Questions tagged [series-expansion]

Questions on dealing with series data and constructing power series expansions of functions.

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fractional series expansion

I would like to perform the following taylor expansion in $\zeta$ for a general positive integer n. It works if I tell mathematica n is a given integer, say 3 (see example) but it fails if I leave it ...
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83 views

NIntegrate with variable in it

I would like to NIntegrate with a variable in the function. Later I will be series expanding it. Can it be done in Matehematica? I am getting errors for a sample ...
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Asymptotic inversion of ExpIntegralEi function

I'm looking at the small-x and large-x asymptotic expansions of the inverse of exponential integral $E_1$ (https://dlmf.nist.gov/6.2#E1) $$\begin{array}{lll} E_1 & = & \int_z^\infty \frac{e^{-...
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1answer
48 views

Simplifying expression using asymptotic values of a function

I have a large expression with bessel function in the result of DSolve. The equation is ...
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47 views

Finding a mapping between two types of (generalized) hypergeometric series

I am given two functions, one is of the form $2F1(a,b,c;z)$, where $2F1$ is a hypergeometric series. The other one is a generalized hypergeometric series $3F2(d,e,f;g,h;w)$, where the characters are ...
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How to evaluate Sum with Singularities?

I derived equation of sum from the following problem, $\int_{a}^{b}\sum_{n=0}^{\infty} cos^n(x)dx$. Using the following definitions, $cos(x)=\frac{e^{ix}+e^{-ix}}{2}$ and $(a+b)^n=\sum_{m=0}^{n}\binom ...
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1answer
30 views

A compact way to expand a series in all variables?

Say I start with an expression with potentially an arbitrary number of variables (input-dependent), for example Exp[x]Sin[y]z^(-1)w, and I want to expand in all variables to a certain power. I could ...
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1answer
50 views

Recognizing the type of hypergeometric series based on the dominant terms

I am solving a (infinitely long) differential equation which has the solution $$ y(r)=-\frac{c}{5}+\frac{l^4c^3}{20r^5}+\frac{l^{6}c^5}{16r^9}+\mathcal{O}(l^8), $$ where I am not sure about the sign ...
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Series expansion for two limits of x [closed]

I have a function f($x$) given by the expression $$f (x) = \frac{\left(1+x\left[1-\sqrt{1+x^2}\right]\right)^2-x+x^3\left[1-\sqrt{1+x^2}\right]^2}{1+x^2\left(1-\sqrt{1+x^2}\right)^2}$$ and would like ...
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Problem using KarhunenLoeveDecomposition

I have a matrix called stochasticData.mat which size is 211302*50 and I need to perform the Karhunen-Loève decomposition on it to calculate the uncorrelated random variables. Note that stochasticData....
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Local Series solution at singular point for system of first order ODEs

I want to calculate Psi[z] in the equation D[Psi[z], z] + A[z].Psi[z] == psi[z] around a given point ...
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1answer
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Comparing the plots of two functions in number theory

Definition. For $x>1$, let $$R(x):=\sum_{n\ge1}\frac{\mu(n)}{n}\,\operatorname{li}(x^{1/n})$$ denote the Riemann prime counting function. If you are not familiar with the mathematical expressions ...
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1answer
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I have a list of coefficients and I am trying to make a power series. How?

I noticed the Series[] command that would be perfect for Taylor polynomials. Unfortunately, I do not have the function available. I just have a list with the ...
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1answer
48 views

How to do a convergence test on a complex series in Mathematica

I set the following to N=5, and want to do a convergence test on u: ...
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Series expansion of action at the boundary

I am using the Riemann Geometry and Tensor Calculus (RGTC) package to compute all tensors associated to the metric components hIN with coordinates ...
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Inequality programming involving sum compositions

$n=3$, $m=3$, $B$ - identity matrix $3 \times 3$ Trying to implement it in Mathematica, but can't figure out how to program the second term. The result is an error. ...
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2answers
55 views

Power series expansion in terms of a function

I have a two variable function z[x,y] = f[x,y] + g[x,y], such that I know the functional form of f[x,y] but not of ...
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Series solution of an ODE with nonpolynomial coefficients

