Questions tagged [series-expansion]

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

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46 views

Limit of an infinite summation

The above is from Maple 2019.1. Is there a way to achieve the same result from MMA12? Tried Limit[Sum[Sqrt[1 + k^2/n^3] - 1, {k, 1, n}], {n -> Infinity}] ...
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0answers
19 views

Fastest way to calculate Taylor expansion coefficients where the coefficients are functions of another variable that is evaluated at several poinst

I have an expression say V which is a function of 18 variables, 17 of which will be Taylor expanded to some order and the last one which will be explicitly ...
0
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2answers
81 views

What series does Mathematica use for Hypergeometric1F1?

I'm trying to get an analytical expression for Hypergeometric1F1[-a, 1/2, X] Provided a is an integer number. I tried adding ...
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0answers
57 views

How to replace variable in result of Series? [closed]

I am trying to replace $\text{xn}$ in $\qquad O(\epsilon )^3+\left(\text{xn}^2+2 \text{xn}\right) \epsilon +\left(-2 \text{xn}^3-4 \text{xn}^2\right) \epsilon ^2-1$ with $O(\epsilon )^2+\epsilon +1$....
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0answers
46 views

Expanding elliptic function of third kind

I want to expand an elliptic function of third kind in terms of $g$, where $n$, $\psi$, and $k$ are functions of $g$. For example, I defined ...
2
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1answer
56 views

Series expansion for `ProductLog[-1,x]`

Is there a way to get Mathematica to expand Series[ProductLog[-1, x], {x, -Exp[-1], 1}]? I'm on Mathematica 12, and it just returns ...
1
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1answer
68 views

Series of a product of series

I am writing a code where I have to multiply large series truncated at some order, say 100. As a result I want again a series up to order 100, and I want to get it in the fastest way. I have tried a ...
1
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2answers
90 views

Coefficient of power $p$ in polynomial expansion : Mathematica doesn't answer me

I would like to compute the coefficient in front of $n^p$ for a polynomial expansion. I wrote the following code: ...
0
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1answer
62 views

Asymptotic series : why doesn't Mathematica compute it in power of $1/x$?

Consider the following : Series[Cos[Pi/2*Sqrt[n/x]] - Cos[Pi/2*Sqrt[(n + 1)/x]], {x, Infinity, 2}] I expect mathematica to find me the $1/x$ perturbative ...
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0answers
21 views

Showing step-by-step answers for power series identities?

We have Sum[1/n^2, {n, 1, Infinity}] (* Evaluate to Pi^2/6 *) ..but is there a way to show how this is derived step-by-step? NOTE: I have seen step-by-step ...
3
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1answer
63 views

Multiply a Sum by a factor

A very simple question: How can I tell to Mathematica that: $\begin{equation} x*\sum_{k=0}^{\infty}\,b_kx^k=\sum_{k=0}^{\infty}\,b_kx^{k+1}\end{equation} $ I tried to multiply but Mathematica gives ...
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0answers
28 views

List of answer to solve with interdependant solutions

I am solving a fairly annoying matrix equation order by order in perturbation around a small parameter. I am solving for a list of free parameter that are $\beta 1[1],\beta 0[1],\alpha 0[1], \beta m1[...
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2answers
78 views

Getting the coefficients of a series that solves a differential equations

I have an example from Stewart's Calculus where the equation $y'' + y = 0$ is solved using power series. The equation ...
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1answer
48 views

Can't get Animate[] to work consitently

Wave on string from [0,L=10]. Endpoints fixed and initial conditions a double sinc: Ti1(x) = Sinc[3*(x - 7)] + Sinc[3*(x - 3)]. Initial velocity Ti2(x)= 0. Evaluating as expansion and trying to ...
1
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2answers
164 views

Dynamic number of parameter fits in Mathematica

I am trying to learn if in Mathematica one can perform fits of analytic functions that are expanded and written as power series. In particular, we can consider the following generic form: $$ f(B,x) = ...
1
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2answers
70 views

Save part of memoized function

I have a function whose performing a series of expansions of a complicated function, which can take a lot of time. To avoid bottlenecks, I use memoization so I need to evaluate it only once. For ...
1
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1answer
27 views

Selecting only the terms whose exponents add up to a specific number

Suppose I have the expression from a truncated series: ...
0
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1answer
45 views

How to work properly with assumptions?

