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.
4,597
questions
1
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1
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Interrupted plot lines when multiple branches of an ODE solution are plotted
I am plotting the following:
...
- 3,466
3
votes
0
answers
52
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How can I get KroneckerDelta to contract an index?
I have a sum like
f = Sum[x[i]^2, {i,1,Infinity}]
and I want to take a derivative of this with respect to one of the variables, e.g.
...
0
votes
2
answers
175
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How to compute the rational approximant of a polynomial?
given the chebyshev series expansion up to the fourth order
$$f(z)= \sum_{0}^4 c_k T_k(z) $$
with some fixed $c_k$
i need to find the coefficients of the rational approximant (2,1)
$$ R_{2,1}= \frac{...
- 2,549
2
votes
1
answer
77
views
How to calculate the values at the points of a 10-point partition (Fourier Series)?
We have the following periodic function.
$$f(x)=\begin{cases}
x+2,& -2<x<0 \newline
2-2x, & 0<x<2
\end{cases}$$
We define the function and we make the plot
...
1
vote
1
answer
88
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Understanding the calculation of the periodic function $f(x)=x-x^3, \quad -1<x<1$
It is given the following periodic function
$$f(x)=x-x^3, \quad -1<x<1$$
This is an odd function, so the coefficients $a_n,a_0$ of the Fourier Series are zero.
Usually, I face function with two ...
2
votes
1
answer
76
views
Using Mathematica Confirm Bessel's Inequality
I am working on the following Fourier series problem
f[x_] = Which[-1 < x < 0, 0, 0 < x < 1, x^2]
Plot[f[x], {x, -1, 1}]
...
3
votes
1
answer
91
views
How to find the derivative of one matrix with respect to another?
I have an expression from a continuum model,which looks somewhat like this excluding a few constants.
Here epsilon is the 3x3 strain tensor and sigma is 3x3 stress tensor. I wish to compute the ...
- 33
5
votes
4
answers
588
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How to calculate the integral only in real domain?
I have the following integral:
Integrate[1/(Sqrt[2] + Cos[4 X] + Sin[4 X]), X,
Assumptions -> Element[X, Reals]] // Simplify
...
- 59
0
votes
1
answer
177
views
AsymptoticIntegrate of a difficult integral [closed]
I would like to use AsymptoticIntegrate to address this problem: to get an asymptotic expression for:
$$\int\limits_0^\infty \frac{\sin (\sin x)}{\Gamma (x+1)} \ln ...
- 38.2k
2
votes
1
answer
91
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Use Mathematica to solve equation derive the probability distributions using the Principle of Maximum Entropy
I am reading the article Deriving probability distributions using the Principle of Maximum Entropy and trying to derive some of the equations in it automatically using Mathematica.
I want to solve the ...
- 123
0
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0
answers
55
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How to increase the precision of numerical derivatives?
I need to make sure about precision of numerical derivatives in my calculations. I use finite element method to obtain a wave function, then I take derivatives of this numerical wave function to get ...
- 1,228
2
votes
2
answers
197
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Stop Mathematica from expanding Dirichlet beta and eta functions
It seems that Mathematica treats DirichletBeta[s] and DirichletEta[s] merely as symbols for ...
- 153
1
vote
1
answer
59
views
How to understand the output of following integration?
If one tries to evaluate the following integral,
Integrate[(u Cos[k])/(
4 \[Pi] (u^2/16 + (-\[CapitalLambda] + Sin[k])^2)), {k, -\[Pi], \[Pi]}]
one gets following ...
- 471
2
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1
answer
92
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0
votes
1
answer
76
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Solve an equation for different values
I was trying to solve this equation for different value of a constant (a) but the solution is not different for those values?!! b and m are positive values.
...
- 447
0
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0
answers
65
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Caputo-Hadamard fractional derivatine nonlinear equation system
I tried it make it solved and plot but im confused, NDsolve, DSolve, ParametricNDSolve didnt work. Please help me, and please write anything.Thank you.
...
