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.

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3 votes
0 answers
89 views

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. ...
4 votes
1 answer
88 views

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 votes
1 answer
80 views

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 ...
1 vote
0 answers
66 views

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 ...
2 votes
1 answer
124 views

Graph for derivative function

Please help. I can't display a graph for the derivative of a function. ...
12 votes
1 answer
1k views

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 $\...
6 votes
1 answer
472 views

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}] ...
9 votes
3 answers
2k views

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 ...
1 vote
2 answers
213 views

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 ...
28 votes
1 answer
533 views

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: ...
0 votes
2 answers
105 views

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 ...
3 votes
1 answer
150 views

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] & ...
5 votes
1 answer
263 views

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 ...
0 votes
1 answer
52 views

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 ...
2 votes
2 answers
230 views

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)...
1 vote
0 answers
36 views

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: ...
0 votes
1 answer
158 views

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=...
9 votes
5 answers
3k views

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): ...
0 votes
0 answers
64 views

Help implementing Lagrange inversion

I'm attempting to find the coefficients of generating functions using Lagrange inversion. I am finding the series coefficients with: ...
4 votes
1 answer
408 views

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 (...
0 votes
1 answer
50 views

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?
0 votes
1 answer
51 views

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 ...
2 votes
1 answer
48 views

How to linearize these terms in many different variables? [duplicate]

How to linearize these terms in q[t] and p[t,y] ...
0 votes
0 answers
54 views

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 ...
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]: ...
0 votes
1 answer
493 views

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 ...
2 votes
3 answers
143 views

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^...
0 votes
2 answers
76 views

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. ...
1 vote
1 answer
52 views

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 ...
0 votes
1 answer
74 views

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: ...
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. ...
0 votes
0 answers
50 views

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 ...
0 votes
1 answer
296 views

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 ...
1 vote
1 answer
281 views

Incorrect results of DSolve

Finding a fundamental solution of a certain linear partial differential operator, I try ...
1 vote
1 answer
257 views

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 ...
1 vote
1 answer
298 views

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 ...
1 vote
1 answer
54 views

First order differential equation with two conditions

I want to produce all possible solutions for this equation: ...
0 votes
1 answer
185 views

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, ...
0 votes
0 answers
47 views

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 ...
1 vote
1 answer
100 views

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}...
0 votes
0 answers
106 views

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 ...
1 vote
1 answer
132 views

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 ...
0 votes
1 answer
84 views

Incorrect results for Integrate[Cosh[x]/(Sinh[x] + Cosh[x]), x] [closed]

Input: ...
0 votes
1 answer
93 views

Change of variable of a numerically solved integral

I have the following function that numerically integrates a function on the interval $\left[ 2, 100 \right]$ ...
1 vote
0 answers
81 views

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 =...
1 vote
1 answer
88 views

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 ...
2 votes
2 answers
85 views

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) ...
2 votes
0 answers
273 views

Symbolic integral with UnitStep functions

I want to symbolically evaluate the following integral (with real positive parameters L1, L2, ...
0 votes
1 answer
141 views

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 ...
8 votes
3 answers
3k views

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? <...

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