All Questions
Tagged with calculus-and-analysis series-expansion
144 questions
0
votes
2
answers
66
views
How do I solve for a variable for a derivative equation?
So I'm entering this as input:
f[p_] := Subscript[l, i] Log[ p] + (N - Subscript[l, i]) Log[1 - p]
f'[p]
...
- 1
1
vote
1
answer
198
views
Integration and expansion of hypergeometric function
I am trying to reproduce the evaluation of the expansion in $\epsilon$ for the function in Appendix B of the paper arxiv.org/abs/0705.0676v3, which involves the hypergeometric function ${}_2F_1$:
\...
- 139
9
votes
1
answer
464
views
Calculating relative error of Ramanujan formula for ellipse perimeter
On this page, they present the Ramajujan's second formula for the perimeter of an ellipse:
$$P \approx \pi (a+b) \left(1+ \frac{3 h}{\sqrt{4-3 h}+10}\right),$$
where $h=(a-b)^2/(a+b)^2$. They expand ...
- 179
0
votes
1
answer
57
views
Convergent Taylor series unrecognized by Sum
I am trying to understand why Sum does not recognize a particular Taylor series as convergent. I have defined a function 'series' like this, that computes the Taylor series of a function 'f' at 'x', ...
- 536
3
votes
1
answer
119
views
Asymptotic integral expansion at infinity [closed]
Define for all $n\in\mathbb{N}$, a definite double integral, $$I_n= \int_{0}^{1}\int_{0}^{1}\left(\frac{x^2(1-x)y^2(1-y)}{1+x^2 y^2}\right)^n\frac{\cos((2n+1)\tan^{-1}(xy))}{\sqrt{1+x^2y^2}}\ dxdy$$
...
- 301
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 ...
- 63
1
vote
1
answer
147
views
Limit of Hypergeometric Functions
Mathematica (12) seems to have a lot of trouble with taking limits involving hypergeometric functions. A simple example I am interested in is the following
...
- 471
2
votes
3
answers
167
views
Find Generalized Series with Symbolic Variable
CoefficientList[Series[Exp[x], {x, a, 3}], x]
Gives the following expression,
$$
\left\{-\frac{1}{6} e^a a^3+\frac{e^a a^2}{2}-e^a a+e^a,\frac{e^a a^2}{2}-e^a a+e^...
- 153
1
vote
0
answers
77
views
Limit giving indeterminate result
I have a function $r_h(v)$ given by,
$$r_h (v) = \left(\frac{m_0}{2 g} e^{-g v_f} \left(-1+e^{-g(v - v_f)} \right) \right)^{1/5}$$
where $m_0$ and $g$ are just numbers. I want to take the limits of ...
- 725
1
vote
1
answer
92
views
Proving an expression from Mathematica which is clearly visible from Plots
I have the following Mathematica code:
...
- 2,963
0
votes
0
answers
204
views
How to solve or test the interval of Uniform Convergence of function series?
How to solve or test the interval of Uniform Convergence of function series? (ref2)
e.g.
$\sum_{k=1}^{\infty}\left(z^{k-1}-z^k\right)$
The convergence interval of this series can be got by ...
- 2,425
1
vote
1
answer
242
views
Discrepancy with Hurwitz Zeta function
I've come across an issue while using Wolfram Mathematica that I don't quite understand.
Consider the following symbolic computation:
...
- 27
2
votes
1
answer
83
views
Why does Series give two different results for given function?
I want to do Series of this function x^12 Cos[y x]^2 Sin[y x]^6 Sin[(-1 + Sqrt[3]) y x]^2 around ...
- 349
0
votes
1
answer
125
views
Can Mathematica estimate this complex function?
Mathematica has given me a function in $x,r$ given by
...
- 35
3
votes
1
answer
75
views
Approximating exponential generating function (EGF) from values of generating function (OGF)
I have a function that can evaluate ordinary generating function, and need to construct an approximation to the exponential generating function by using a few calls to the ordinary generating function....
- 6,697
2
votes
2
answers
302
views
Series expansion using binomial theorem in Mathematica
The series expansion of $\displaystyle\left(1-\frac{a}{x}\right)^{1/3}$ using the Binomial theorem is given by
$$\displaystyle\left(1-\frac{a}{x}\right)^{1/3}=1-\frac{a}{3x}-\frac{a^{2}}{9x^{2}}-\...
- 179
5
votes
3
answers
125
views
Why earlier terms generated from AsymptoticDSolveValue change when increasing the order?
I was comparing my hand solution with Mathematica's. I noticed the particular solution terms generated for a Frobenius series solution change depending on the order. This should not happen. As when ...
- 151k
4
votes
2
answers
147
views
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\...
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)...
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 ...
- 7,333
0
votes
0
answers
48
views
Taylor series expansion of a challeging type of polynomial with two summation signs
I have a few questions about series expansions of a particular and difficult type of polynomial written in terms of two summations signs.
