Questions tagged [calculus-and-analysis]
Questions related to the calculus and analysis branches of Mathematica, including, but not limited to, limits, derivatives, integrals, series, and residues.
4,587
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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....
- 6,705
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1
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On a formula in Wolfram alpha
I need to understand a formula for the Hypergeometric function $_3F_2(a,b,c;d,e;1)$
I searched and got the following from Wolfram alpha ( see:- second formula in https://functions.wolfram.com/...
- 330
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1
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Analytical Solution in Generalized Heat Equation
Given the following problem
$$u_{t}-\frac{v}{2}\cdot u_{xx}+\frac{v}{2}\cdot x^2 \cdot u(x)=0 $$
$$u(x,0)=f(x)$$
Where $ f(x)=y_1(x)+0.2y_4(x)+0.01y_6 (x) $
$y_n$ are the Eigenfunctions that defined ...
- 127k
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3
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Intersection points of two-variable polynomials
Let $h$ and $g$ designate two multivariable polynomials:
$$\!\!h(x,y)=-96\! \left(32 x^2\!+\!8 x (3\!-\!8 y)\!+\!40 y^2-28 y+5\right) \left(10 x^2+x (4 y-2)+\quad\quad\quad\\ \quad\quad+(1-2 y)^2\...
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1
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1
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Evaluate 3D- and 5D-constrained integrals for absolute separability probabilities
In a recent posting,
TwoQubits
user JimB, employing a change-of-transformations put forth by N. Tessore, was able to confirm a formula for the "two-qubit absolute separability Hilbert-Schmidt ...
CommunityBot
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Why do MMA13.2 and MMA12.3 give different results of the same integral? [closed]
MMA13.2
f13[z_] =
Integrate[(
L W z (L^2 + W^2 + 8 z^2))/((L^2 + 4 z^2) (W^2 + 4 z^2) Sqrt[
L^2 + W^2 + 4 z^2]), z] // FullSimplify
MM12.3
...
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2
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4
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250
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Evaluate using Mathematica or otherwise the value of $f'(0)$
If we have$$f(a)=_3F_2\left(\frac{1}{2},\frac{a+1}{2},\frac{a+2}{2};
\frac{a+3}{2},\frac{a+3}{2};1\right)$$
where $a\geq 0$ and $_3F_2$ is the Hypergeometric function with unit argument.
Evaluate ...
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Fractional Differential Equation
I am facing the following problem.
If u[x_,t_]=t+y[t]x,
Then,
D[u[x,t],t]
Gives us
1+xy'[t].
But if I use the command for Fractional Derivative like
FractionalD [u[x,t],{t,a}], then Mathematica print ...
- 21k
3
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1
answer
74
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Assume asymptotic value in a limit?
Is it possible to give Mathematica a limit about's function at infinity?
For example I would like to
...
- 139k
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1
answer
113
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Can't evaluate limit
I'm trying to evaluate the following piece of code to know if it's constant but whenever I run it it just simply shows the formal limit expression as output.
Can someone help me ? Thanks !
...
CommunityBot
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1
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147
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Symbolic partial differentiation of a Lagrangian function including a summation
I'm using Mathematica to double-check that I correctly derived first order conditions for the models I'm researching. Before I get started, I wanted to actually teach myself how to use Mathematica, ...
CommunityBot
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0
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102
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Simplify or FullSimplify doesn't work on Gamma function [migrated]
I have this expression:
Gamma[n - 0.691 + 2.35 I] Gamma[n - 0.691 - 2.35 I]
which returns real values for all $n$. However, neither Simplify nor FullSimplify can ...
- 30.2k
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0
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40
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How can we calculate an integral when Integrate does not converge? [closed]
I tried to calculate the probability density of a wavefunction, but got not result. Does anyone have an idea of an alternative approach to get the probability density of it?
Here are the details:
<...
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1
answer
161
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Solving analytical integral
I am trying to solve the following integral
\begin{equation}
\left[\int_{0}^{t/2}d\tau-\int_{t/2}^{t}d\tau\right]\left[\int_{0}^{t/2}d\tau'e^{-\left|\tau-\tau'\right|/\tau_{c}}-\int_{t/2}^{t}d\tau'e^{-...
- 139k
2
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2
answers
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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}}-\...
- 139k
4
votes
2
answers
186
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Smallest positive real solution with InverseWeierstrassP
How can I get the smallest positive real solution with
InverseWeierstrassP[p, {g2, g3}] for real $g_2, g_3$ (if it exists)?
