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.
4,921
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Workarounds for unevaluated results of ContourIntegrate of non-analytic functions
There is a new useful command ContourIntegrate since 13.3.
I find that the command is well-done, but even the Sun has spots.
...
- 24.5k
4
votes
1
answer
88
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Strange result simplifying higher order BesselJ
Consider the following integral:
$$\int_0^1 Z_n^m(r)\ J_m(\rho r) r dr$$.
The solution of this should contain a single Bessel function:
$$(-1)^{(m-n)/2}\ J_{n+1} (\rho)/\rho$$ (see https://www.osti....
- 2,887
2
votes
1
answer
80
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How to make Assumptions in IntegrateChangeVariables?
There is a new useful command IntegrateChangeVariables since 13.1.
The command has the Assumptions option which is very poorly ...
- 3,548
1
vote
0
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66
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How do I integrate with respect to one variable when an integrand has many variables?
I am trying to integrate the following integrand:
(μ*a*curr/4 π)*((x - a*Cos[ϕ] + y - a*Sin[ϕ])/
((x - a*Cos[ϕ])^2 + (y - a*Sin[ϕ])^2)^(3/2))
with the following ...
- 18.9k
2
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1
answer
124
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Graph for derivative function
Please help. I can't display a graph for the derivative of a function.
...
- 152k
12
votes
1
answer
1k
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Integral with HeavisideTheta takes too long to evaluate
Bug introduced in 9.0 or earlier, persisting through 13.2 or later
I tried to compute $$\int_{-1}^1 d x_1 \int_{-1}^1 d x_2 \int_{-1}^1 d y_1 \int_{-1}^1 d y_2 \theta(x_1 x_2 + y_1 y_2)\,$$ where $\...
- 13.1k
6
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1
answer
472
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Symbolic integration of a sigmoid function
I'd like to obtain a symbolic result for the following integral:
Sigmoid[x_] := 1/(1 + Exp[-x])
Integrate[Sigmoid[x]*Log[Sigmoid[x*w]], {x, -Infinity, Infinity}]
...
- 65.8k
9
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3
answers
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Mathematica gives wrong answer for an integral
Bug introduced in 9.0 or earlier, persisting through 13.2 or later
When I execute the following,
Integrate[ Exp[-w^2 + I w^3], {w, -∞, ∞}]
I get
...
- 13.1k
1
vote
2
answers
213
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DSolveValue differential equation list of arbitrary values
I have been solving arrays of differential equations, and recently moved to the general case, where my initial conditions are arbitrary constants (either real or complex). However when I moved from ...
CommunityBot
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28
votes
1
answer
533
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Are greek symbols causing different evaluation?
I've updated today to Mathematica 9.0.1.0 from version 8 and found something that absolutely confuses me.
Let us define a piecewise function:
...
- 63.8k
0
votes
2
answers
105
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Integration result depending on variable name
From previous threads I have read on the subject, it seems that Mathematica is known to simplify expressions to varying degrees depending on alphabetical order of chosen variable names. But none of ...
- 63.8k
3
votes
1
answer
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Do symbolic integration of function including \[ScriptCapitalL]
I have done a symbolic integration in mathematica 13.2 as follow
Integrate[-((Sqrt[-r(r - 2 l)] l)/(r(r - 2 l) (-r + l))), r] // Simplify[#, r > 0] &
...
- 63.8k
5
votes
1
answer
263
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Integral depends on choice of symbols
I am trying to perform integrals of the kind $\int dx\frac{x^2-q^2}{z-x+i\eta}$. Mathematica, however, gives back different results whenever I replace the parameter $z$ by $w$, for instance, as ...
- 63.8k
0
votes
1
answer
52
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How to chose appropriate finite limits for NIntegrate?
I want to NIntegrate a function using finite limits so it gives a result as close as possible to a result which I would get making the limit ...
CommunityBot
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2
votes
2
answers
230
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The center of mass of a semiellipsoid
I am trying to find the center of mass of a semiellipsoid using cylindrical coordinates.
r^2/a^2+z^2/b^2 < 1, z < 0 the density = 1.
