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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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How to Handle First Variation with Symbolic Computation in Mathematica?

Background I am currently working on a problem related to surface tension and found a research paper that appears to have the information I need. I am attempting to replicate the calculations in this ...
Mikoto's user avatar
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2 votes
2 answers
97 views

Why does Replace function not work the first time I try it?

Please explain why in the following mathematica code, the cos term was not replaced in the first try, but got replaced in the second try? ...
Prem's user avatar
  • 121
2 votes
1 answer
89 views

How to handle undefined Integral results for excluded conditions in Mathematica

I have a mathematical function where both input parameters x and y are real numbers within the interval [−π,π]: ...
Xuemei's user avatar
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0 votes
1 answer
43 views

LineIntegrate seemingly does not work

The scalar line integral function seemingly does not work, the both expressions remain unevaluated: ...
Anixx's user avatar
  • 3,630
3 votes
1 answer
119 views

How to run Integrate[Sqrt[(x - a)^2 + (y - b)^2 + (z - c)^2], Element[{x, y, z}, Ball[{0, 0, 0}, 1]], Element[{a, b, c}, Ball[{0, 0, 0}, 1]]]

I find that MMA cannot run the integral below ...
metroidman's user avatar
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0 answers
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Issue solving recurrence relation

I´m trying to solve the following recurrence relation: ...
Cardstdani's user avatar
1 vote
1 answer
40 views

Allan Variance integral for a first order Markov process

I am using Mathematica to solve this definite integral: 4*(q*T)^2*Integrate[(Sin[Pi*f*t])^4/(Pi*f*t)^2/(1 + (2*Pi*f*T)^2), {f, 0, Infinity}] $T$ and $t$ are real ...
George Pavlath's user avatar
3 votes
2 answers
283 views

Problem computing a limit

I have the following function in Mathematica: $\frac{x! \sum _{i=1}^j (-i+j+1) S_{j+1}^{(-i+j+2)} x^{-i+j+1}}{(x-j)! \left((1-x)_j\right){}^2}$ Defined as: ...
Cardstdani's user avatar
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Troublesome output of summation's derivative

I am trying to compute a derivative of a nested sum, but I can't avoid Mathematica to output Kronecker deltas and unexpected cases on complex numbers. Consider the following setting: ...
FDP's user avatar
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1 vote
2 answers
129 views

Convergence of $\left\{n!\sum_{k=1}^{n!}\frac{1}{k^{\frac{3}{2}}}\right\}$ as $n\to \infty$ using Mathematica or otherwise

Using Mathematica or otherwise, I need to find the convergence or divergence of$$\lim_{n\to\infty}\left\{n!\sum_{k=1}^{n!}\frac{1}{k^{\frac{3}{2}}}\right\}$$ where $\{x\}$ denotes the fractional part ...
Max's user avatar
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2 answers
116 views

NDsolve does not solve equation

I have a question for system of ordinary differential equations, because Mathematica does not give the solution. To be more specific, it does not give any sign that there is some problem. There is the ...
Miroslav Jovic's user avatar
1 vote
1 answer
94 views

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 ...
GJ P's user avatar
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1 vote
1 answer
170 views

What methods can quickly determine the coefficient of determination of the data?

The given data is: x = {0.04, 0.06, 0.04, 0.08, 0.08, 0.05, 0.05, 0.07, 0.07, 0.06}; y = {0.25, 0.4, 0.22, 0.54, 0.51, 0.34, 0.36, 0.46, 0.42, 0.4}; Is there a ...
csn899's user avatar
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3 votes
0 answers
129 views

Trivial Mathematica integral on a cube not computing

I was trying to evaluate some intergal over some cube with variable side length, but for some reason Mathematica spit out the input. The interand was sufficiently nice that I couldn't believe it didn'...
tommy1996q's user avatar
5 votes
3 answers
714 views

Mathematica integral wrong

I need to evaluate the following integral: ...
tommy1996q's user avatar
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0 answers
50 views

Numerical Integral and Partial Differentiation Of a Complicated Expression

I need help plotting Figures 1 and 2 solving EQ 36 and 37 from https://arxiv.org/pdf/2310.12445. here I am trying to find \Gamma,\phi and their Q values for ...
Argha Debnath's user avatar
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1 answer
53 views

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 ...
Michael Hamel's user avatar
1 vote
2 answers
146 views

Confused about the output of `CosIntegral`

I am interested in the properties of the cosine integral function, in this case the anti-derivative of Cos[Pi*x]/x, which Mathematica evaluates to ...
Richard Burke-Ward's user avatar
0 votes
1 answer
49 views

Checking the sign value of the derivatives of a complicated function

Consider the following function: ...
ppp's user avatar
  • 765
0 votes
2 answers
80 views

Integration with respect to functions

I observe some strange behaviour of Mathematica when pulling functions into the integrator. I know that Mathematica's capabilities are limited when computing Stieltje or Lebesgue Integrals. But even ...
oyy's user avatar
  • 187
4 votes
3 answers
250 views

How is the result of this integral obtained by the function `Integrate`?