I'm trying to find an analytical solution which depends only of $i,j$ and $a$ variables, for a serie considering some initial assumptions. The series is $$s_{i0}=s_{0j} = 1,\\ s_{11} = 1-a,\\ s_{12} ...
1
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1answer
35 views

Numerically approximating the series of the solution to a PDE

Suppose I had a PDE such as the heat equation in two variables, and I want to solve it with mathematica, and ask it to return me a series expansion of the solution. For example, ...
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0answers
203 views

Mathematica 12 crashes upon taking the Log of a double series

I recently upgraded from Mathematica 10.4 to 12.0. Unfortunately, I am now experiencing crashes with code that was stable in 10.4. The code involves extensive manipulation of the Series function ...
3
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1answer
29 views

Collect more than one symbol and series

I have a complicated expression in function of 2 variables A and f that appears in all the possible combination. For example ...
2
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1answer
67 views

Series keeps unwanted terms

This expression doesn't simplify properly. I have this series. In[33]:= temp2[[8]] Out[33]= SeriesData[s, 0, { t[2] t[4], 0, Rational[3, 2] x[1] x[4]}, 0, 3, 1] ...
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1answer
49 views

Plotting the percentage difference between two curves

I would like to plot the percentage difference between the following functions: ...
2
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1answer
115 views

Mathematica just takes infinite time to solve this

Can You help? I don't even care about the exact solution. I will be satisfied by the series expansion of the result around a=0. I tried to solve by series expansion and the same result .. infinite ...
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0answers
25 views

Expanding the complex conjugate of a Mathieu function as a series

In my calculations, I found this equation: ...
1
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1answer
52 views

Inverting a system of series

The problem is as follows, I have some series of expansion $$ t_1=\sum_{n_1,n_2}a^{(1)}_{n_1,n_2}x_1^{n_1}x_2^{n_2},\quad t_2=\sum_{n_1,n_2}a^{(2)}_{n_1,n_2}x_1^{n_1}x_2^{n_2} $$ and I want to express ...
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3answers
51 views

Expanding non-commutative terms

How can I expand the following equation with non-commutative terms: ...
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1answer
55 views

Product of power series

Suppose p[i_] = Series[2/x + 1 + a[i]*x + 1/b[i]*x^2, {x, 0, 3}] How could I find the coeffients of $\prod_{i=1}^np_i$? For example, I would like to define <...
4
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3answers
243 views

Alias for root of a polynomial

I need to work with a variable $u$ such that $u^2 + u + 1 = 0$. I don't want to find a root of the polynomial $u^2 + u + 1$. Rather, I have to work with $u$ symbolically so that a (polynomial) ...
1
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3answers
138 views

Why doesn't Mathematica evaluate the series with recursion relationship as expected?

I define a series $dg(i)$ as the $i$th derivative of a function $g[t]$ for $i>0$ and known the first term $dg(1)=(t-x)g(t)$. In mathematica, the code is: ...
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2answers
90 views

Series expansion at infinity not giving expected result [closed]

I have the function: F = (4 + 2 x + x^2 - exp(x) (4 - 2 x + x^2))/(2(-1 + exp(x)) x^2) I want to find the series of F at <...
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0answers
29 views

What is the coefficient of E^(k t) in this moment generating function?

nn = 10; Range[0, nn]! CoefficientList[ Series[Product[Cosh[k t], {k, 1, 3}], {t, 0, nn}], t] returns the first 10 moments of the Wilcoxon signed rank statistic ...
7
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1answer
230 views

How could this asymptotic expansion be obtained?

