Questions tagged [series-expansion]

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

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Knowing the expansion of a function, how can we find its expansion using the inverse of x? [migrated]

If we have a function like: $$\text{f[x$\_$]:=}\sum _{i=0}^{\infty } a_ix^i$$ where we can find / know the $a_i$ coefficients, but not really for which function it will converge. How can we find $f[...
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Truncate series with real powers that are not fully numerical

Take a function with the following expansion A3[z_]:=Sum[Subscript[a3x, jj]*z^jj, {jj, 0, 2}] + z^r Sum[Subscript[a3NAx, jj]*z^jj, {jj, 0, 1}]; where ...
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1answer
80 views

How do you find the Inverse of Elliptic Integral of Second Kind when modulus is large

So I tried to take the inverse of EllipticE when modulus is large, in Mathematica, but the solution gives wrong answer. ...
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Mathematica integrates centered functions, but can not integrate shifted ones

Mathematica seems to integrate this function: $\int \limits_{-\infty}^{\infty} d w\, \frac{\sin ^2\left(\frac{1}{2} wt \right)}{w^2} \frac{\frac{\gamma ^2}{4}}{ \left(w^2+\frac{\gamma ^2}{4}\right)}$, ...
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34 views

How does Mathematica derive the asymptote of the function f(k) = (c1 + k)^(c2 + k) / (c3 + k)^(c4 + k)? [closed]

I want to derive the asymptotic of the function $f(k) = (c1 + k)^(c2 + k) / (c3 + k)^(c4 + k)$. The following is derived via a Taylor series expansion: ...
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1answer
76 views

AsymptoticDSolveValue multiple solutions

I'm trying to solve the following ODE asymptotically. $$y(x)^2 y'(x)^2-\left(\sqrt{2} x\right)^2 y'(x)^2+y(x)^2=0$$ From ...
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1answer
55 views

Expanding Determinant as Multilinear Alternating Function

I am working with the expression $$\det\big{|}f(-kx), f(-(k-1)x),\cdots,f(0),\cdots, f((k-1)x), f(kx), g(x)\big{|},$$ where $f,g\colon\mathbb{R}\mapsto \mathbb{R}^{2k+2}$, and want to use the Taylor ...
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2answers
49 views

How to do series expansion for functions which have symbolic parameters?

I would like to find the series expansion of (E^(x^k/k!) Gamma[k, x])/Gamma[k] for $k$ being a positive integer, up to the order of $x^{2k+1}$. Mathematica ...
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20 views

Multivariate Lagrange inversion

I know that the function InverseSeries (Reference here) provides an interface for the Lagrange inversion formula. However I can't find anything on the ...
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40 views

Bad Integral evaluation for Piecewise function

I have been trying to evaluate this symbolic function: f[ρ_, R_, α_, yp0_, yp_] := R*((ρ - R*Cos[α])^2 + (R*Sin[α])^2 + (yp-yp0)^2)^(-(1/2)); Mathematica can ...
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62 views

Series can not expand the following root object about infinity. Is there another way to expand it?

I have the following equation which I want to solve it for r: ...
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Need help with Limit (DirectedInfinity)

I tried to compute the following limit: ...
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1answer
44 views

Plot function f(x) vs first three terms of fourier expansion

I have a function f(x): f[x_] := Sum[(2*(2-2Cos[n*[Pi]]-n*[Pi]Sin[n*[Pi]])/(n^(3)*[Pi]^3))*Sin[n*[Pi]*x]] I want to plot this function vs. the sum of the first ...
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How can I reduce $\sum_{i=1}^n (i^2-i+1/4)$ to a function of $n$? [closed]

How can I reduce $\sum_{i=1}^n (i^2-i+1/4)$ to a function of $n$?
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2answers
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Build general form of an infinite sequence

Please, I would like to build the general form of an infinite number sequence $$\dfrac{8}{35}, \dfrac{5}{21} ,\dfrac{8}{33} ,\dfrac{35}{143},\dfrac{16}{65} ,\dfrac{21}{85} ,\dfrac{80}{323},\...
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1answer
255 views

