Questions tagged [series-expansion]

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

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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
39 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 ...
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2answers
161 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) = ...
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2answers
68 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 ...
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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: ...
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1answer
42 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} ...
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1answer
34 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
174 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 ...
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1answer
27 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 ...
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1answer
62 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
45 views

Plotting the percentage difference between two curves

I would like to plot the percentage difference between the following functions: ...
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1answer
111 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: ...
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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
49 views

Expanding non-commutative terms

How can I expand the following equation with non-commutative terms: ...
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1answer
54 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 <...
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3answers
241 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) ...
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3answers
135 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
89 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
27 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 ...
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1answer
229 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} \,...
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1answer
56 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 $...
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2answers
50 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 ...
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1answer
56 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 ...
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1answer
508 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
79 views

Result with assumptions contradicts previous result

Without assuming anything on the argument of the complex number inside the Gamma function ...
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1answer
43 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 ...
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1answer
88 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 ...
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1answer
30 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 ...
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2answers
46 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 ...
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1answer
84 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^...
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1answer
90 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\...
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0answers
48 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 ...
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0answers
114 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. ...
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0answers
66 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)...
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1answer
43 views

Using Series with Refine or Assuming to restrict the power

I have a system of differential equations which contain a singular point. To avoid the singular point, I am expanding the coefficients and solutions in a power series around that point. Due to the ...
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1answer
102 views

Power series representation of MeijerG function, $G_{m,n}^{p,q}(x)$ [closed]

I've been experimenting with Mathematica and I keep getting the following (where $G_{m,n}^{p,q}(x)$ is the MeijerG function): Is it possible to express those $f_{i}(x)$ as a power series in $x$? ...
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2answers
127 views

Multivariate series for approximating implicit system

I'm trying to approximate the solution of an implicit set of equations by means of a Taylor series. I have managed to do so for a solution expressed in terms of a single independent variable, by using ...
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1answer
80 views

Divergent series not correctly plotted

I have a problem about the plotting of a function which is defined as the power series $$F(\eta)= \left[1+\frac{10.75}{\eta^{15/4}}+O\left(\frac{1}{\eta^{15/2}}\right)\right]^{-7/4} \biggr[1 + \frac{...
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0answers
33 views

Series expansion with fractional exponents

The following series expansion Series[1 + Sum[b[n] (x^(1/4))^n, {n, 1, 3}], {x, 0, 1}] gives terms up to O(x^{5/4}). I would ...
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0answers
60 views

Summation involving 2F2 hypergeometric function

Trying to simplify the following sum: $$ \sum_{i=0}^n\frac{z^i}{(n-i)!}\,\frac{1}{(1+a)_i\,(1-a)_i}\sum_{j=0}^i(-1+a)_j\,(-1-a)_j\frac{(-z)^j}{j!}, $$ where $n=1,2,\ldots$, $z>0$, $0<a<1$, ...
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1answer
73 views

Finding the limiting cases for the root of a function

I'm sorry my title is not descriptive; the function I am interested in is too long to put in there. What I am studying is the real, positive roots of the following function: $f(\epsilon) = (\Delta^2-\...
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1answer
51 views

Using Mathematica to find series expansions for partial derivatives of the generalized Riemann zeta function

I am trying to use Mathematica to find a suitable series expansion for the expression $$ \zeta ^{(1,0)}\left(-1,1-\frac{i}{2}\right) - \zeta^{(1,0)}\left(-1,1+\frac{i}{2}\right),$$ which ...
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1answer
57 views

FindGeneratingFunction gives up too easily [closed]

I am trying to automatically find a generating function from the coefficients of a simple rational function using Mathematica's FindGeneratingFunction: ...
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3answers
150 views

NDSolve fails at the regular singular point of a second-order ODE

Here is the ODE I want to numerically integrate, odey=-l (1 + l) R[y] + (k - y) (-2 Derivative[1][R][y] + (k - y) (R^\[Prime]\[Prime])[y]) If we rearrange it in ...
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0answers
35 views

Inverting a series

How do I invert the following, \[Rho]=r + b0 Sum[Pochhammer[1/2, k]/(k! ((1 - q) k - 1)), {k, 0, \[Infinity]}] + b0^(1 - q)/(2 q) r^q + O[r^(2 q - 1)] to get $r(...
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1answer
60 views

Taylor's theorem approximation [closed]

I'm struggling to determine an estimate for a function (e^-x) using the taylor theorem and getting a truncation error as well. I've tried using the series function but that doesn't let me apply a=0.