I must precise that I am a very limited user of Mathematica (I can only run it from time when going at university). Working this problem, I found that $$\sigma_n=(1)^n\frac{\pi}{2} \big( j_{0,n+1} \,...
1
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1answer
57 views

Finding Coefficients of the Product of Sums

Is there any way to get Mathematica to find the coefficients of the product of sums? As an example (the problem I am trying to solve): Coefficients for a Taylor expansion of $e^{z^2}$ centered around $...
1
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2answers
52 views

Specifying annular region while obtaining Laurent series

I'm trying to obtain the Laurent Series of $f(z)=\frac{z}{(z+1)(z-2)}$ about $z=-1$ in the annular region $|z+1|>3$. I've been trying to use the following code ...
1
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1answer
63 views

Using the Sum function square the sums of numbers

How do I use the Sum function for adding and squaring consecutive terms? It's an exercise from Wolfram Challenges. I want to write my own function that uses the Sum function to get the sum of (1+2)^2 ...
1
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1answer
515 views

Plotting a Maclaurin series

Can anyone plot the degree-8 Maclaurin polynomial for the function $(x+1)/((x^2)-9)$ and function $(x+1)/((x^2)-9)$ on the same plot. Please help, I've tried everything I could think of.
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2answers
80 views

Result with assumptions contradicts previous result

Without assuming anything on the argument of the complex number inside the Gamma function ...
0
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1answer
45 views

My integral doesn't evaluate

i'm doing something called Sommerfeld expansion i got somehelp online source i will show after code. Sommerfeld expansion to integrate Fermi-Dirac equation to find total number of particles N for ...
0
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1answer
105 views

Fastest way to simplify rational functions

I am using Series to approximate function of two variables: Series[f[x,y],{x,0,m}] the function is a complicated sum of ...
2
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1answer
33 views

Maximally expand tensor series?

I am trying to obtain a power series expansion in some real parameter, but all the terms are arbitrary products of tensors. I.e. I want to expand an expression containing sums/products/powers of this ...
1
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2answers
57 views

Series expansion of integral

I'm looking for a way to do a series expansion of $$\frac{\mu_0bI_0}{4\pi}\int_0^{2\pi}\frac{\cos\left[\omega\left(t-\frac{1}{c}\sqrt{r^2+b^2-2rb\sin(\theta)\cos(\phi)}\right)\right]}{\sqrt{r^2+b^2-...
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1answer
48 views

How can I plot an infinite series with two variables with legends?

I want to sketch the graph of this series
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0answers
30 views

Can I use inbuilt machine learning to guess the n-th term of the inverse series?

Consider this concrete example. InverseSeries[ Series[ PolyLog[s, z], {z, 0, 10}]] From this output is there a way of writing down a formula for the n-th term? ...
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0answers
36 views

Unexpected behavior of Series

I noticed something strange today. Consider the code Series[x + x^2, {x, 0, 2}] This of course outputs the usual series cut of at the second power of x. If I run ...
1
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1answer
86 views

How to expand a composite function into series?

I need to expand such a function $$g[y,z(x,y)]=\frac{-y (z+1)^4-z^4-4 z^3+8 z+8}{z+1},\tag{1}$$ into powers of $x$ and $y$. Among $x,y,z$ there is a constraint equation, for example $$(3 y+3) z^4+z^...
2
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1answer
95 views

Solving for the recursion relation for the expansion coefficients of the asymptotic expansion of an ODE

I want to solve for the asymptotic solution of the following differential equation $$ \left(y^2+1\right) R''(y)+y\left(2-p \left(b_{0} \sqrt{y^2+1}\right)^{-p}\right) R'(y)-l (l+1) R(y)=0$$ as $y\...
2
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0answers
49 views

Symbolic perturbation expansion for quantum mechanics using Hellmann-Feynman derivaties

I am interested in some quantum mechanical perturbation expansion for energies. Actually I want to implement these terms $E_n^{(k)}$. As is stated below one can do that using CAS. I would be ...
4
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0answers
117 views

Another example where using FullSimplify gives different result than Simplify

I believe this question is very similar to Result of Series[expression] is different when I simplify the expression, however, due to my lack of Mathematica experience, I am reluctant to call it a bug. ...
2
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0answers
67 views

Eliminating higher order trigonometric terms

I am interested in eliminating higher-order trigonometric terms from a long symbolic expression. Specifically I want to reproduce this simplification that is done (in a tutorial I am working through)...