Series vs Asymptotic in 12.1

The functionality of Series and Asymptotic (new in V12.1) is very similar. In fact, they are both listed in the Asymptotics ...
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1answer
85 views

Fourier series using mathematica [duplicate]

I want to check my homework answers. So I am wondering if there is a way you can find Fourier series using mathematica of this function g(x)=x(1-x) on interval [0,1)? If so, can someone please show me ...
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1answer
68 views

Mathematica code to Math symbols and equations

Can anyone please help me determine what the following Mathematica code is doing in terms of Math equations: (I'm having trouble putting the code here, so I attached in a picture) Is the code doing ...
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47 views

Taylor series approximation of Error function on complex domain

I am using Taylor series to approximate error function on complex domain. For a given $z\in D\subset \mathbb{C}^1$, we know the Taylor series of error function (around point $z_0=0$) is uniformly ...
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1answer
68 views

Computing the entropic risk measure symbolically in MMA

Using the link Definition, it is possible to compute the entropic risk measure as follows: $$ EVaR=\text{inf}_{z>0}\{z^{-1}\ln\left(\frac{M_X(z)}{\alpha}\right)\}, $$ wherein $X$ is a random ...
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2answers
69 views

Finding the smallest value n from which the difference between the sum of an infinite series and a partial sum is less than 0,001

Basically I was told that this "the Solve function generally doesn't handle infinite series very well. Choose some values of n and see how large the error is, ...
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2answers
159 views

`Series` gives wrong result

Bug introduced in 10.0 or earlier and persisting through 12.0. The following code shows that Series gives different results depending on whether one simplifies the ...
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29 views

Recurrence relation of coeeficients of power series solution to DE

Say I have a DE $$ -\phi \left(\phi \left(\left(6975 \phi ^2-3704 \phi +160\right) \omega '(\phi )+\phi \left(\left(6975 \phi ^2-4688 \phi +266\right) \omega ''(\phi )+\phi \left(2 \left(...
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Form a power series from a list

Say I have a list L = {0,1,2,4,6} how can I form a power series with coefficients in L x + 2x^2 + 4x^3 + 6x^4 ?
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2answers
270 views

How to obtain the exact solution of this equation in terms of an infinite series of rational numbers

How to obtain the analytic solution of the number series form of this equation ...
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0answers
49 views

Series expansion of Lerch transcendent still buggy?

This series expansion of a Lerch transcendent seems fixed in V12. However, the following still fails: From the definition of a Lerch transcendent, ...
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1answer
214 views

Implement Baker-Campbell-Hausdorff expansion

I wish to calculate the recursive formula, $$\mathrm{e}^{-A_1 \Delta t/2}\mathrm{e}^{-A_0 \Delta t/2}B\mathrm{e}^{A_0 \Delta t/2}\mathrm{e}^{A_1 \Delta t/2} $$ with the BCH expansion to second order ...
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2answers
180 views

Fast way to the Taylor series expansion coefficients of multivariable function?

Is there a fast method to get the coefficients of Taylor series expansion of function $f(x_1,x_2,...,x_d)$ with maximal summed partial derivative up to $n$, where $d,n$ can be relatively large (for ...
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0answers
41 views

Number-theoretic notation in Wolfram Cloud for iOS

I am using Wolfram Cloud Mathematica, and I want to write an equation using the mathematical symbols for Sum (like a capital sigma, but no the Greek letter) and <...
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1answer
64 views

Strange behaviour of infinite sum (H[n]- Series[H[n]])

Bug report filed 14.01.2020 A support case was created with the ID [CASE:4371991] EDIT It is easy to show that the workaround "limit of finite sum" proposed in the solution by user64494 leads to ...
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1answer
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Series of implicit function (Puiseux expansion) - problem

This two one-line codes should represent the same thing, i.e. the first root of polynomial in $y$: ...
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1answer
49 views

Inverse of matrix up to some order

Let $A(t,s)$ be a matrix of any size (potentially large), whose entries are polynomials functions wrt $(t,s)$ of order $N$. I would like to compute the inverse $X$ of $A$ up to the order $N$ that is $...
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58 views

Smooth approximation near a non differentiable point

Let $f:\mathbb{R}_+\rightarrow \mathbb{R}$ be a function differentiable for $x>0$ but non differentiable at $x=0$ (for instance $f=\sqrt{\cdot}$) and $g$ be a polynomial function. I know how to ...
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1answer
75 views

Finding ODE series solution coefficients

I am trying to solve an ODE by subbing in a series form and then looking individually at the coefficients of different powers of the variable. I'm looking at a general form of equation: $$\frac{\...
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79 views

How to calculate the series of this function?