I know that the center of mass is (1/M int,int,int rcos(theta)...
- 47.3k
1
vote
0
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36
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How to assume a function is holomorphic?
For example, for a holomorphic function $f(z)=f(x,y)$, it's well know that $\ln |f|$ is harmonic. I want to check it with mathematica with the following code:
...
- 11
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1
answer
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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=...
CommunityBot
- 1
9
votes
5
answers
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Digits of Pi in colored spiral
In How to make the digits of π go around in a spiral like this?
it is described how to plot pi in a spiralform (in my case as binary number):
...
- 42k
0
votes
0
answers
64
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Help implementing Lagrange inversion
I'm attempting to find the coefficients of generating functions using Lagrange inversion. I am finding the series coefficients with:
...
- 24.5k
4
votes
1
answer
408
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Implementing positivity constraints over a six-dimensional hypercube
This question involves the same subject matter as my previous one (How can one achieve the most accurate estimates of certain six-dimensional integrals under specific constraints?), but with another (...
CommunityBot
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0
votes
1
answer
50
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Plotting an integral with solving for a specific constant [closed]
Given an Integral I(t) which is a function of a constant tc, this constant is determined with the condition I(tc)=0, how can I add this condition?
- 46.2k
0
votes
1
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51
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Numerical Integration without giving all values [closed]
Is there a way to solve expressions numerically, while keeping other variables.
For example, I want to Integrate the expression below for $\theta p,\theta q$ while keeping other things as it is. The ...
- 24.5k
2
votes
1
answer
48
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How to linearize these terms in many different variables? [duplicate]
How to linearize these terms in q[t] and p[t,y]
...
- 213
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votes
0
answers
54
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Seeking Wolfram Mathematica Techniques for Detailed Step-by-Step Solution of Integral Equation Transformation [closed]
Hello Mathematica Community,
I am working on a mathematical problem where I need to transform an integral equation into a simpler form using Euler's Formula and the Error Function. My current ...
- 1
0
votes
1
answer
62
views
How to linearize this equation in two different variables?
How to linearize this equation in two different variables p[t] and q[t,y]:
...
- 213
0
votes
1
answer
493
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Principal value of a double integral numerically
I have several integrals of the form :
where and .
where a,b,wc and T are constants
and PV denotes the Cauchy principal value I tried to integrate one of this form using NIntegrate however when ...
CommunityBot
- 1
2
votes
3
answers
143
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Find Generalized Series with Symbolic Variable
CoefficientList[Series[Exp[x], {x, a, 3}], x]
Gives the following expression,
$$
\left\{-\frac{1}{6} e^a a^3+\frac{e^a a^2}{2}-e^a a+e^a,\frac{e^a a^2}{2}-e^a a+e^...
- 40.6k
0
votes
2
answers
76
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Integrate with `Max` and `Min` under assumptions runs forever
I'm trying to solve an integral under certain conditions on the parameters. The following is my code.
...
- 24.5k
1
vote
1
answer
52
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Leading significant digits
Given that the initial area in a Sierpinski triangle is A_0 and the subsequent areas
$$(A_n) =A_0 (3/4)^n$$ ,
what is the formula to calculate the leading ...
- 47.3k
0
votes
1
answer
74
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What does "{True,True}" mean in the result of IntegrateChangeVariables?
I want to change variables in the following integral:
Integrate[q*D[z[x, y],x]^2, {x, -∞, ∞}, {y, -∞, ∞}]
as follows:
...
- 24.5k
0
votes
0
answers
98
views
Integral of complex exponential with assumptions
I feel confident this problem has been posed before but alas I cannot find it.
...
- 18.9k
0
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0
answers
50
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Express the double integral in terms of the function values at discrete nodes
Consider the following integral:
$$
\mathcal{I}[p_{1},t] = \int \limits_{0}^{p_{1}}dp_{2}\int \limits_{0}^{p_{2}}dp_{3} \ \text{integrand}
$$
where
...