Basically, I have a second-order differential equation for g[y] (given below as odey) and I want to obtain a series solution at $...
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Series expansion is separately expanding numerator and denominator

I am trying to expand a function in power series, but Mathematica is expanding the numerator and the denominator separately. ...
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55 views

Get rid of of $O(\epsilon^2) $ terms [duplicate]

So basically I have the expression on the form: (4 ϵ^3 a b )/((-1+ϵ)^2 (a-b)^2) or $ \frac{4 \epsilon^3 ab }{(\epsilon -1)^2 (a-b)^2} $. I guess this is a math ...
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1answer
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Series expansion with criteria on the coefficients

I will first do an illustrative example. Suppose I have the following function: $ f(\vec{x},\vec{t})=\frac{x_1x_2}{(1-x_1 x_2^{-1} t_1)(1-x_2x_1^{-1} t_2)}$ I want to expand it with respect to $(t_1,...
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2answers
144 views

Asymptotic expansion around infinity for inverse cdf of normal distribution

I'm trying to get a asymptotic expansion as $x\rightarrow\infty$ for a particular expression. I have ...
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1answer
31 views

Reading off coefficients as array

Suppose I have a series expansion with non-associative characters, i.e., $1**2**3**4**5 + 2**3**4**5**1 + \cdots$ Then I want to make some array which produces $A[1]= \{1,2,3,4, 5\}, A[2]=\{2,3,4,5,1\...
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68 views

Is there a easy way to make a Taylor Expansion in MMa? [closed]

We know from special relativity that:$$E^2=m_0^2c^2+p^2c^4$$$$E=\sqrt{m_0^2c^2+p^2c^4}$$$$p=m_0v$$$$E=m_0^2c^2(1+v^2)^{1/2}$$Now I know that I can use a Taylor Series to approximate the square root ...
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Series expansion from the red book on special functions by Richard Askey

I want to check my calculations via mathematica. In the book I am reading there's this expansion: $$\frac{(1+\frac{1}{j})^x}{1+x/j}=1+\frac{x(x-1)}{2j^2}+\mathcal{O}(1/j^3)$$ though I get instead of ...
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Series expansion of PolyLog[2, 1/z]

There is a well known identity involving the Dilogarithm: $$ \mathrm{Li}_2(1/z) = - \mathrm{Li}_2(z) - \frac{\pi^2}{6} - \frac{1}{2} \log^2(-z) $$ As far as I understand it should be valid for all $z \...
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How to nicely expand a Gauss Hypergeometric function?

Does anybody know how to obtain the z->1 expansion for the Gauss Hypergeometric 2F1(a,b;c;z) on Mathematica as shown here ? I tried to use Series with the assumption c-a-b non-integer, but the ...
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1answer
38 views

LogicalExpand to find coefficients in power series

I am attempting to use the LogicalExpand command to find an equation for each coefficient in a power series. The documentation gives the following example of this usage: ...
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Non unique asymptotic solution of a second-order ODE

I have the following code for the series solution (via Frobenius method) of the differential equation ode around $y=\infty$. The solution and its derivative are <...
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1answer
51 views

Collecting coefficient list of arbitrary polynomial

Say I have a polynomial like: 1+x^(n)+3x^(n+1)+3nx^(3n+4) I want to extract the coefficient list {1,1,3,3n}. I've been toying ...
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How to get a frequency equation from limited power expansion of differential equation solution?

I am trying to extract frequency that is variable depended from nonlinear coupled differential equation. I managed to get a solution in form of power series expansion up to 8th term and possible more....
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Series coefficient not working for abstract powers

I'm trying to extract coefficients of some complicated polynomials. If I try to write SeriesCoefficient[1/(x-1)^4,{x,0,m}], this works fine, everything as expected. ...
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3answers
124 views

How to get out coefficient of term in series?

Suppose I have a function $f(s,t) = [(1-t^2)(1-s^2t^2)]^{-1/2}$. Is there a way to get the general coefficient in this power series of the form $s^{2k} t^{2n}$?
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1answer
53 views

Syntax for series with parameter [closed]

I am totally new to this - I cannot find how I can find a series limit that has also parameters, ie like $ a_n = \sqrt{(kn+2)} + \sqrt{(n)} , ~~k \in (0,+\infty )$ edit : cross-posted here https:/...
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In a power series, how do we keep the $\frac{1}{n!}$ term without simplification for all $x^{n}$?

