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.
5,020
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Plot wavefronts of Laguerre-Gaussian modes
I am trying to plot wavefronts of Laguerre-Gaussian modes, and want to show the changing number of twists with the change in l values. If anyone has the code or any insights. Kindly help me about it. ...
0
votes
1
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114
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For parametrized functions, can we determine function range directly? [closed]
H[t_] = Log[t] - (t - 1)/(λ t + μ)
The definition domain and the parameter ranges for the function H are:
...
1
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1
answer
337
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Why can't Mathematica do an easy integral?
Mathematica can't do this integral:
$\int \sqrt{1-x^2} \cos ^{-1}(x) \cos ^{-1}\left(\sqrt{1-x^2}\right) dx$
However if I replace $\cos ^{-1}\left(\sqrt{1-x^2}\right)$ with $\frac{\pi }{2}-\sin ^{-1}(...
1
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2
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146
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How to do edge detection in Mathematica?
I have some pictures of water level and I want to do edge detection, I used the following command, but this command is vertical edge detection and I want horizontal edge detection and water level. ...
0
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1
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64
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How to make a density plot for an isotropic function written in cylindrical coordinates?
$f(\rho,z)=2\rho^2+\frac{2}{3}z^2 \quad$ is a function in the cylindrical coordinate system $(\rho,\theta,z)$
The function $f(\rho,z)$ is isotropic because it does not depend on the angle $\theta$. ...
4
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1
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170
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Calculating integral with exponential function and parameter
I want to calculate this integral but Mathematica does not solve it.
Why? Can it be solved by Mathematica?
...
3
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1
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175
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Extremum of the graph of a function has integer coordinates
Graph of the fuction $y=\dfrac{(x-26)(x+9)}{(x+14)(x+19)}$ has maximum point and minimum point are (-16,-49) and (-4,-1) whose ...
3
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1
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114
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How can I check all results in this code about the extremum of a function?
I am trying to find three integer numbers, a, b, m, so that the function
$$x^3/3 + a (m + 1) x^2/2 + b (m^2 - 5 m) x$$ have $x=2$ is a maximum point of the function....
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0
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28
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Generator of limits of successions according to type of indeterminacy. Correct syntax
I am trying to make a limit generator with 3 columns, number of the exercise, the second column is the limit (perfectly written with n-->00), and the third column is the result of the limit.
I have ...
2
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1
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53
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How to explore the unchanged eigenvalue with the largest real part?
The provided code below employs the Chebyshev collocation method to calculate eigenvalues. I'm interested in examining the eigenvalue with the largest real part while ensuring it remains almost same ...
4
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1
answer
666
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What are some Mathematica packages used for general relativity?
Could you suggest any Mathematica packages that are used for General relativity calculations? Id like to write code to solve the Schwarzchild Lagrangian equation.
2
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2
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289
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Performing maximum likelihood estimation for SIR Model
The code first produces a plot of the dataset, then simulates the SIR model using the parameter values we found in the manual calibration to the full dataset of the total number infected .
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5
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3
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368
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Integral involving $\cosh$
How can I calculate the following integral using Mathematica?
$$\int_{-\infty}^{\infty} \frac{\mathrm{d} x}{\left(1+x^2\right) \operatorname{ch} \frac{\pi x}{2}} = 2 \ln 2$$
I tried:
...
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1
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88
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Proving an expression from Mathematica which is clearly visible from Plots
I have the following Mathematica code:
...
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0
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69
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Find knee by intersecting line method
I have an x-y data list. The data is here. It looks like below:
I want to mark the knee location. I mean I need to find the point of greatest slope change and get the x coordinate of this point. ...
0
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1
answer
78
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Writing a sum-of-squares function or SIR model fits
I am trying to write a function that calculates and returns the (sum-of-squares) value (SSQ) for the fit of a simple SIR model, parameterized with given values of $\beta$ and $\gamma$, to any dataset.
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1
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54
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Factors of $(-1)^\alpha$ for $\alpha\notin\mathbb{Z}$ when evaluating integrals
Sometimes when I try to use Mathematica to evaluate some integrals depending on some parameter $\alpha\in\mathbb{C}$ the result ends up involving $(-1)^{\alpha}$. Now this is ambiguous if $\alpha\...
2
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2
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123
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173
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1
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97
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Difficulties in solving a system of differential equations with a matrix differential equation
I'm trying to solve a system of differential equations that looks like this:
...
2
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0
answers
104
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Fractional derivatives (FractionalD) of an InterpolatingFunction to retun another InterpolatingFunction
Question
I want to calculate the FractionalD of an InterpolatingFunction object, however, evaluating
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4
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2
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168
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Finding area under a region of a Parametric Plot using mesh functions
I have the following Mathematica code:
...
3
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2
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110
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Integrating to solve for normalization factors
For $f(x)$ solve for its normalization factor as follows:
$$
\left\{ \begin{array}{l}
f\left( x \right) =A\exp \left( -\frac{\left( x-x_0 \right) ^2}{2\sigma ^2} \right)\\
A=\frac{1}{\sigma \...
1
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1
answer
151
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What is mathematica command for finding eigenvalues of a generalized eigenvalue problem?
The provided code below is designed to compute the eigenvalues for the problem I inquired about earlier in this post: How to find eigenvalues?. I believe my code is mostly accurate, with the exception ...
1
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1
answer
105
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Plotting infinite sum with Bessel function and generalized G-function [closed]
How can I plot this function with sum of Bessel functions
\begin{aligned} \tau (r,t)&= gt \biggl(\frac{r}{R} \biggr)^{2}+ \frac{2g}{\lambda ^{\eta }R} \sum_{n=1}^{\infty } \frac{J_{2}(rr_{{n}})}{...