I have to calculate the series of the function F[r_] := 1 - a - a*r^(5 + n)/(r^(8 + n) + 1); for r->Infinity for generic ...
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1answer
29 views

replacement of series expression

I want convert $$x^k\sum_{i=1}^\infty a[i]x^i$$ to $$\sum_{i=1}^\infty a[i]x^{i+k}$$ by Mathematica,since it's timing consuming and worth a little to do it by hand. from ...
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2answers
41 views

How to obtain the linear terms of the expansion?

Consider the following function, which is actually linear on the going-to-expand points. I want to use Mathematica to expand the function near these points, and keep only the linear terms, i.e. the ...
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2answers
109 views

Expand in series an equation of two variables

I am sorry if this has been asked, I don't really know how to formulate the search to find it. We have some system of equations (that are equal to zero), for example (not sure how to enter ...
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5answers
137 views

How do I collect different exponents together?

Suppose I have: ...
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1answer
47 views

Double Fourier series - value at a point [closed]

I would like to receive the value of the series at point x->1,y->1. Where is the mistake? ...
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37 views

Why can Mathematica detect a linear term in a series expansion only in special cases?

The series expansion Series[Sqrt[2 - 2 Cos[d k]], {k, 0, 5}] Is in odd powers of Abs[k], or more precisely ...
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53 views

Dealing with positive power variables in Taylor expansion and assumptions

I would like to compute the Taylor expansion of monomials whose powers are nonnegative rational variables. First, here is a function which encodes the multivariate Taylor expansion at the points <...
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35 views

Series expansion of explicit functions

For the following input, zv[u_, v_] := v + zv1[ u, v]/u; mvu[u, v] := D[zv[u, v], u]; Series[mvu[u, v], {u, 0, 1}] the output is ...
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30 views

Differentiation of a series with unknown coefficients

I have a function $f(x)$ expressed as $$f(x) = \sum_{k=1}^n {a_k}sin(kx)$$ and its derivative with respect to $x$ is then $$f'(x) = \sum_{k=1}^n k{a_k}cos(kx)$$ I am actually new to Mathematica and ...
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1answer
55 views

Is priority important in `Series` expansion?

I have a strange case here! The story begins with this equation ...
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4answers
286 views

How to convert this term to a Hypergeometric function?

term=8*(-1)^(1/4)*Sqrt[b]*q0^(3/2)*\[Kappa]* EllipticF[I*ArcSinh[((-1)^(1/4)*Sqrt[b]*r)/Sqrt[q0]], -1] This is a physical term and it is not convenient to appear ...
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1answer
41 views

Issue With Example of Series Inversion

I am having a problem with the Mathematica InverseSeries command. Looking at the information page here, we have the following example; ...
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0answers
62 views

How to print highest power of a polynomial first [duplicate]

Is there any built-in way to print the highest power of polynomial first and followed by the rest? ...
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1answer
167 views

Taking the power of a series gives a terribly complicated expression

I have a series $$\Phi_2=\sum_{0\leq n\leq N}a_nq^n+O(q)^{N+1}$$ whose coefficients $a_n$ are Laurent polynomials in the variables $X_1,X_2$. I want to find $\Phi^3$. This is should be pretty ...
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3answers
114 views

Inverting series with symbolic coefficients?

I am trying to invert the series symbolically. Is this possible in Mathematica? Example 1 - Let $p = u + au^2 + bu^3$, where $a,b$ are symbolic variables. I am trying to invert the series around $u=...

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