When I use Mathematica to calculate $$ \int_0^{+\infty } \frac{\log (1-\mathrm i x)}{1+x^2} \, \mathrm dx , $$ I get different results using functions NIntegrate ...
Soriak's user avatar
  • 411
2 votes
2 answers
96 views

Partial Derivative after Numerical Integration of a Complicated Expression with Singularity at Zero

I want to generate Fig no. 2,3,4 from https://arxiv.org/pdf/2212.08237 with the help of the following equations. In Fig 3 t_(opt) is the time Q_T reaches its max value Q_T(opt) for each T. Here is ...
Argha Debnath's user avatar
1 vote
1 answer
61 views

StreamPlot3d for the magnetic field of a loop

I am trying to go beyond StreamPlot in the case of a current loop that produces a magnetic field. My code works for StreamPlot, but does not end when I use the same field expressions with StreamPlot3D....
Coti's user avatar
  • 43
3 votes
3 answers
347 views

Integration of the product of two exponential functions

How can I find the value of this integral? Integrate[E^((-b)*t^a - k*t), {t, 0, Infinity}]
Emad kareem's user avatar
0 votes
1 answer
44 views

Integrate diverges for convergent integral, and returns unnecessary condition

I'm trying to solve an infinite integral using Mathematica. For some reason, Mathematica claims the integral diverges although when solving to a finite limit L and ...
Yoav Zack's user avatar
  • 433
2 votes
1 answer
35 views

Describing the region of integration $\mathbb{R}_+^n$ for arbitrary $n$

Let $f_n:\mathbb{R}^n \to \mathbb{R}$. I would like to know an elegant way of writing the integral $$ I_n = \int_{\mathbb{R}^n_+} f_n(x)~\mathrm{d}x $$ in Mathematica. For fixed $n$, the best I could ...
P.S. Dester's user avatar
0 votes
5 answers
366 views

Possible bug in partial derivative [closed]

In 14.0 on Windows 10 let us consider D[Min[x, y], x] Piecewise[{{1, x - y <= 0}}, 0] As I understand it, the result is ...
user64494's user avatar
  • 27.3k
9 votes
8 answers
737 views

Numerical approximation of the integral by using data

I want to use the numerical approximation of the integral of a function given a list of data: $$\int_a^bf(x)dx\approx\sum_{k=1}^N\frac{f(x_{k})+f(x_{k-1})}{2}(x_{k}-x_{k-1}),$$ where $f(x_0)=f(a)$ and ...
Patrick El Pollo's user avatar
5 votes
5 answers
250 views

Integration involving Piecewise function and DiracDelta function

I want to calculate an integration, which reads where $\delta\left(q_{23}^{01}\right)=\delta\left(1+q_1-q_2-q_3\right)$ and $\mathrm{min}(1,q_1,q_2,q_3)$ means the minimum of $(1,q_1,q_2,q_3)$. What ...
so_sure's user avatar
  • 421
1 vote
1 answer
60 views

Checking the formula for solving the Poisson equation with an integral over all of space

EDIT: According to 2.2.3 Rigorous derivation, if we find the integral $u(x,y) = \int \int G(x,y,m,n)f(m,n) dm dn$ then we obtain a function that satisfies the Poisson equation $\nabla^2u(x,y)=f(x,y)$. ...
ayr's user avatar
  • 2,444
1 vote
2 answers
113 views

How can I check integration $\int_0^c \exp\left(-ax+\frac bx\right)~dx$?

I am trying to integrate $$\int_0^c \exp\left(-ax+\frac bx\right)~dx$$. I proceed manually in this way (according to https://math.stackexchange.com/questions/2978887/primitive-of-exp-a-times-x-frac-bx?...
Abhishek Ghosh's user avatar
3 votes
1 answer
81 views

How to add a custom analytical integration to integration library

How can I add an unknown analytical integral in Mathematica to its integration library so that the Integrate routine, e.g. ...
mheld's user avatar
  • 31
3 votes
3 answers
164 views

Issue in HypergeometricPFQ function:

I have a solution from integral: A = Integrate[x^n*(1 + x)^n*Exp[-n*x^2] /. n -> 1, {x, 0, Infinity}] //Expand (*1/2 + Sqrt[π]/4*) %//N (*0.943113*) Then I ...
Mariusz Iwaniuk's user avatar
1 vote
2 answers
128 views

Solution of an implicit integral equation

The present question is pretty similar to this one. Here I want to find the solution of the following implicit integral equation: $$ P_H(u)=\int_{-\infty}^\infty dJ \int_{-\infty}^\infty du_1 \int_{-\...
Ruth Murphy's user avatar
0 votes
1 answer
113 views

NIntegrate can't solve this complicated multiple integration with absolute value in Exp and Cos functions

I'm trying to use NIntegrate to calculate this complicated multiple integration, which has absolute value in the Exp and Cos functions. $$ \int_{-\infty}^{\infty} dx_1 \int_{-\infty}^{\infty} dx_2 \...
qshi47's user avatar
  • 1
4 votes
3 answers
404 views

Why is DSolve unable to solve this first order homogeneous ode with IC?

