All Questions
Tagged with calculus-and-analysis series-expansion
144 questions
4
votes
0
answers
84
views
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 ...
3
votes
3
answers
263
views
What is the formula for this numerical series?
I'm developing a questions game. My goal is that the score for each correct answer will increase as the user answers more questions. Initially there are 15 points for each correct answer. Every 4 ...
3
votes
2
answers
264
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 ...
3
votes
2
answers
258
views
Can Mathematica simplify an ordinary differential equation (ODE) by assuming a power series solution and obtain the recurrence relation?
I have an ordinary differential equation and want to solve it using power series as ψ[x_] = Sum[Subscript[a, n] x^n, {n, 0, ∞}] to obtain the recurrence relation ...
3
votes
2
answers
312
views
SumConverge does not give a valid result
I wanted to see if the function SumConverge worked for the following summation:
$$\sum_{n=1}^{\infty}\frac{\sin^2(n)}{n^p}$$
We can clearly see that for $p>1$ the series converges and for $p\le1$ ...
3
votes
1
answer
119
views
Asymptotic integral expansion at infinity [closed]
Define for all $n\in\mathbb{N}$, a definite double integral, $$I_n= \int_{0}^{1}\int_{0}^{1}\left(\frac{x^2(1-x)y^2(1-y)}{1+x^2 y^2}\right)^n\frac{\cos((2n+1)\tan^{-1}(xy))}{\sqrt{1+x^2y^2}}\ dxdy$$
...
3
votes
1
answer
645
views
Multi-dimensional PadeApproximant
I'm trying to approximate a decaying at infinity 2D function f[x_,y_] with a rational function of 2 variables. For a 1D function like that ...
3
votes
1
answer
79
views
Order of evaluation of Exp and Normal on result from Series
This may be more math related than Mathematica related, but I thought this might be of interest to the group.
I'm trying to work with some Taylor Series approximations of functions that are ...
3
votes
1
answer
75
views
Approximating exponential generating function (EGF) from values of generating function (OGF)
I have a function that can evaluate ordinary generating function, and need to construct an approximation to the exponential generating function by using a few calls to the ordinary generating function....
3
votes
1
answer
88
views
Expanding in small but non zero quantity
I am trying to deal with a function which diverges at $r=0$. I encountered the following function
$$f(r)=\int_0^1\frac{x^2(2-x)}{1-x+rx^2}\mathrm dx$$
The paper says that in the limit $r\ll1$, it ...
3
votes
1
answer
177
views
Generating function for residues of a complicated function
I have a rather complicated function involving 3F2 Hypergeometric functions (see below), which has infinitely many poles. I can extract the residues individually. But it would be great if I could ...
3
votes
0
answers
85
views
Asymptotic expansion for a function containing irrational exponents
I'm trying to find an asymptotic expansion of a function containing irrational exponents in the form of a power series (which also might contain irrational exponents). It works correctly if I request ...
3
votes
0
answers
140
views
Cannot Understand nth Derivative of x/ArcTan[x]
The nth derivative of x/ArcTan[x]:
f[x_, n_] = D[x/ArcTan[x], {x, n}]
Evaluates to:
I cannot get this general from to return ...
3
votes
0
answers
231
views
Negative result of a integral of positive function
Let's try this: I have a function of two arguments, $e$ and $\omega$.
First I integrate some function of $(e, \omega)$:
...
3
votes
0
answers
142
views
Error when simplifying a series expression
I have the following function which I call FF[q_,y_,u_] and this function is well known to have a reasonable Taylor expansion in all three variables. For example, ...
2
votes
2
answers
331
views
Asymptotic integral computation takes too long
I am trying to understand how grows the function $k\mapsto\int_{0}^{\infty} \left({1 - \left(1 - \exp(-t/k)\right)^k}\right)dt$ for $k\to\infty$, and I expect a result asymptotically equal to $k\log(k)...
