Questions tagged [series-expansion]

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

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Mathematica 13.0 appears to automatically reverse nested series which contain logarithms. Is there a way to prevent this behavior?

When performing manipulations on nested series, I need the series to retain the nesting order I originally defined. In Mathematica versions 8.0 and 10.4, this was almost always the default behavior. ...
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Can Mathematica solve an ode asymptotically as x goes to infinity?

Given the following ode for $x\rightarrow\infty$: $$\left(x f(x)^3 \left(\frac{(x f^\prime)^\prime}{x}\right)^\prime\right)^\prime=0,$$ in the sense of "asymptotics", the equal sign is ...
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Plotting contours of a two-variable function containing a sum

I'm trying to use Mathematica to plot contours of a rather intricate two-variable function. The equation describes the velocity profile for laminar flow in a tube of rectangular section, namely: $$ {w^...
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Expanding a Matrix Vector product in powers of epsilon

I am attempting a perturbation expansion in Mathematica. As part of this, I would like to expand a matrix-vector product where the vectors are given in powers of epsilon. Eventually, I'd like to ...
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4 votes
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Numerically solving nonlinear PDE $u_t = G(u,u_y,u_{xy},u_{xx}u_{yy})$ with unbounded initial condition

I am trying to numerically solve some nonlinear partial differential equations similar to the example given below, for which I have been unable to obtain stable numerical solutions due to some ...
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6 votes
3 answers
314 views

Neglecting higher order terms in a Lagrangian

I have a lagrangian which is modified by variable change. I want to neglect all the 4th order and higher terms in the new lagrangian. The code being used is given below: ...
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NonlinearFit with series coefficients

I have a set of data: ...
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1 answer
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How to plot a graph for the solution to a differential equation

I am very new to Mathematica and am struggling to get a plot for a differential equation I need solving. I am doing a simplified version of the lane-emden equation for n=1 so have the follwoing ...
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Calculate an n-order determinant by FindSequenceFunction

Calculate an n-order determinant: $\left|\begin{array}{cccccc}1 & 2 & 3 & \cdots & n-1 & n \\ n & 1 & 2 & \cdots & n-2 & n-1 \\ n-1 & n & 1 & \cdots ...
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2 votes
2 answers
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Using the generalised binomial theorem to expand an expression

I would like to use Mathematica to compute the following expansion: $(1+x)^\rho= 1 + \rho x +\dots$ for some $|\rho|<1$ as for example explained here. I tried the Series expansion functions ...
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Taylor expand around a vector

I have a matrix $A(\vec{v})$ whose entries are scalar functions of a vector $\vec{v}=(v_{x},v_{y})$. To be concrete I have something like $A=\begin{pmatrix}f_{1}(\vec{v})& f_{2}(\vec{v})\\f_{3}(\...
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Why the coefficient function is very fast

When looking for the coefficients of an desired series, I found that the Coefficient function is very fast compared to other functions and methods. In the following summary, we find the different ...
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5 votes
2 answers
162 views

Best way to handle numerical integration and power series with large numbers

Due to the fact that almost everything I do in my research is analytic, I am quite unfamiliar with numeric calculus, so I was wondering if anyone could give me some advice on the most efficient way to ...
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2 votes
1 answer
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Why would `SeriesCoefficient` not work on an inequality

Here is a humble function with a series of powers of $x$, but when I express the function as an inequality it is no longer able to solve. Is there a reason for this behavior? ...
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How to write code for SeriesCoefficient to work for non integral coefficients?

I have a function of $r$ which I expand at $\infty$ using Series. It is a complicated and messy function, with a parameter $0 \leq \epsilon < 1$. After expansion,...
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Assumptions for FourierSeries

I want to calculate the Fourier series of the following function. $u(t)=\left\{\begin{array}{lc}0, & -\frac{T}{2} \leqslant t<-\frac{\tau}{2} \\ h, & -\frac{\tau}{2} \leqslant t<\frac{\...
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Successive solutions using previously found [closed]

is there a way to use previous calculated values of solve? solving equations based on asymptotic expansion $x^2+x-\varepsilon=0$ $x=x_0+\varepsilon x_1 + \varepsilon^2 x_2 + \varepsilon^3 x_3$ ...
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3 answers
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FindSequenceFunction on trigonometric series

I want to get the sine series general expression of the following two functions by FindSequenceFunction. (1) $f(x)=\left\{\begin{array}{l}0,-2 \leqslant x<0, \\ ...
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-1 votes
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How to use FindSequenceFunction to obtain the general expression of Fourier series?