- 4,681
0
votes
1
answer
296
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How to obtain the convolution of these functions?
I am starting to use Mathematica. I wanted to compute
$$\nabla F*\phi(x),\;x\in\mathbb{R^2}$$
where $F(x)=\ln\|x\|$ and $\phi(x)=\chi_{B(0,2)}(x)$.
I am a new user of this software, so I don't know ...
CommunityBot
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1
vote
1
answer
281
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Incorrect results of DSolve
Finding a fundamental solution of a certain
linear partial differential operator, I try
...
CommunityBot
- 1
1
vote
1
answer
257
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Plot solution of an integral with parameter
In this article, they solve the following integral
$$ \mathcal{I}_{o b s}\left(\nu_0\right)=\int_{\gamma_i} \frac{(A(r))^{3 / 2}}{r^2} \sqrt{\frac{1}{A(r)}+r^2\left(\frac{d \varphi}{d r}\right)^2} d r ...
- 42k
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
- 1
1
vote
1
answer
54
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First order differential equation with two conditions
I want to produce all possible solutions for this equation:
...
- 137k
0
votes
1
answer
185
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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
- 1
0
votes
0
answers
47
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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 ...
- 691
1
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1
answer
100
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Plot after applying NIntegrate
I'm trying to integrate and graph $I$ vs $b$ the following integral (eq. 24 in this paper) in wolfram mathematica
$$ I = \int_{2}^{\infty} \frac{g_i^3 \mathcal{K}_t \mathrm{~d} r}{r^2\left|\mathcal{K}...
- 95
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votes
0
answers
106
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Integrate function differing from RUBI's Int function in version 13.3.0
I had Mathematica Version 12.3.1 Student Edition and Integrate and RUBI's Int functions produced the same answers for my purposes with good results. I updated my version to 13.3.0 Student Edition and ...
- 9
1
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1
answer
132
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How to separate parameters from a list based on certain conditions?
The following codes create a table of parameters Br and Dr based on corresponding parameters b and c. How to separate/divide the list/table of Br and Dr for which corresponding b or c are negative ...
CommunityBot
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1
answer
84
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0
votes
1
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93
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Change of variable of a numerically solved integral
I have the following function that numerically integrates a function on the interval $\left[ 2, 100 \right]$
...
- 49.6k
1
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0
answers
81
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Intensity pattern [closed]
I'm trying to get the following intensity pattern from this paper
In the Newtonian case, the equations for the intensity (eq. 9 and 10 in the article), I put them in Mathematica with the change $b^2 =...
- 95
1
vote
1
answer
88
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Definite integral of real function outputting complex values
I'm trying to integrate the function $f(\omega)$ from $-\infty$ to $\infty$, where
$$
f(\omega) = \begin{cases}
\frac{(1-2 \omega )^2}{64 \cos (\tau (-\omega +\phi
+1))-80} & 0<\omega \leq ...
- 24.5k
2
votes
2
answers
85
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How to prevent evaluation without simplification?
I am trying to compute a sequence of terms, which have a form similar to $g(z)$ at different points $z$. For $z$ close to 0, I have observed that the terms are evaluated as 0/0 (indeterminate form) ...
- 22.9k
2
votes
0
answers
273
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Symbolic integral with UnitStep functions
I want to symbolically evaluate the following integral (with real positive parameters L1, L2, ...
0
votes
1
answer
141
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Concept of tangent and residual in AceFem and automation
here trying to solve the pde: ut + ux = 0 in acefem. I have two questions regarding tangent and residual.
Tangent shows the s$$ and residual shows ...
CommunityBot
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votes
3
answers
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Why doesn't Mathematica recognise two integrals as equal?
The following expression
$$\int_{0}^{t} f(t_0) \,dt_0 - \int_{0}^{t} f(t_1) \,dt_1 = 0$$
is zero because it is just a change of integration variable.
Why doesn't Mathematica give zero in this case?
<...
- 24.5k