It should be remarked that I have also read Mathematica's ...
- 179
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 ...
6
votes
3
answers
196
views
Extracting a logarithmic divergence of an expression using Series
Consider the following expression:
...
- 1,553
6
votes
3
answers
417
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:
...
- 2,963
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 ...
- 2,425
5
votes
1
answer
134
views
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 ...
5
votes
2
answers
214
views
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{\...
- 2,425
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, \\ ...
- 2,425
-1
votes
2
answers
124
views
How to use FindSequenceFunction to obtain the general expression of Fourier series?
I want to get cosine series of the following functions.
$f(x)=\left\{\begin{array}{cc}\cos x, & 0 \leqslant x<\frac{\pi}{2} \\ 0, & \frac{\pi}{2} \leqslant x \leqslant \pi\end{array}\right.$...
- 2,425
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 ...
- 2,425
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,
...
- 2,425
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:
$\sin x=\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2 n+1) !} x^{2 n+1}$
...
- 2,425
6
votes
1
answer
221
views
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{...
- 2,425
0
votes
0
answers
295
views
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:
$...
- 2,425
6
votes
1
answer
233
views
Using Integrate and then Series seem to produce a wrong result
Bug introduced in 11.1.0.0 or earlier and persisting through 14.0 or later
Run this:
...
1
vote
1
answer
127
views
Series expansion of QPochhammer symbol
Mathematica has an internal function QPochhammer[a,q,n] which is given by
$$\text{QPochhammer}[a,q,n]=\frac{\text{QPochhammer}[a,q]}{\text{QPochhammer}[a q^n,q]}=\frac{\prod_{k\geq 1}(1-a q^k)}{\prod_{...
- 143
0
votes
0
answers
120
views
fractional series expansion
I would like to perform the following taylor expansion in $\zeta$ for a general positive integer n. It works if I tell mathematica n is a given integer, say 3 (see example) but it fails if I leave it ...
- 41
1
vote
2
answers
134
views
Series expansion for two limits of x [closed]
I have a function f($x$) given by the expression
$$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}$$
and would like ...
- 255
1
vote
1
answer
176
views
I have a list of coefficients and I am trying to make a power series. How?
I noticed the Series[] command that would be perfect for Taylor polynomials. Unfortunately, I do not have the function available. I just have a list with the ...
- 113
2
votes
3
answers
320
views
How to get out coefficient of term in series?
Suppose I have a function $f(s,t) = [(1-t^2)(1-s^2t^2)]^{-1/2}$.
Is there a way to get the general coefficient in this power series of the form $s^{2k} t^{2n}$?
- 381
13
votes
1
answer
390
views
Series with ArcTan gives wrong symbolic answer in Wolfram Language
Bug introduced after 9 and persisting through 13.1. Resolved in 13.2
Recently, I have found a very bad problem with Wolfram Language. It gives the wrong answer for a quite simple expression!
When ...
- 231
1
vote
1
answer
135
views
Asymmetric multivariable Taylor expansion
I want to expand a two-variable function up to asymmetric orders in two expansion variables, i.e.
$$f(x,y) = T[f(x,y)] + \mathcal{O}(x^2,y^3,xy,xy^2).$$
Note that, while quadratic terms in $y$ are ...
- 13
4
votes
1
answer
157
views
How to expand Lie characters?
The following involves characters of affine Lie algebras, and I will be using as reference the book on CFT by Francesco et at (here are some screenshots if useful). But hopefully the post will be self-...
0
votes
0
answers
46
views
Series of inverse functions, unclear numerical constant
I was answering another question here and came up with this simple illustrative example that should have an analytic solution. Indeed it has, but I do not understand it. In particular, where 85 is ...
- 19.8k
1
vote
1
answer
124
views
I failed to evaluate double integral
I try to evaluate this symbolic integral and evaluate its two series expansions according to certain variables, the plot the output providing some numerical values.
This is a relativistic rotational ...
4
votes
1
answer
1k
views
How to get the Taylor series of implicit functions
Given that the equation $x+\frac{1}{2} y^{2} +\frac{1}{2} z+\sin (z)=0$ can determine an implicit function $z(x,y)$ at {0, 0}, I now need to expand the implicit function $z(x,y)$ to a fourth-order ...
2
votes
1
answer
94
views
Mathematica integrates centered functions, but can not integrate shifted ones
Mathematica seems to integrate this function:
$\int \limits_{-\infty}^{\infty} d w\, \frac{\sin ^2\left(\frac{1}{2} wt \right)}{w^2}
\frac{\frac{\gamma ^2}{4}}{ \left(w^2+\frac{\gamma ^2}{4}\right)}$,
...
- 153
0
votes
0
answers
46
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 ...
1
vote
0
answers
63
views
