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.
5,173
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How to enforce $\alpha < 0$ and $k \neq 1$ when solving a system of linear equations?
I want to solve a system of linear equations $$\begin{cases}
\alpha C + \beta &=0 , \\ ( e^{\sqrt{-\alpha}} - k ) A + ( k - e^{-\sqrt{-\alpha}} ) B &=0, \\ ( e^{\sqrt{-\alpha}} - 1 ) A + ( 1 - ...
CommunityBot
- 1
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Plot with solutions of NDSolve
I am trying to substitute the values of phi[z] and phi'[z] back into my equation for Hubb[z] to then plot and compare. The issue I am having is that when I am substituting back into the Hubb[z] ...
- 54.6k
1
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1
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Issue with variable within Sum[] bounds
I have the following sum that starts from a variable lower bound index $n$:
$$\sum _{y=n}^x \frac{n^2! \left(n^2\right)^{-x} \mathcal{S}_x^{(y)}}{\left(n^2-y\right)!}$$
I tried to compute a closed ...
- 14.9k
3
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2
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Sum[] function not computing the sum
I have the following sum:
$$\sum _{y=0}^x \left(1-n^2! \left(n^2\right)^{-y} \mathcal{S}_y^{\left(n^2\right)}\right)$$
Which, in Wolfram language can be written as:
...
- 20.9k
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1
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136
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Integration Technique
I have 6 huge expressions I want to integrate. Is there any way to break them piecewise and then send them for integration?
Two sample expression are:
Expression 1:
...
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0
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Problem with Plot3D in Mathematica for plotting functions involving parabolic cylinder, exp, and error functions
I have the following problem to make plots of two functions:
Eq.1
$$
\begin{align*}
A_1(a, \Delta, \omega_0)
&= 1 - \frac{2\Delta^2}{\omega_0} + \frac{a^2}{\omega_0^2}
- \left[1 + \frac{...
- 2,590
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2
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235
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Incorrect result for Integrate[] over Region
I was doing a calculation that involved integrating rational functions over a triangular region, and eventually noticed some of my results didn't make any sense. After some investigating, I was able ...
CommunityBot
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1
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Replicating Complex Plot (branch cuts, poles, shading)
I am trying to recreate the following 2 plots:
where the function being plotted is $|e^{2ipR}S_0^{\text{well}}(p)|$ where $|...|$ is the absolute value, $R=1$, and $S_0^{\text{well}}(p)$ is given as
\...
- 2,185
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2
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187
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1
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1
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I want to plot the group delay of the following mathematica code but it takes too long time
I am working on a Mathematica project where I want to plot the group delay. The problem is this code is running too slow. Is there any possibility of speeding up the code?
Here is the original code:
<...
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2
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Evaluation leak in Derivative[]
Bug introduced in 4, persisting through 14.1.
I usually only report bugs, but in this case, two reason prompted me to post this:
Given that derivatives are computed internally by various commands, ...
- 241k
1
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3
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Calculation of an integral with a Bessel function
How can I calculate below integral? Is it possible to calculate it in Mathematica? If yes, how? Am I doing something wrong?
...
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0
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How to represent arithmetic operations (addition, subtraction, multiplication, and division) in column form?
Can Mathematica generate the vertical format for arithmetic operations such as addition, subtraction, multiplication, and division?
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1
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How to do matrix calculus in Mathematica?
I would like to differentiate Ax with respect to A where A is a matrix. Now, obviously, I know the derivative is x but for more nasty expressions, is there a way to differentiate expressions with ...
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2
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63
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Same integral giving different results
I am trying to solve the following integral using Mathematica
$\int_{0}^1 dz \int_{0}^{1} dz' \exp(-k|z-z'|)\cos[\pi p z] \cos[\pi q z]$,
with $p,q\in \mathbb{Z}$. To do so, I am doing the following:
<...
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1
answer
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Poincare section keeps running without giving results
I am trying to find the Poincare section as given in Figure 7 of this paper.
This is the code that I am using for it:
...
- 47.7k
1
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1
answer
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How to Solve Partial Derivative with Summation (for Linear Regression)?
I'm demonstrating linear regression so I would like to calculate derivatives 1 & 2 using Mathematica, something like this
This is based on the textbook notation in Johnston's Econometric Methods ...