I am trying to visualize the Euler numbers coming from the generating function: \begin{align*} \sum_{n\geq0}E_{n}\frac{x^{n}}{n!} & =\text{sec}(x)+\text{tan}(x)\\ & =1+x+\frac{x^{2}}{2!}+2\...
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Series with ArcTan gives wrong symbolic answer in Wolfram Language

Bug introduced after 9 and persisting through 12.3.1 Recently, I have found a very bad problem with Wolfram Language. It gives the wrong answer for a quite simple expression! When calculating ...
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2answers
411 views

How to “prepare” expression for Taylor expansion

I find myself regularly in a situation where I have an expression like $$\frac{m^2+M^2}{(m^2-M^2)^2}$$ with the assumption that $M\gg m$ and the need to expand it up to order $\mathcal{O}(M^{-2})$. By ...
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1answer
55 views

Asymmetric multivariable Taylor expansion

I want to expand a two-variable function up to asymmetric orders in two expansion variables, i.e. $$f(x,y) = T[f(x,y)] + \mathcal{O}(x^2,y^3,xy,xy^2).$$ Note that, while quadratic terms in $y$ are ...
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2answers
193 views

Cut off higher order terms does not work for first order [duplicate]

I want to cut off terms starting with higher order than epsilon^0. It works fine for any other value but unfortunately, I need to truncate values with higher order than 0 or 1. But it seems not to ...
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2answers
91 views

Indeterminate solution of 2nd order nonlinear differential equation

I seek to prove a solution in power series in y or by numerical methods for the second order nonlinear differential equation: ...
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How to make a code to find Taylor series symbolic solution to four coupled nonlinear differential equations?

I am trying to modify the existing code developed by Michael E2 in this question here. His solution was for one differential equation. I like his code because it has ability to solve nonlinear ...
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Discrepancy in the series expansion of BesselK[ν, z]

I am trying to expand the modified Bessel function of the second kind $K_{\nu}(z)$ for small values of the argument $z$ keeping $\nu$ fixed. Mathematica 12.2.0 says that it is ...
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2answers
95 views

Does this sum converge, and why?

Mathematica says the following sum Sum[(mm Gamma[mm])/ Gamma[-(1/2) + mm] - (mm^(3/2) - (3 Sqrt[mm])/8 - (7 Sqrt[1/mm])/ 128), {mm, 1, \[Infinity]}] ...
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How to expand Lie characters?

The following involves characters of affine Lie algebras, and I will be using as reference the book on CFT by Francesco et at (here are some screenshots if useful). But hopefully the post will be self-...
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59 views

Finding series coefficients

Given b[0] := 1; Sum[Binomial[n, k]*((2*n - 2*k - 1)!!)^2*b[k + 1], {k, 0, n}] == ((2*n + 1)!!)^2; is there a way to find the coefficients ...
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2answers
114 views

Real roots of an infinite series consisting of Harmonic number

I know that the following equation, as a function of $s$, has two real roots: $$ \sum_{n=1}^{\infty}e^{s(1-s)H[n]}=\frac{1-r}{r}e^{s^2}-1 $$ for $0<r<1$. Is there any simple way to find these ...
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Extremely memory consuming Expand

Expand floods all my 64GB RAM in MMA 12.1 (Windows) just by sorting the powers of 16 variables. Somebody with >64GB RAM could run it. A memory saving ...
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45 views

Taylor expansion of expected value of a function with multiple random variables

The expected value of a function of multiple random variables can be approximated by a Taylor expansion. How this can be done in MMA is described in other posts (Link1, Link2). Let's assume we have a ...
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2answers
78 views

Non-linear equation

I have to solve this equation but the problem is X is function of x it means X[x] $$X(X-a)+ b e^{-2 X t}=B,$$ a,b,B are constants. How we can I get some result for ...
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How to revert behavior of SeriesData to pre 12.1

In Mathematica 12.0 and earlier, SeriesData[x, 0, {1/u + Log[x/y]}, 0, 3, 1] used to preserve its list of expressions in the form it was given. Now, in ...

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