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0
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1
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41
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0
votes
1
answer
83
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Integral of non-singular cosine apparently doesn't converge
After running the following code I get an error that simple integrals are not converging. I make a false assumption that e>0, and proceed to evaluate the integral over e. Mathematica doesn't ...
- 141
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0
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38
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5
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2
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300
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Integrating an expression, that Mathematica cannot solve, via hints or substitutions?
Integrating the following expression, if possible, requires some hints or substitutions which I cannot find.
...
- 163
6
votes
3
answers
308
views
Finding the mean and variance of a distribution
I have tried unsuccessfully to use Mathematica to derive both the expected value (mean) and variance of a particular statistical distribution of interest to me.
The distribution is a discretized ...
- 185
7
votes
2
answers
226
views
PDE involving derivative at boundary, with a boundary condition at infinity
I am trying to find the function $T(z,t)$ which solves this differential equation:
$$\frac{\partial T}{\partial t}=\frac{\partial^2 T}{\partial z^2}+St\left[ \exp\left [ -\frac{\left( x_f-Ut\right )^2}...
- 515
8
votes
2
answers
673
views
Inertia tensors for non-typical rigid bodies
Calculate inertia tensors
This topic inspired me to experiment with calculating tensors of more complex shapes of rigid bodies (I did not find them in the Mathematica database).
For simple shapes of ...
- 2,304
2
votes
1
answer
103
views
4
votes
2
answers
146
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Plotting R0 against two parameters on a 3D plot with planes for certain R0 values
I have been trying to replicate the following graph from the paper. The authors have plotted R0 on a 3D graph against the two parameters, gammaa and betas with two planes for R0=1 and R0=2.
The ...
- 99
3
votes
3
answers
250
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Сan this integral be solved exactly (using Integrate)?
Is it possible to solve exactly the following integral?
...
- 771
0
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1
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48
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Why does an error occur in one case, but not in the other, and how to fix it?
Why do errors appear for this Kx[11, 15] when this Kx[11, 14] is considered without error? How to fix it?
...
- 771
2
votes
1
answer
37
views
limits of DisplayForm integral get swapped when copying, but only visually
I tried doing the following today (Dec 28, 2022) in my cloud notebook (running on Firefox 102.5.0esr on Debian 11):
Integrate[Sin[x]/x,{x,0,Infinity}]//Hold
and the ...
- 181
0
votes
0
answers
80
views
Maximization Problem
I am trying to find the solution of the following problem.
$ \max_{n_{\in [1,M]}} \frac{n \left(\frac{k (n-2 (\rho -1) \tau )}{\frac{k n (n-2 (\rho -1) \tau )}{2 (n+1)^2}+1}-2 \rho (\tau -1)\right)}{...
- 70
1
vote
2
answers
87
views
Euler's Method from Python to Mathematica [duplicate]
The initial value problem is :
$$P'(t)=0.7P(t)(1-\frac{P(t)}{750})-20, P(0)=30$$
The time step is set to $Δt=7$ days
For the algorithm we have:
$f(t,P)=0.7P(1-\frac{P}{750})-20$
with step $h=7$
$t_0=...
0
votes
0
answers
63
views
Why do I get different results for this complex line integral?
I want to compute line integral of $f(z) = \frac{1}{z(z-1)^{n+1}}$ over a closed curve $\gamma: t \mapsto 1 + \frac{1}{2} e^{i t}, t \in [0,2\pi]$ in the complex plane in two different ways but the ...
- 153
1
vote
2
answers
94
views
Why does NDSolveValues give different results?
I'm trying to solve this differential equation to obtain the density matrix elements $\rho_{22}$.
$$\frac{d\rho}{dt}=-i\Big(H_{0}(t)\rho(t)-\rho(t)H^{\dagger}_{0}(t)\Big)$$,
for non-Hermitian ...
- 11
0
votes
0
answers
29
views
why ctrl-% for upper limit of integral does not work on macOS?
I'm trying to enter an integral "visually" on mathematica 13.2 on macOS ventura 13.1. I did esc-intt-esc, and then used ctrl-dash (which the Mathematica documenation calls ctrl-underscore, ...