- 67.6k
1
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0
answers
62
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How can I speed up symbolic integration of this highly oscillatory function?
I am trying to figure out how to speed up this symbolic integration. I have tried parallelization and numerical integration (both I am failing to implement).
Here is the function:
...
- 5,370
0
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1
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28
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3
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2
answers
155
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Unexpected behavior of SetDelayed and Derivative
Let
A = x^2
and define a function
f[x_] := A
Computing D[f[x],x] gives ...
- 241k
5
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2
answers
354
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An integral using Mathematica or otherwise
Consider the unit square integral $$I=\int_{(0,1)^5}\frac{x(1-x)y(1-y)u(1-u)v(1-v)w(1-w)}{(1-(1-xyuv)w)^2}\ dxdydudvdw$$
Using Mathematica or otherwise I need a closed form of I, possibly in terms of ...
- 14.9k
3
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1
answer
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Many contradictory results for a single integral
I am interested in solving the integral $$
\int_{\mathbb{R}} dx \frac{e^{i a x^2}}{(x - b - ic)^d(x - b + ic)^d}
$$
for $a,b,c \in \mathbb{R}$ and $d \in \mathbb{N}$. As an early step, I have asked ...
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2
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139
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Incorrect limit involving hyperbolic Kepler equations
I’m trying to work with the Kepler equations for a hyperbolic trajectory in the case of a repulsive force (see Wikipedia for the attractive case). The trajectory can be described by the right branch ...
- 161k
3
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0
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Comparing simplification result by Mathematica with Maple [closed]
I use both Maple and Mathematica. Was porting small code from Maple to Mathematica.
In one small computation I noticed this very strange difference and is really serious difference which can affect ...
- 27.7k
2
votes
1
answer
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LaplaceTransform and DSolve handles PDE well in version 14, but not in 13.3
I'm facing an error when trying to solve a simple Laplace-transformed PDE with one basic boundary condition, followed by applying the InverseLaplaceTransform. The ...
- 67.6k
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1
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How we can apply a pure function for dot and prime
I want to apply a pure function. If Y[t,r]
How can I replace the derivative of Y with respect to ...
- 161k
3
votes
0
answers
69
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FilledPolarCurve: Area and Integrate give different values
Version 14.1 introduces FilledPolarCurve as a new way to make regions.
In some cases, Area and integrating 1 over the region give different values for the region's area. Is this a bug?
Example:
...
- 241k
0
votes
1
answer
53
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Parametric differential equations
I'm totally beginner in Wolfram. And I trying to do some research. If it's not difficult for you, please help me.
I have a system of three differential equations with four parameters: beta, delta, p, ...
- 55.5k
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1
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How do I display my output as a complex number with decimals for the real and imaginary components? [closed]
This is my code:
c=Integrate[t/(E^t-1),{t,-Pi,Pi}]/(2Pi) a[j_]:=Integrate[t*Cos[j*t]/(E^t-1),{t,-Pi,Pi}]...
- 161k
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1
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2-loop diagrams dimensional regularization with Feyncalc
I attempted to evaluate the following diagram using FeynCalc, following the method outlined by Pierre Ramond in his QFT book, but I encountered difficulties:
\begin{equation}
\Sigma(p)=\frac{\lambda^2\...
- 3,562
2
votes
1
answer
64
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Help dsolve obtain this solution to standard second order ode with missing x and y
Could someone find a smart way to help DSolve solve this second order standard ode? This is second order nonlinear ode with missing $x$ and missing $y$.
...
- 55.5k
-3
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1
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I have a error on my code but I can't find it!
I have a problem by my code I can't find it yet!!
...
- 59.8k
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NDSolve for non-linear Poisson equation: the case of a pn diode
I am solving the physical problem of a pn diode under no bias voltage. This involves solving a rather 'complicated' non-linear Poisson's equation in 1D
\begin{equation}
-\frac{\partial^2\phi(z)}{\...
- 585
1
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2
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Integration of linked PDFs over probability simplex
I'm trying to integrate the following expression:
f[a1,p1+p2+p3+p4]*f[a2,p2+p3+p4]*f[a3,p3+p4]*f[a4,p4]
Where
...
CommunityBot
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2
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364
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Incorrect results of DSolve
Finding a fundamental solution of a certain
linear partial differential operator, I try
...