The documentation just says, that the ...
- 55.8k
2
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3
answers
175
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Evaluate $\int_{0}^{\pi/2} \cos^a (x)\sin(ax) dx$ using Mathematica
Consider the integral $$I(a)=\int_{0}^{\pi/2} \cos^a (x)\sin(ax) dx$$
where $a\geq 0$
I am trying to evaluate the above integral using Mathematica using the following code :
...
- 12.4k
0
votes
1
answer
33
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Modified Rational approximant of a polynomial (v2)
ok this time I will try to post only the essential information, I hope it doesn't end like the previous thread.
the objective is similar to the previous one but now I cannot use the Padèapproximant ...
- 2,549
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1
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I can't solve this problem [closed]
$$\sum_{k=1}^{\infty}\ln(\frac{(k + 1) (k + 3)}{(k + 2) ^ 2})$$
The answer is = ln(2/3)
Can you please help me?
- 330
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1
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Calculate wavelet coefficients with Mathematica
Let $\operatorname{rect}(x)$ be a constant function equal to 1 with support $[-1 / 2,1 / 2]$. Define a step function using the sequence as
$$ f(x)=\sum_{k=1}^4 a_k \operatorname{rect}(x-k)$$
where $...
- 37.4k
1
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1
answer
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Green function and differential equation
I'm trying find the Green's function for a special operator.
I have the homogeneous equation
x (x - 1) y''[x] - 2 I Zm y'[x] - L(L + 1) y[x]==0
for $x>1$ ...
- 42.9k
2
votes
0
answers
77
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Why does this finite integral blow up in Mathematica?
I am having some trouble understanding the output of a particular integral. I think I am misunderstanding something about how Mathematica is used, or about how the Integrate function works, and I ...
- 21
1
vote
3
answers
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Evaluate $\int_{0}^{\pi/2} \cos^a (x)\sin(bx) dx$ using Mathematica [closed]
Consider the integral $$I(a,b)=\int_{0}^{\pi/2} \cos^a (x)\sin(bx) dx$$
where $a\geq 0$ and $b\geq 0$
I am requesting a code in Wolfram Mathematica so as to write the non negativity restrictions of $...
- 139k
3
votes
1
answer
101
views
Is this a bug in IntegrateChangeVariables?
Bug introduced in 13.1, persisting through 13.2.1.
...
- 1,408
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1
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Mathematica doesn't seem to be able to compute the Fourier transform of the Haar orthonormal basis over $L^2(\mathbb{R})$
Consider
F[x_, n_, m_] := 2^{n/2}*WaveletPsi[HaarWavelet[], 2^{n}*x - m]
A paper I am reading uses a different convention of the Fourier transform than Mathematica'...
- 21k
1
vote
1
answer
56
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How to take the derivative of a function in `Im[]`?
I tried to take the derivative of a long expression containing $\text{Re} \text{Li}_3(e^{2\pi ix})$, the trilogarithm while $x$ is real. Since the real part function is linear ,it should be like this
$...
- 21k
7
votes
3
answers
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Integrating a function over a surface integral
From a first principles bandstructure calculation I get an energy scalar field in three dimensions $E(x,y,z)$. It's now easy to plot a constant energy (contour)-surface for dedicated values $\epsilon=...
1
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1
answer
111
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How to separate parameters from a list based on certain conditions?
The following codes create a table of parameters Br and Dr based on corresponding parameters b and c. How to separate/divide the list/table of Br and Dr for which corresponding b or c are negative ...
CommunityBot
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2
answers
58
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How to print particular cases with Integrate?
I would like to perform the following simplification:
...
- 55.8k
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1
answer
73
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Could Mathematica analytically work with operators (such as $\nabla$) and vectors?
I have the following expression: $$(\nabla+\vec{P})^2$$
where $P=(r\cos\theta sin\phi,r^4\sin\theta,r\cos^3\phi)$
I would like to open the brackets and simplify the expression.
Are there in ...
- 42.9k
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10
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What is the equation to plot a vertical line?
Please help. Trying to find area between three curves, e^-x, x = 2, y = 1. Can't find out how to plot x = 2. Don't want to use <...
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0
answers
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Efficient powers of DPR1 matrices
I'm looking to compute the following quantity for $A$ where $A$ is a convergent positive definite $d\times d$ diagonal + rank1 matrix (DPR1).
$$f(s)=\operatorname{Tr}(A^s)$$
Earlier answer answer by ...