This is first order homogeneous ode from undergraduate textbook, which can be solved using standard $y=u x$ change of variables. But DSolve can't. Could someone figure out why? I show below that ...
Nasser's user avatar
  • 147k
1 vote
0 answers
66 views

Integration by parts for Green's first identity

How should one implement $$\int _{\Omega }\nabla u\cdot \nabla v\,d\Omega \ =\ \int _{\Gamma }v\,\nabla u\cdot {\hat {\mathbf {n} }}\,d\Gamma -\int _{\Omega }v\,\nabla ^{2}u\,d\Omega\,,$$ Symbolically ...
FreeMind's user avatar
  • 141
0 votes
0 answers
44 views

Problem Derivative (D) of function defined with Piecewise or UnitStep functions - SIMPLE BEAM PROBLEM, 3 SUPPORTS

I am solving a simple structural beam problem using slope-deflection approach - 3 support A(left- fixed), B(middle- pin), C(right - pin). Beam AB - uniform load, Beam BC - concentrated load in ...
SAK's user avatar
  • 63
2 votes
2 answers
164 views

How to calculate this improper integral?

How to calculate the Integral involving MarcumQ function whose Integral interval is (0, +Infinity)? Any help (code or reference) would be greatly appreciated. ...
138 Aspen's user avatar
  • 1,593
0 votes
0 answers
62 views

Analytic expression of the $n$-th derivative of a function of two variables

I am hoping to get Mathematica to find an analytic form for the $n$th derivative of the following function of two variables: $$f(a,b)=\frac{1}{\left((1-a) b\right)^{-k} \left(1-(1-a) b\right)^{k-n}+\...
tyogi's user avatar
  • 297
7 votes
2 answers
378 views

How to find stationary points graphically?

I have the following Mathematica code: ...
codebpr's user avatar
  • 2,433
2 votes
2 answers
214 views

Numerical derivative of numerical function

I am trying to calculate the derivative of a numerically defined Hamiltonian. This is a minimal working example (on a simple example): ...
blahblah's user avatar
  • 111
2 votes
1 answer
70 views

Bugs with `Integrate` when dealing with complex situation?

I'm trying to calculate $\int_0^{\infty } \exp \left(t \left(-x^3+(1+i) x\right)\right) dx$ with mathematica, and different assumptions on t give different results: ...
Jie Zhu's user avatar
  • 1,811
3 votes
1 answer
463 views

What tools does Mathematica have for solving indefinite integrals?

I haven't used Mathematica regularly in about 15 years, so I'm a bit out of date on what the latest tools are for indefinite integration. I have Mathematica 11 on my computer. through searching online,...
Machinus's user avatar
1 vote
4 answers
173 views

Can Mathematica solve $\int \frac{\cot^2\left(\frac12\sec^{-1}(x)\right)}{\sqrt{\tan\left(\frac12\csc^{-1}(x)\right)}} \, dx$?

Can Mathematica solve integral $(1)$? $$\int \frac{\cot^2\left(\frac12\sec^{-1}(x)\right)}{\sqrt{\tan\left(\frac12\csc^{-1}(x)\right)}} \, dx.\tag{1}$$ This is ...
Emmanuel José García's user avatar
2 votes
2 answers
67 views

How to get derivation of a function with respect to another function?

I often have this need: I have some functions that have the same variable and have inclusion relationships, like this. f[t_] := t^2 + 1 g[t_] := Exp[-I Sin[f[t]]] ...
tsukatsuki_sorano's user avatar
0 votes
0 answers
61 views

Solving a set of non-linear coupled differential equations

Here is a set of first-order differential coupled equations with ...
SciJewel's user avatar
  • 575
1 vote
2 answers
121 views

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 ...
kahless96's user avatar
9 votes
5 answers
308 views

How to help DSolve obtain this solution to $y'(x)=\sec (y(x))$

V 14 ode = Cos[y[x]]*y'[x] == 1 ic = y[0] == 2 DSolve[{ode, ic}, y[x], x] Anything can be done to help DSolve obtain the following solution? Hand solution \begin{...
Nasser's user avatar
  • 147k
1 vote
1 answer
53 views

Laplace transform of special function

The Confluent hypergeometric function of first kind (aka Kummer's function) is defined as $${\mathbf{M}}\left(a,b,z\right)=\frac{1}{\Gamma\left(a\right)\Gamma\left(b-a% \right)}\int_{0}^{1}e^{zt}t^{a-...
K.K.McDonald's user avatar

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