2
votes
3
answers
181
views
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, \\ ...
2
votes
2
answers
369
views
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:
$\sin x=\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2 n+1) !} x^{2 n+1}$
...
2
votes
3
answers
167
views
Find Generalized Series with Symbolic Variable
CoefficientList[Series[Exp[x], {x, a, 3}], x]
Gives the following expression,
$$
\left\{-\frac{1}{6} e^a a^3+\frac{e^a a^2}{2}-e^a a+e^a,\frac{e^a a^2}{2}-e^a a+e^...
2
votes
3
answers
320
views
How to get out coefficient of term in series?
Suppose I have a function $f(s,t) = [(1-t^2)(1-s^2t^2)]^{-1/2}$.
Is there a way to get the general coefficient in this power series of the form $s^{2k} t^{2n}$?
2
votes
2
answers
302
views
Series expansion using binomial theorem in Mathematica
The series expansion of $\displaystyle\left(1-\frac{a}{x}\right)^{1/3}$ using the Binomial theorem is given by
$$\displaystyle\left(1-\frac{a}{x}\right)^{1/3}=1-\frac{a}{3x}-\frac{a^{2}}{9x^{2}}-\...
2
votes
2
answers
218
views
Finding number of terms determined by the error for sum of series
So inf is the actual numerical value taken of the sum from $n=0$ to $n=\infty$ for $(-1)^n/(3n+1)$. Now I'm trying to find how many terms I need so that the sum of ...
2
votes
2
answers
149
views
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,
...
2
votes
1
answer
186
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 ...
2
votes
2
answers
884
views
Series expansion of integral
I'm looking for a way to do a series expansion of
$$\frac{\mu_0bI_0}{4\pi}\int_0^{2\pi}\frac{\cos\left[\omega\left(t-\frac{1}{c}\sqrt{r^2+b^2-2rb\sin(\theta)\cos(\phi)}\right)\right]}{\sqrt{r^2+b^2-...
2
votes
2
answers
402
views
Asymptotic solution of the saddle-point equation
As an example we have the following equation:
$$\sum _{j=1}^{\infty } \frac{r^j}{\left(1-r^j\right)^2}=n$$
Sum[r^j/(1 - r^j)^2, {j, 1, Infinity}] == n
I'm ...
2
votes
2
answers
148
views
Strange failure of Series and Derivative
I just spend three hours and posted two Questions trying to figure something out, and it turned out all the confusion was caused by this mysterious quirk. I want to expand g[x,v] in v at v=0, using ...
2
votes
4
answers
364
views
Series expression of a RootSum object
Mathematica was not able to calculate the definite integral of a trigonometric function of mine:
...
2
votes
1
answer
83
views
Why does Series give two different results for given function?
I want to do Series of this function x^12 Cos[y x]^2 Sin[y x]^6 Sin[(-1 + Sqrt[3]) y x]^2 around ...
2
votes
2
answers
257
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 ...
2
votes
1
answer
208
views
Problem with SeriesData
Problem:
I need to find the leading order term in an expansion whose leading order behavior is a priori unknown. I can of course go with Series and try different orders, say ...
2
votes
3
answers
349
views
Mathematica failing to series expand a simple analytic function
Bug introduced in 5.0 or earlier and persisting in 11.2
Mathematica is doing funky things with the function
...
2
votes
1
answer
2k
views
Asymptotic forms of Bessel function [closed]
I want to replace Bessel functions by asymptotic forms, so the question is: can I find the best ones with help of Mathematica? And if it's possible, how can I do it?
Update
How can I get with ...
2
votes
1
answer
94
views
Mathematica integrates centered functions, but can not integrate shifted ones
Mathematica seems to integrate this function:
$\int \limits_{-\infty}^{\infty} d w\, \frac{\sin ^2\left(\frac{1}{2} wt \right)}{w^2}
\frac{\frac{\gamma ^2}{4}}{ \left(w^2+\frac{\gamma ^2}{4}\right)}$,
...
