Questions tagged [series-expansion]

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

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79 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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2answers
51 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
96 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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2answers
254 views

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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0answers
37 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
77 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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0answers
42 views

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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2answers
32 views

Maintain percentage between elements of a series

Doing some code and I am struggling to find the next formula. I have a series that starts at a number X. I want to have a increase of 10% between each consequent element of my series. 10 -> 11 ->...
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2answers
73 views

Recursive sequence with RecurenceTable

I want to compute elements of a recursive sequence and use them as coefficients of a power series. However, the (i+1)-th element depends on all previous elements. Writing this as a sum in ...
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2answers
91 views

Series expansion of a multivariate function

I am new to Mathematica. I would like to understand if this output of $$\text{Series}\left[\left\{u v,\frac{u^2}{2}+w^2,\log \left(\frac{1}{u^2+1}\right)\right\},\{u,0,1\},\{v,0,1\},\{w,0,1\}\right]$$ ...
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How to get an asymptotic of the real-valued branch of the inverse function?

Consider function $f:\mathbb R^+\to\mathbb R^+$, defined as $f(x) = x + x^2\left(1 + \log x\right)$. I need to find an asymptotic approximation of its inverse function $f^{\small(-1)}\!:\mathbb R^+\to\...
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Calculating the coefficients in a double Laurent series

I am interested in calculating the coefficients in the Laurent series of the following Hankel function $H_{0}^{(1)}(\sqrt{(3+x-y)(3+1/x-1/y)})$ I am familiar with the Coefficient command and its ...
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2answers
58 views

Defining a function as a truncated taylors series [closed]

I am trying to numerically find the error of a truncated series. To do so I want to first define a function that is the truncated sum, I cannot seem to do that. All I am tried is not working. ...
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1answer
65 views

Wrong result with symbolic calculation in Mathematica

I am looking at the expansion of a time series Series [$\left(\sum_{s=1}^\infty\frac{r}{(1+k)^s}x^{s+1}\right)^{-1}${x, 0, 2} ...
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23 views

Series expansion in variables with different indices [duplicate]

The Series[] command is useful for reformatting algebraic expressions and getting rid of higher-order terms: ...
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0answers
88 views

How to remove the smallest term from asymptotic expansion?

It is well-known that $e^{-1/x}\sim o(x^n)$ as $x\to 0^+$ for any $n\in\mathbb{N}$, thus if I do an asymptotic expansion for a function, say $f=1/(1-x)+e^{-1/x}$ as $x\to 0^+$, I expect to receive an ...
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2answers
43 views

Replacing function by another one each time it appears

I have expression involving Cos of some parameters. I would like to replace those Cos by their infinite series each time they do appear. I tried the following which doesn't work: ...
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38 views

Series of inverse functions, unclear numerical constant

I was answering another question here and came up with this simple illustrative example that should have an analytic solution. Indeed it has, but I do not understand it. In particular, where 85 is ...
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1answer
69 views

expand function as power series of another function

Suppose I have a well-known function $f(x)$ and I want to know if this function can be expanded into power series of another function $g(x)$, like $f(x)=\sum_n a_n g^n(x)$, where $a_n$ is the ...
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37 views

Recognizing Padé approximant in Simplify/Refine

Having used Maple before, I know for a fact that there we can just type in a function, then separately define its Padé approximant (or Taylor) and then use simplify ...
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0answers
40 views

How to reduce the computation time of this series expansion?

I have a complicated function gg[Delta,l][z1,z2] which is given in the sample code below. I would like to expand this function for as many ...
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2answers
53 views

Computing the coefficients of series

I am trying to fully expand the following $$\sum_{n=0}^{\infty} \frac{q^{\frac{n(n+1)}{2}}}{(q;q)_n}$$ Which when expressed in Mathematic is: q^(n*(n + 1)/2)/QPochhammer[q, q, n] I would like to be ...
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28 views

Expanding a Function into a Series and Plotting it

I'm trying to expand the function: f(x)=cos(2x)sin(x/2) into two series (let x0=0) of small x, with the first series being 5 terms and the second series being 15 terms. Then plot the function and the ...
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0answers
63 views

Two equations identical but opposite signs have the same Maclaurin series in Mathematica: Is that possible? [closed]

I have two equations Sqrt[1 - 2 y] y^(5/2) Sqrt[y - 4 Sqrt[1 - 2 y] y^(3/2) + 2 y^2] and ...
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48 views

Errors with assigment and no errors with replacement rule, why? [closed]

So I have an expression, let's call it f11 which is a function of t of this form f11= a1*t^(-4)+ a3*t^(-3)+... and so on up until a finite positive power of t. In ...
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2answers
30 views

Power expansion in terms of a fraction of two variables

I have the following complicated expression: ...
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3answers
280 views

Force Mathematica to display `Series` in factorial notation

The Series expansion for Sin[Pi * x] is Series[Sin[Pi * x], {x, 0, 10}] Pi * x - (Pi^3 * x^...
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1answer
45 views