5
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2
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177
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Integral with DiracDelta gives incorrect answer
If I calculate
Integrate[DiracDelta[x - y] DiracDelta[x - z], {x, -∞, ∞},
GenerateConditions -> False]
I get this right answer, viz.
...
1
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0
answers
164
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How to show the limit of two functions in a code approach zero?
I don't know how to shorten the program, the entire code is required to understand everything:
...
1
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1
answer
92
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Integral taking forever to compute
I have this complex function $wig2$ which I am trying to double integrate with respect to 2 variables $q$
and $p$. Using the code below, I have tried to run it several times; however, it takes forever ...
1
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1
answer
163
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How to find eigenvalues?
I have the following system of differetial equations, which are linearized based on steady state solutions. How to discretize them and find eigenvalues or use any other method without discretization ...
2
votes
1
answer
90
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How can I let Mathematica know that the Laplacian of a function is zero?
Suppose I have a long algebraic expression, and in there somewhere I have derivatives of a function V[x,y,z].
I want to set
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2
votes
1
answer
70
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Computing multiple variable integral
I have a 2 variable function named 'wig', and I have to take double integral of its absolute value with respect to the variables p and q. Then subtract 1 from the final answer and 3D plot it with q ...
2
votes
1
answer
102
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Eigenfunction condition of time-independent Schrödinger equation
I'm trying to get the eigenfunctions and eigenvalues of the following one-dimensional Schrodinger equation: $$ -\frac{1}{2}\frac{d^2}{dx^2}\psi(x)+V(x)\psi(x)=E\psi(x)\,,$$ where the external ...
0
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1
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88
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Different NIntegrate results for the same formula
What I do is Integrate a formula and then NItegrate the output of the previous integration. The problem is I get different ...
0
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1
answer
46
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How to define the following rule for all objects in a compact way?
Consider the following symbolic definitions:
...
0
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0
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48
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Manipulating Dirichlet characters and L functions
I read some basic knowledge about characters and L functions, and would like to play around with them in MMA.
I tried to do the following things, but ending in minor success. (MMA notation used)
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3
votes
2
answers
105
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Eigenvalues and classification of critical points [closed]
I started with a function (x,y) and tried to write the code to work out the eigenvalues and classify the critical points.
The output it all good up until I try to use Which[] to classify the critical ...
4
votes
2
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230
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Issue evaluating integral: seems to lead to incorrect answer
I am trying to compute the integral
$$ F(i, j) = \int_0^1 \phi_i(x) \phi_j(x) dx, \quad \mbox{where}~~ \phi_k(x) = \sqrt{2} \sin\Big(\frac{2 k - 1}{2} \pi x\Big) $$
The integer $k \geq 1$. The answer ...
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4
answers
176
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Evaluation of an integral using Mathematica or otherwise
I need to find a closed form (in terms of known functions) using Mathematica or otherwise of $$\Re\left(\int_{\frac{1}{2}}^{1}\frac{\tan^{-1}\left(\frac
{1-x}{\sqrt{-i-x^2}}\right)}{\sqrt{-i-x^2}}\ dx\...
0
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0
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87
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PolynomialGCD::lrgexp when integrating
I want to calculate the exponential in the last line of the following code, but the code never completes and just produces errors about PolynomialGCD.
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2
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130
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Can't integrate a highly oscillatory integrand
I have a long formula
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2
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0
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27
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Silly question about holding an expression in a sum [duplicate]
Sorry about this dumb question in advance. Suppose I evaluate this :
test[x_, y_] := Laplacian[x^2 + y^2, {x, y}]
test[x, y] /. {x -> 1, y -> 1} (*4*)
Now if ...
5
votes
2
answers
367
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Find where the slope is greatest
I have an x-y data list.
I now need to denoise the data and make it smooth. The next step is to find the position with the largest slope and determine the x coordinate of this point.
The data list is ...
3
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1
answer
102
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2
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2
answers
168
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How to correctly solve functional equation
I want to solve the below functional equation:
Let $f : (0, ∞) \rightarrow R$ be a differentiable function satisfying the equation $$f(xy) = e ^ {xy - x - y} (e ^ y f(x) + e ^ x f(y))$$ for all $x, ...
4
votes
1
answer
63
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Confusing cases in Piecewise
The following
FullSimplify[Integrate[Exp[I ω t] Sin[ω x]/ω, {ω, 0, ∞},
Assumptions -> t > 0 && x > 0]]
gives a piecewise output of the form
I ...
2
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0
answers
54
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Plotting diagram of complex equation [closed]
I have this equation:
q1[t_] := -I*γ*Sin[θ] + ((γ*Cos[θ])/n^2)*(I*ξ[t]*y+z*α[t])
and the main code is the following with the definition of each parameters:
...
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2
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457
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Calculating a limit using Wolfram cloud or otherwise
Using Wolfram cloud or otherwise, I need to show $$\lim_{n\to\infty} \frac{n\ 4^{2n}}{e^{2n}}\ \left\{\binom{2n}{n} d_{2n}a_n\right\}\leq \frac{3}{4}$$ where $$a_n:= -\sum_{j=0}^n\binom nj^2(2(H_{n-j}-...
0
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2
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132
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Complex integral with branch cuts
I am struggling with a complex double integral with multiple branch cuts. Even the single variable complex integral I find quite complicated due to the branch cut and the special functions involved in....
1
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1
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79
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How can I differentiate Hermite polynomial with respect to $n$?
How can I differentiate the function below (where $\beta$ and $\gamma$ are constants) with respect to $n$?
...
0
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1
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37
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Volume of revolution with InterpolatingFunction
I have an $(x,y)$ dataset read in from a file. The data do not follow any sort of equation, but have the general shape of $y=x^2$. The data below are a placeholder ...