Questions tagged [series-expansion]

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

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85 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
199 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
209 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
701 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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92 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
151 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
316 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
65 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
260 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
81 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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37 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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42 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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2answers
189 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
216 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
105 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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128 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
74 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
66 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
143 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
186 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
114 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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203 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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186 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
87 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
126 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
66 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
256 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
43 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
122 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.
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1answer
91 views

How to get better series expansion of expressions involving exponential

I tried to compute the series expansion of this equation r1 = 1/2 (1 + 2^(-1 - k) (1 + (3/5)^(1 + k))^(1 + k)) Series[r1, {k, Infinity, 1}] // Normal // Expand ...
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96 views

Ignoring higher order terms in solving a system of equations

I have used Series & Normal commands a lot to ignore higher order terms in final results. However, I have no idea how to solve a system of equations ignoring higher order terms. For example, ...
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131 views

Bug in SumConvergence

Bug introduced in 10.0.1 and fixed in 12.0.0 Version 11.2.0.0 on MacBook Pro: ...
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1answer
129 views

Express result of calculation in terms of Gamma functions only

I would like to to express the result of my integration just in terms of Gamma functions. The following integral is at hand: $$ \int_0^1dz\int_0^1dy(z(1-z))^{-\epsilon}(1-y)^{1-2\epsilon}y^{-1-\...
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1answer
86 views

Harmonic number of $r$ with order $r$ [closed]

Mathematica recently offered me this expression as the result of an evaluation: HarmonicNumber[r, -r]. I initially thought this must be equivalent to ...
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2answers
191 views

Hessian matrix product in Taylor expansion of vector function

I am trying to get the 2nd order coefficient of the Taylor expansion at $\pmb{x}=\pmb{0}$ of ...
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2answers
277 views

Calculating the series expansion of a theta function

I have defined the q-theta function as follows: $$\theta(x;q) = \prod_{k=0}^{\infty} (1-q^k x)(1-q^{k+1}/x)$$ I want to calculating, using this, the series expansion of the following series: $$\...
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2answers
440 views

Expansion of the Meijer G Function

I'm trying to do the integral Integrate[ B^2*BesselK[0, ko*ρ]^2*2 π*ρ, {ρ, a, ∞}] which I figured should be relatively simple as the integral of a Bessel ...
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0answers
43 views

Define a function with singular Taylor expansion

With an undefined symbol f command Series[f[x], {x, 0, 1}] returns f[0]+f'[0] x+O[x]^2 ...
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4answers
255 views

Approximate the solutions as a series

I would like to solve the following equation $y^2=x^2+ax^2y^2+by^2x^3+cy^3x^2$ where $a,b,c$ are small, so $y\approx x+O(x^3)$. I would like to have a series approximation of the solution rather than ...
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1answer
74 views

Recombining vectors after taking derivatives

I want to map two expansions order by order and to solve to get unknowns coefficient. We define $X_{\mu}= \frac{(x-y)_{\mu}}{(x-y)^2}$. The quantity I need to expand in the end is $t_{\mu\nu}= \frac{...
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2answers
167 views

Finding series expansion of solution of algebraic equation

I have the following algebraic equation: ...
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2answers
80 views

Code to generate series expansion ran fine in M10, not in M11

I have a simple code to get a planet's true anomaly as a function of time, using Newton's method and a series expansion. (Note that I purposefully iterate Newton's method two more times than the order ...
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4answers
119 views

help with simplifying function

I am computing the 4th order talyor approximation of this function $$\big(\frac{b}{g-x}\big)^{0.25}$$ The analytic textbook result is: $$b^{0.25} \big(\frac{1}{g^{0.25}}+\frac{0.25 x}{g^{1.25}}+\...
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1answer
95 views

Taylor series and falling factorials [closed]

I'm new in Mathematica and I have following problem to solve: If $f$ is any function $f: R \to R$ which can be represented as a Taylor series in some close neighbourhood $S$ of $0$ , is it true that ...
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0answers
123 views

Finding symbolic series coefficients

Is there any way to get the symbolic form of a series? For simple series this seems possible e.g. SeriesCoefficent[Exp[x],{x,0,n}] tells me that the general ...
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2answers
181 views

How to do this series expansion with Mathematica

Consider for large integers $n$ the expression $\sin \left(\pi \sqrt{4 n^2+n}\right)$. Since $\sqrt{4 n^2+n}=2 n \sqrt{1 + \frac{1}{4 n}}$ we can use the standard series for the square root and ...
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2answers
156 views

Calculating the n-th term of the series expansion of a special function [closed]

I am trying to calculate the $n^{\text{th}}$ term of the following polynomial: $$\, _2F_1\left(-n,n+3;\frac{3}{2};x\right)$$ To do this I calculate: ...
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1answer
136 views

Taylor Series of numerical solution of FindRoot

I have found the inverse to the following equation by using FindRoot. I want to have a Taylor approximation (say starting from ...
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2answers
291 views

Confused about the solution obtained from vector linearization

I am trying to linearize a vector expression $$\frac{(\mathbf{u}+d\mathbf{u})\times(\mathbf{v}+d\mathbf{v})}{\vert(\mathbf{u}+d\mathbf{u})\times(\mathbf{v}+d\mathbf{v})\vert}$$ Here is my code ...
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0answers
68 views

How to invert a series which has both integer and half integer powers

I have a series which has both integer and half integer powers. $\frac{1}{2}-\frac{\sqrt{x}}{4}+\frac{x}{8}-\frac{x^{3/2}}{16}+\frac{x^2}{32}-\frac{x^ {5/2}}{64}+\frac{x^3}{128}-\frac{x^{7/2}}{...

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