- 6,705
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1
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Is there any hope to compute analytically $y=f(x)$ for which $g(x,y)=0$?
I have a two-variable function $g(x,y)$ with $0<x\leq\frac{\pi}2$ and $\frac32<y<5$. I am looking for $y=f(x)$ for which $g(x,y)=0$. Using ContourPlot
<...
- 42.9k
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0
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Calculating double integrals from a very big function
I need to compute integrals (for different value of i and j) of a function named se in this post. We need to double differentiate from se related to i and j and then compute double integrals over the ...
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0
answers
41
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Functions for translating a vector from one coordinate system to another
I have a vector $\vec{H}$ with components in Cartesian coordinate system $(x,y,z)$: $\vec{H}=(0,0,H)$. I would like to obtain coordinate of this vector in Spherical coordinate system $(r,\theta,\phi)$....
- 727
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0
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Making subscript partial derivative notation work using Notation` [duplicate]
I want to write derivatives like $u_{xx}$ and have this be parsed as D[u,{x,2}]. Immediately the Notation` method came to mind, though I'm not sure how I could implement this. I thought of using ...
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Reduce does not give me the proper answer
I am considering the function
f[T_, c1_, c2_] := \[Pi] (rh[T]^2 (L^2 - rh[T]^2))/(G (L^2 + 3 rh[T]^2)) (1 + (64 \[Pi] G (c2 - c1))/L^2) + 64 \[Pi]^2 c1,
with $G=L=1$...
CommunityBot
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1
vote
2
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134
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Doesn't analytically integrate sensibly let alone correctly
(This is induced x velocity on vortex sheet y=1+gtax/g+tx by strength 1+ac/X)
...
- 42.9k
4
votes
3
answers
102
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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 ...
- 226k
6
votes
1
answer
156
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How to use Slater Type Orbitals as a basis functions in matrix method correctly?
This question is a continuation of my previous series of questions about basis functions.
I would like to find the minimum energy of Coulomb potential motion using matrix method.
$H=-\frac{1}{2}\Delta-...
- 41k
2
votes
2
answers
427
views
Bug in integrate in in 13.1.0?
Bug introduced in 13.1, and fixed in 13.2.
I'm trying the following locally (13.1.0 for Mac) and getting incorrect result
...
- 11.8k
2
votes
1
answer
70
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Mathematica calculates the parameter value when the first derivative is 0
I want to get the value of the corresponding parameter t when dil=0 or dvc=0. What method can I use?
...
- 127k
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Asymptotic returns expression unchanged instead of first-order approximation
I'm using Asymptotic to get first-order expansion of expr=Gamma[0, s/10] - Gamma[0, s], and it returns ...
- 6,705
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votes
2
answers
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Calculating an integral with a seemingly complicated integrand
I would like to calculate the following integral:
$$ I=\int_{0}^{1} \min (1,(u-1)^{n-1}(u-v)^{n-1}) dv $$
where $u$ is a constant taking values between $1$ and $2$ and $v$ is variable whose range is $(...
- 55.8k
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Double integral problem with infinite integral
double numerical integral as show in the follow:
$-\frac{4 \sqrt{\frac{3}{8}-\epsilon ^2} \text{sgn}(w) \sin \left(\frac{3 w}{2}\right) \cos (100 w) \left(\frac{4 \epsilon \left(-\frac{10000}{3} \...
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vote
0
answers
56
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Potential Bug of Floating Point Notation in an Integrated Radical
Mathematica version 13.2, tested on MacOS and Windows.
I recently stumbled into what I think is to be a bug with symbolic integration with radicals. Upon testing, I found some cases that I feel ...
- 105
0
votes
1
answer
33
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How to find the difference equation from system response or primitive equation?
If I have a system response or the primitive equation , how can i find its difference equation by mathematica?
eg:
...
- 8,517
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votes
0
answers
23
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Why complementary solution of an ode changes when adding term on the RHS?
I found a case where AsymptoticDSolveValue gives correct solution to a second order ode when the RHS is zero. i.e. complementary solution is correct.
As we all ...
- 127k
3
votes
1
answer
61
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Why two the same integrals give different values?
Why two the same integrals give different values?
...
- 41k
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0
answers
86
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How to solve double integral involving Bessel functions?
I want to solve the following double integral 'p' which involves Bessel function of first kind and of order zero.
By solving I mean, how can I get a simple expression with only k and R.
- 41