand η(τ)/η(3τ)?

    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...
  13. D

    Series expansion of xln((x+1)/x)

    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}))...
  14. L

    Exponential Power Series Expansion

    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...
  15. A

    Exploring Geometric Series Expansion with Higher Powers

    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...
  16. S

    Series expansion of logarithmic function ln(cosx)

    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...
  17. S

    Series expansion of logarithmic function

    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)-...
  18. C

    Multi-Variable Second Order Taylor Series Expansion: Ignoring Terms

    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...
  19. Telemachus

    Troubleshooting Fourier Series Expansion for Piecewise Function

    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}...
  20. S

    Continuum Conversion of Lattice Points via Taylor Series Expansion

    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?
  21. T

    Power Series Expansion Homework: Multiplication & n-k Addition Method

    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...
  22. Q

    What Should (x-a) Look Like in This Series Expansion?

    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...
  23. S

    Difference Between Laurent & Taylor Series Expansion of Complex Diff. Eq

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

    Power Series Expansion for 1/1+x

    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
  25. J

    Mastering Laurent Series Expansion: A Layman's Guide

    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.
  26. C

    Power Series Expansion about Point

    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...
  27. H

    Bessel function series expansion

    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...
  28. P

    Series expansion of an integral

    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...
  29. S

    Power series expansion for Log z

    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...
  30. J

    Taylor Series Expansion - Don't understand how to use

    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...
  31. J

    Laurent Series Expansion of (z^2-1)^(-2)

    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...
  32. N

    Series expansion of integral (ln(x))^2/(1+x^2) dx from 0 to infinity

    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...
  33. R

    Fourier Series Expansion of coshx

    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+...
  34. R

    All Laurent series expansion around 1.

    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...
  35. Saladsamurai

    Partial Taylor Series Expansion

    "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...
  36. P

    Taylor Series Expansion About the Point i

    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...
  37. Y

    Question on orthogonal Legendre series expansion.

    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...
  38. S

    Power Series Expansion and Residue Calculation for log(1-z)

    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.
  39. Y

    Can anyone explain this regarding to fourier series and bessel series expansion?

    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...
  40. E

    How to calculate the Laurent series expansion of 1/(1-z)² in the region 1<|z|?

    [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...
  41. M

    How Do You Calculate the nth Coefficient of a Taylor Series for sqrt(x) at a=1?

    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...
  42. L

    Taylor Series Expansion for the Relativistic Factor of Momentum

    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...
  43. W

    Laurent series expansion help

    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...
  44. V

    Laurent Series expansion for the following function

    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...
  45. M

    Series Expansion for Inverse Laplace Transform of Irrational Functions

    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}...
  46. N

    Does f(t)=1 Have a Fourier Series Expansion?

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

    Does every continuous function has a power series expansion on a closed interval

    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?
  48. Q

    Power series expansion and largest disc of validity

    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...
  49. J

    Nonlinear ODE by an infinite series expansion

    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
  50. R

    What does this happen to my series expansion?

    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)}...
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