CommunityBot
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1
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2
answers
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How to find the local maximum and minimum of a function
I am trying to find the local maximum and minimum of a function. The code below works if function is an exponential but when function involves a sinusoid, it does not give numerical values and instead ...
CommunityBot
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votes
1
answer
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Uncaught SystemException returned to top level calling integrate in V 14.1
Report to WRI [CASE:5169570]
May be someone with better debugging skills than me in Mathematica internals could figure what causes this. This integral worked OK in V 14.0 but now in V 14.1 it gives
<...
- 4,206
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1
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Simplifying an ODE in Bessel notation
I have the following Mathematica code:
eq[r_] = 3 (1 + 2 a) g1 r y[r] - 2 (2 y'[r] + r y''[r]);
DSolveValue[eq[r] == 0, y[r], r]
This gives me the ...
- 2,513
4
votes
1
answer
93
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Best way to implement the "plus-distribution" for very long expressions?
I'm at my wits end here. Is there an efficient way to implement the following distribution into MMA:
$$\int_0^y dx \frac{f(x)-f(y)}{y-x}$$
where $f/1-x$ can be analytically integrated. The difficulty ...
- 109k
2
votes
3
answers
187
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How to fix this "wrong" analytical integral evaluation?
Integrate[(1/Sqrt[(x + 1)^2 + 1])*(1/Sqrt[1 + x^2]), {x, 0, 1}]
yields
...
- 241k
5
votes
1
answer
258
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Closed form of an integral using Mathematica or otherwise
Define $$I=-\int_0^1\int_0^1 \frac{x^2(1-x)y^2(1-y)(2(1-xy)+(1+xy)\log(xy))^3}{(1-xy)^7}\ dxdy $$
Now using Wolfram Alpha
$I\approx 0.00133186$.
Using Mathematica or otherwise, I need to find a ...
- 5,042
1
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1
answer
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Integration of Gaussian functions takes ages
I am trying essentially to compute a fourier transform with Integrate.
1]Is integrate faster or slower than FourierTransform?
2]Does anyone know how to efficiently calculate these integrals?
My code ...
CommunityBot
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0
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How to use mathematica for differential geometry
I would like to learn how to efficiently use mathematica for differential geometry and tensor analysis. I googled it and mostly found some old references to packages. I wonder if there is a good ...
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Integration or numerical integration over one variable out of many variables
I have a function defined as
AFB = (6*I6parSMM)/(2*(3*I1cparSMM + 6*I1sparSMM - I2cparSMM - 2*I2sparSMM));
where the I'parSMM's are functions of other variables s, ...
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1
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Can this previously solved three-dimensional constrained integration also be solved with certain added products in the integrand?
In Solved3DConstrainedIntegration
the constrained three-dimensional (Hilbert-Schmidt-metric-based HSmetric) integration problem for the absolute separability probability of the two-qubit (quantum bit)...
CommunityBot
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1
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3
answers
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How to set integral starting from 0 in the symbolic solution to ODE
I have the following ODE
Bfull'[t] == - (p0/(
p0 + (1 - p0) E^Integrate[α[s], {s, 0, t}])) * α[
t] (S[t] - Bfull[t]) + α[t]*c + δ * Bfull[t]
where α and S ...
- 241k
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votes
1
answer
64
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Finding an increasing or decreasing function
I have a problem where I am given the derivative of a function, asked to graph it, then state where it is increasing or decreasing using Mathematica.
f'(x)=(x^2-3^x)/2x 0<x<4
I've tried the line ...
CommunityBot
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How to calculate the average stream velocity from the segmentation of video frames?
I have a 10 second video of a river flowing. I want to calculate the velocity of the river flow using optical flow. In this operation, I used the ImageDisplacements ...
- 27.7k
7
votes
2
answers
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Wrong result for simple real-valued integral (spurious imaginary part)
Bug introduced in 13.3 or earlier and fixed in 14.1
There's a question on math SE (question 4879248) about this integral:
...
- 30.5k
2
votes
1
answer
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How can I find a condition for convergence of an integral? [closed]
Wolfram|Alpha says that
$$ \int_0^\infty \frac{x^n}{(x-1)(x-2)} dx=\pi (1-2^n)(\cot(\pi n)+i)$$
for $\text{Re}(n)>-1$. However, for $n=0$, it says the integral doesn't converge, even though it has ...
- 30.5k