Optimizing a series expansion for high order in $x$

I would like to expand the following function at $x \sim 0$ up to some high $x_\text{max} = \Delta_\text{max}$: $$16 \sum_{\Delta=1}^{\Delta_\text{max}} \sum_{s=0}^{\Delta-2} f_{\Delta,s} \frac{(s+\...
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37 views

Series expansion of hypergeometric function with two variables

I have a function $g(x,y)$ that contains a product of hypergeometric functions, both involving the variables $x$ and $y$. I try to do a series expansion in the two variables as recommended in this ...
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66 views

Transforming a tensor expression

I have the expression $\qquad\sqrt{\det[\gamma^{\mu \nu} F_{\mu \nu}]}$ Basiclly, I need to expand this expression in some form like : $trF^4 + (trF^2)^2 + \sqrt{\det[F_{\mu \nu}]}$, where $\gamma^{\...
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41 views

Incorrect truncation of a series expansion

What is wrong here? fExp = f0 + f1 Sqrt[s] + f2 s + O[s]^(3/2); {s fExp + O[s]^(3/2),s f + O[s]^(3/2)/.f->fExp} Returns ...
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27 views

How to perform a series of an Integral with Respect to variable x which affects limits of integral

I am trying to approximate the effect of a parameter on an integral through a series expansion. The integrand depends on this parameter, but so do the limits of integration: $I(x)=\int_{t_1(x)}^{t_2(x)...
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1answer
49 views

Solving a system of differential equations (Use of Dsolve?)

I am hoping to solve the equations as follows, where 1H*, 3M* and 3E* are functions of t and all others are constants. I tried the code, ...
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3answers
109 views

Higher order Around, for large error propagation

TLDR question: How to redefine Around to work with higher order approximation. Motivation From the documentation Around ...
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2answers
129 views

Expanding logarithms in the negative domain

I am trying to expand some functions $f(z,\bar{z})$ containing logarithms for $z = x + i k x$ with $x<0$, but the results are not always consistent. As a simple example, consider the following code:...
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1answer
80 views

I failed to evaluate double integral

I try to evaluate this symbolic integral and evaluate its two series expansions according to certain variables, the plot the output providing some numerical values. This is a relativistic rotational ...
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0answers
27 views

Series Expansion Using Results from Reduce

I have a list of relations from using the Reduce function and I want to use them to get a series expansion in each variable. Here's what my relations look like: ...
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30 views

Iteratively Expanding Solutions in Multiple Variables

I've been working on a physics problem which requires finding the limiting behavior of variables as another variable goes to either zero or infinity. What I have is relations between 16 independent ...
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3answers
87 views

Make a series with coefficients taken from a list of values

I want to create a function $u(x) = \sum_{j=0}^9 a_j \cos{j\pi x}$ where the $a_j$s come from a list of random numbers. I tried the following ...
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0answers
41 views

AsymptoticSolve and Series not working in limit $\to \infty$. How to solve this functional polynomial relationship?

I want to solve for $J_d$ as a function of $n_d$ for $\eta\gg1$ and $\eta\gg\eta_0$ in the following equations by eliminating $\eta$ from the two equations. \begin{equation} J_d = J_0\cdot\left[\...
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1answer
31 views

Define a function of multiple arguments at a given point

Trivia I am using Taylor series expansions to solve a system of PDEs for a number of functions $f_1(z, \phi), f_2(z, \phi) ...$ The expansions are generated by ...
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1answer
114 views

How to get the convergence radius of the result of Series? [duplicate]

The Taylor expansion of function is very useful, but the convergence radius of power series results after Taylor expansion is also important. But the result of ...
4
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1answer
373 views

How to get the Taylor series of implicit functions

Given that the equation $x+\frac{1}{2} y^{2} +\frac{1}{2} z+\sin (z)=0$ can determine an implicit function $z(x,y)$ at {0, 0}, I now need to expand the implicit function $z(x,y)$ to a fourth-order ...
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1answer
65 views

Problem with Taylor expansion of a function

I need to make a Taylor expansion of the following expression: ...
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1answer
44 views

How to find expansion coefficients in Fourier-Legendre

I am trying to find the coefficients for the Fourier-Legendre expansion of a potential. My goal was to obtain the coefficients as expressions in terms of x and y. I followed the example given on the ...
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0answers
33 views

Help with a custom coordinate transformation

I'm just starting to use Mathematica and can't wrap my head around the task I have at hand. It comes from field-theory. (Sorry I had to insert formulas as images instead of proper LaTex as it would ...
1
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2answers
65 views

Why can't MMA find the series of this function at 0?

I want to expand the following two functions into series at x = 0, but MMA(Version 12.1.1) runs all the time and cannot return results: ...
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0answers
59 views

Why is this causing a Memory Allocation Failure? [closed]

This code causes an error. Why? It works fine if I cut the polynomial ff off at 12 terms instead of 15, but I can't understand why that would make a difference. Is the Series[] function just ...
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0answers
42 views

series expansion unpredictably dividing by leading coefficient

When expanding the simple expression expr = y^3 as a series in x, with expand = y -> Sum[Subscript[y, j] x^j, {j, 1, 20}]; ...

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