- 181
2
votes
0
answers
153
views
Decoupling many second-order linear ODEs
I am doing mesh/loop analysis on a fairly complex electrical circuit, and the result is a system of 45 second-order coupled differential equations such as
$$ (\ddot{q}_{23} - \ddot{q}_{24}) L_{3} + (\...
- 21
0
votes
1
answer
75
views
Why Doesn't Mathematica Evaluate This Integral? [closed]
I am using Mathematica (the online version, but not WolframAlpha) and I'm trying to estimate the integral $\int_{-1}^1 e^{\frac{i}{1+x^2}}\,dx$. To do so, I've entered: ...
- 143
1
vote
0
answers
128
views
NIntegrate with big error estimates
I'm trying to calculate a 4-dimensional integral using standard wolfram mathematica methods, but I can't get rid of a big calculation error.
My integral
...
- 11
4
votes
4
answers
142
views
Evaluate the following simple continued fractions
A continued fraction is a fraction whose numerator is an integer and whose denominator is an integer added to a fraction whose numerator is an integer and whose denominator is an integer added to a ...
0
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1
answer
74
views
1
vote
1
answer
52
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Convergence with Maximize - Iterations, Precision
I have problems with some applications for finding convergence - below is one example of failure for {x=0.35,y=0.35,z=0.3}, but it works fine for {x=0.3,y=0.3,z=0.4}.
How can I increase the iterations ...
- 197
1
vote
1
answer
149
views
Why is there an error in one case and not in the other?
Why does the error occur for P[34, 14] whereas for P[33, 14] has no error? How to get rid of this error?
...
- 771
1
vote
0
answers
66
views
Integral of (Elementary) Volume Form over Orientable Surface of Genus 2?
Per this post on math.stackexchange.com, I am told it is impossible to find an elementary formula for a coordinate patch covering of the orientable surface of genus 2 (or higher genus) (or the non-...
- 263
4
votes
2
answers
348
views
Unexpected indefinite integration result
Consider following expression:
$\int_0^1 \sqrt{4194 x^4-6804 x^3+3960 x^2-972 x+90} \, dx$
If i use N[] to calculate it, i get following result:
$N\left[\int_0^1 \sqrt{4194 x^4-6804 x^3+3960 x^2-972 x+...
- 43
3
votes
1
answer
40
views
How to troubleshoot NIntegrate non-numerical error?
I am trying to evaluate a 2d integral (denoted as x and y here) with a function that involves a numerical integral at each point....
- 33
1
vote
1
answer
218
views
0
votes
0
answers
41
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Encountered a singular Jacobian while solving nonlinear equations
I am solving the nonlinear questions in equation 4.1a-4.1c given in the paper. (https://arxiv.org/pdf/cond-mat/0008249.pdf) but no success.I posted similar questions in the link Encountered a singular ...
- 51
9
votes
1
answer
650
views
$\int \frac{\tan^2 x}{1+x^2} dx$
Motivated by this integral, which does not evaluate in Mathematica, I'm wondering if there is some trick or substitution that will solve the indefinite integral.
My motivation is to find integrals ...
- 38.2k
1
vote
1
answer
75
views
How to remove/ (avoid) the # sign in "Solve" output [duplicate]
I am trying to find an equation of A with variable B using the Solve function.
I have the first derivative of firm A's profit with respect to location a, which is equal to zero.
I have the first ...
- 11
3
votes
1
answer
87
views
Encountered a singular Jacobian while solving nonlinear equations
I am solving the nonlinear questions in equation 4.1a-4.1c given in the paper.
(https://arxiv.org/pdf/cond-mat/0008249.pdf) but no success.
...
- 51
0
votes
0
answers
53
views
Integrate the absolute value of a function along the orbital period
I need to integrate the absolute value of this functions
1.f1=
...
2
votes
1
answer
83
views
Resolve a custom implicit equation
Consider the following non-linear equation:
$$
k x = \frac{\frac{9}{4}-y^{22/25}}{2\ 2^{19/25} y^{22/25}-\frac{9}{4} \left(10 y+e^{-y}\right)^{22/25}-1}
$$
where k ...
- 117