I want to get cosine series of the following functions. $f(x)=\left\{\begin{array}{cc}\cos x, & 0 \leqslant x<\frac{\pi}{2} \\ 0, & \frac{\pi}{2} \leqslant x \leqslant \pi\end{array}\right.$...
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Splitting a sum in parts for finding a telescoping sum?

Have here a sum of which it can be split into several sums. How to do this from the sum notation itself at once ? Note :I do not want to use the command "Apart" to first split the fraction ...
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3 votes
2 answers
169 views

Series solution of a differential equation

Calculate the series solution of a differential equation: $\frac{\mathrm{d} y}{\mathrm{~d} x}=-y-x$, ( $\left.y\right|_{x=0}=2$) AsymptoticDsolvevalue can calculate ...
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3 answers
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How to make a definition and evalulation for a sequence [closed]

Here a example of sequenze (unfortanely i could not copy/paste as Latex ?) How to input this in MMA ? EDIT: example 2 : for n = 8
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2 votes
2 answers
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Series for $(1+x)^{m}$ with specific notation

I'm trying to get mathematicas series function for $(1+x)^{m}$ to output a result that look like this: $(1+x)^{m} = \sum_{n=0}^{\infty} \frac{m !}{n !(m-n) !}x^{n}$ However, ...
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2 votes
2 answers
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Series for Sin[x] with specific notation

I'm trying to get mathematicas series function for Sin[x] to output a result that look like this: $\sin x=\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2 n+1) !} x^{2 n+1}$ ...
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6 votes
1 answer
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Getting terms and only evaluate specific parts of a series

How to write the first five terms of this series in the following form by MMA code? $\sum_{n=1}^{\infty} \frac{1 \cdot 3 \cdot \cdots \cdot(2 n-1)}{2 \cdot 4 \cdot \cdots \cdot 2 n}= \frac{1}{2}+\frac{...
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2 votes
3 answers
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How to map integration and multiplication to a serie?

*For further study of series, they can be put into a different form. Normal[Series[1/(1 - x), {x, 0, 10}]] This serie 1/(1-x) = 1 + x + ... ,must be first ...
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How to get the n-order Taylor expansion of bivariate function?

I want to get the n-order Taylor expansion of a bivariate function at point (x0,y0): f[x_, y_]: = E^(x + y); {x0,y0}={0,0}; The result calculated by hand is: $...
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Result of expansion changes based on when I define a quantity

I have code where I evaluate a series expansion symbolically. I want to do the same thing without explicitly specifying the point before the expansion. Naively, I think this should not change my ...
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2 votes
1 answer
116 views

Solving coupled ODE by analyzing solution near zero

This question is related to an earlier question I had asked, regarding coupled first order odes. I'll add the system of odes with their boundary conditions again here. . The comments for the previous ...
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How does Mathematica take a Series expansion at Infinity?

I have a program that requires a series expansion at infinity and at a finite horizon, and I have posted two questions about simplifying these expressions or obtaining the series expansion recently. ...
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135 views

Mathematical Expression takes too long to even display

I have the following expression, that I obtain in the following way: ...
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1 vote
1 answer
77 views

Series expansion not returning anything

I have a differential equation, which I want to expand as a series at a finite $r$ value - the horizon, and at $\infty$. The functions involved are definitely cumbersome, and I have recreated my code ...
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Series solution of differential equation

My equation reads: I expand the solution: For example: to zeroth order differential equation reads: where the index i+i is understood modulo 2. At order $\Lambda^4$ equation reads: Proposing the ...
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6 votes
1 answer
207 views

Using Integrate and then Series seem to produce a wrong result

Run this: ...
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0 votes
1 answer
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Series development of laurent in a defined domain

I am trying to correct some bills for laurent series with mathematica, but the output I am getting at the moment is not the best. For example, I have this function $$\frac{1}{z^2 + 9}$$ to develop at ...
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0 votes
3 answers
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How can I represent a series with a summary?

