Questions tagged [series-expansion]

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

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50 views

Trying to find a value for a Taylor series that matches the plot of the function [closed]

The function is g[x_] := (1/Sqrt[2*Pi])*E^((-x^2)/2) and I'm trying to find an $n$ for Series[g[x], {x, 0, n}], n], that matches ...
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Does Mathematica assume real variables in this case?

When we have a function like $f(x) = x^2-1$ and we expand it in a power series about some $x = x_0$, does Mathematica automatically assume that $x$ is real valued?
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How to verify series solution to an ode generated by AsymptoticDSolveValue?

To verify solution returned by DSolve, one can use the method shown in howto/CheckTheResultsOfDSolve.html and look for True (...
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2answers
63 views

Series solution to an ode does not satisfy initial conditions. Frobenius series. AsymptoticDSolveValue

I was trying to verify my solution to this ode using power series method. The expansion point is x=0. This ode has removable singularity, so Frobenius series and ...
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2answers
184 views

Check the convergence of double sum

I have the following double summations: Sum 1 : $\sum _{p=0}^{k-1} \left(\frac{\sqrt{\frac{(p+1) \Gamma \left(p+\frac{11}{4}\right)}{\Gamma (p+2)}}}{(p+2) \sqrt{\Gamma \left(\frac{11}{4}\right)}}-\sum ...
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AsymptoticSum does not give any output

I am trying to get leading terms in terms of $p$ of the following expression $\sum_{j = p+2}^{\infty} \frac{\sqrt{\Pi_{n=2}^{j} (1+(0.75/n)) }}{\sqrt{j}(1+j)} $. I know that this sum converges and is ...
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72 views

Conjugate Puiseux expansions in Mathematica

In a paper "A quantitative version of Runge’s theorem on diophantine equations" ACTA ARITHMETICA LXII.2 (1992), P. G. Walsh developed a method to solve all 2-variable diophantine equations ...
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62 views

Trouble while trying to re-run open source code

I am a physics student and this is my first time working with Mathematica. I am trying to run the notebook mentioned here, titled "Amplification factors of the superradiant scattering of a ...
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Is this behavior of the Series function correct

While series expanding a function, I got an unexpected result, which seems like an error to me. I have boiled it down to this minimum (non) working example: ...
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53 views

Taylor Expansion of and Exponential function but multivariable

I have tried find the Taylor series expansion for my multivariable function, f[x_, y_, z_] := Exp[I*(x^2 + y^2 + z^2)^(1/2)] of order 3. This is what I tried, ...
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MMA does not provide the correct asymptote for an integral function

Given is the function $$f(x)=\int_0^\infty \mathbb{exp}\left(-\frac{x^2}{2t^2}-t\right)\mathbb{d}t$$ Mathematica returns for the asymptotic behavior $x\to\infty$ using ...
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Series expansion of a function up to linear terms [duplicate]

I have the following: \[CapitalSigma] = r^2 + a^2 Cos[\[Theta]]^2; \[CapitalDelta] = r^2 - 2 M r + a^2 - k/3 r^2 (r^2 + a^2); grr = \[CapitalSigma]/\[CapitalDelta]; ...
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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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1answer
91 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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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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57 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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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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2answers
64 views

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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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
60 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
84 views

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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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
32 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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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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151 views

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
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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
58 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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131 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
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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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1answer
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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
423 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
62 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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