2
votes
1
answer
173
views
Working with Limit on a Root expression
I am trying to diagonalize a somewhat larger matrix that is dependent on several parameters. Applying, say, Eigenvalues to that matrix is quite straightforward, but ...
2
votes
1
answer
263
views
Unable to evaluate the Limit of an expression at Infinity, with assumptions on parameters
I need some help in evaluating the following limit with assumptions:
...
2
votes
1
answer
167
views
Series applied to an infinite sum does not work
When I have
Sum[(-1)^n x^(n^2) y^n, {n, 0, ∞}]
and I try evaluating
...
2
votes
1
answer
344
views
Evaluate an Exponential involving an Integral Operator
How would I tell Mathematica to evaluate something like
\begin{align*}
\exp\left[\int_a^b dx\right]f(x)\equiv f(x)+\int_a^b f(x)dx+\frac{1}{2!}\int_a^b\int_0^{x}f(x')dx'dx+\frac{1}{3!}\int_a^b\int_0^{...
2
votes
2
answers
1k
views
About generating power series
For an arbitrary function $f(x,y)$ I am defining functions LogMT1 and LogMT2 as follows,
...
2
votes
1
answer
799
views
About high dimensional integrals
I want to be able to do high dimensional integrals like,
(..naively I wrote it as this..)
...
2
votes
0
answers
197
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 ...
2
votes
0
answers
186
views
Formal Power Series and 0^0
I'm having the following problem: I define
series = Exp[Sum[Subscript[J, n]/n t^n, {n, 1, \[Infinity]}]]
and it's all fine. Also when I ask Mathematica
...
1
vote
2
answers
134
views
Series expansion for two limits of x [closed]
I have a function f($x$) given by the expression
$$f (x) = \frac{\left(1+x\left[1-\sqrt{1+x^2}\right]\right)^2-x+x^3\left[1-\sqrt{1+x^2}\right]^2}{1+x^2\left(1-\sqrt{1+x^2}\right)^2}$$
and would like ...
1
vote
1
answer
247
views
Maclaurin Series - Table
I'm starting to learn Mathematica. I have to solved this eq and draw the graph. It is developing a series of Taylor in about x0 == 0.
This is my equation to solve
<...
1
vote
2
answers
139
views
Finding an elementary function growing asymptotically as the integral of a sequential product
I am trying to understand how grows the function $f:\mathbb{N}\to\mathbb{R}$ Integrate[(1-Product[1-Exp[-2*j*t/(k*(k+1))], {j, 1, k}]), {t,0,\[Infinity]}] for $k\to\...
1
vote
1
answer
537
views
Laurent series 0 < |z-3| < 3
I wanna check my laurent series exercises on Mathematica, but can't seem to find a command or program to achieve the result of such type of interval.
$f(z)=\frac{1}{(z-3) z},\\1<|z-3|<3$
The ...
1
vote
3
answers
229
views
How to express the function $a(t)$ knowing a parametrization $a(\eta)$ and $t(\eta)$?
I have this function :
\begin{equation}\tag{1}
a(\eta) = \sqrt{\sin{2 \eta}},
\end{equation}
and this time variable :
\begin{equation}\tag{2}
t(\eta) = \int_0^\eta a(\eta') \, d\eta'.
\end{...
1
vote
1
answer
242
views
Discrepancy with Hurwitz Zeta function
I've come across an issue while using Wolfram Mathematica that I don't quite understand.
Consider the following symbolic computation:
...
1
vote
1
answer
135
views
Asymmetric multivariable Taylor expansion
I want to expand a two-variable function up to asymmetric orders in two expansion variables, i.e.
$$f(x,y) = T[f(x,y)] + \mathcal{O}(x^2,y^3,xy,xy^2).$$
Note that, while quadratic terms in $y$ are ...
1
vote
1
answer
151
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.