As the title suggests, I'm trying to represent a series through a simple summation. For example, the function Series[Exp[x], {x, 0, 10}] obviously gives me the ...
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Fourier expansion of Dedekind eta function with rational arguments

I want to be able to compute the Fourier series expansion in q=Exp[2πiz] for DedekindEta[(az+b)/(cz+d)] with integer a,b,c,d, but not requiring ad-bc=1. Just replacing z in terms of q and doing ...
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1 vote
1 answer
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Taylor series with a nonatomic variable [closed]

My series needs to be taylored (groan) in two ways. First, my function is actually a matrix, but that is trivial, since Series is threadable. But second, I want to ...
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2 votes
2 answers
71 views

Series function and square roots [closed]

sometimes when using the Series[] function to expand something we can encounter terms that go as fractional power of the small variable we are using to expand. Imagine the final Series expansion is ...
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7 votes
1 answer
343 views

Help with Double Sum (lattice sum) over all integers m,n of 1/(a+m^2+n^2)

Im researching electric fields in periodic arrays of charges, and encountered this summation that I can't find any published work on. Has anybody encountered a solution to $\sum_{m,n=-\infty,\infty}\...
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1 vote
1 answer
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Series expansion of QPochhammer symbol

Mathematica has an internal function QPochhammer[a,q,n] which is given by $$\text{QPochhammer}[a,q,n]=\frac{\text{QPochhammer}[a,q]}{\text{QPochhammer}[a q^n,q]}=\frac{\prod_{k\geq 1}(1-a q^k)}{\prod_{...
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1 vote
2 answers
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Vanishing order of a power series (or polynomial) [duplicate]

Is there a simple and efficient way to compute the vanishing order of a power series, i.e. the degree of its smallest nonzero coefficient? It seems like this is a basic operation that should be built-...
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0 votes
0 answers
42 views

Create a ContourPlot3d graph of a function (x,y,z) that contains series

As I said, I want to create a three-dimensional contour plot of a function that contains series of x and y. I just can't figure out how to do it. Here is the code: ...
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0 votes
1 answer
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How to get the first term of a Taylor Series? [duplicate]

How can I get the first non-zero term of a Taylor Series, when I don't know beforehand what power it will be? Example: Series[Sin[a]^6],{a,0,6}] returns $a^6$, ...
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9 votes
1 answer
258 views

Undocumented behaviour of Series in Mathematica 12.3

I have discovered a very inconvenient behaviour of SeriesData in a new version of Mathematica v12.3.0: it automatically expands brackets in the series coefficients! Example: For the input ...
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1 vote
0 answers
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Does Mathematica assume real variables in this case?

When we have a function like $f(x) = x^2-1$ and we expand it in a power series about some $x = x_0$, does Mathematica automatically assume that $x$ is real valued?
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3 votes
1 answer
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How to verify series solution to an ode generated by AsymptoticDSolveValue?

To verify solution returned by DSolve, one can use the method shown in howto/CheckTheResultsOfDSolve.html and look for True (...
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0 votes
2 answers
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Series solution to an ode does not satisfy initial conditions. Frobenius series. AsymptoticDSolveValue

I was trying to verify my solution to this ode using power series method. The expansion point is x=0. This ode has removable singularity, so Frobenius series and ...
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  • 108k
1 vote
2 answers
215 views

Check the convergence of double sum

I have the following double summations: Sum 1 : $\sum _{p=0}^{k-1} \left(\frac{\sqrt{\frac{(p+1) \Gamma \left(p+\frac{11}{4}\right)}{\Gamma (p+2)}}}{(p+2) \sqrt{\Gamma \left(\frac{11}{4}\right)}}-\sum ...
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3 votes
1 answer
234 views

AsymptoticSum does not give any output

I am trying to get leading terms in terms of $p$ of the following expression $\sum_{j = p+2}^{\infty} \frac{\sqrt{\Pi_{n=2}^{j} (1+(0.75/n)) }}{\sqrt{j}(1+j)} $. I know that this sum converges and is ...
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