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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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Strange behaviour of integrals with Cos, Sin, and Exp

Bug introduced in 8.0.4 or earlier and persisting through 14.0 During the study of the problem How to solve this integration? I have discovered a strange behaviour of some integrals. I would ...
Dr. Wolfgang Hintze's user avatar
14 votes
0 answers
342 views

Is there a way to teach integrate new solutions?

I have an integral which I can solve, but integrate cannot: ...
Sebastik's user avatar
  • 181
12 votes
0 answers
254 views

Reduce: Var is not a valid variable

Bug introduced in 7.0 and persisting through 13.1 or later. Fixed in 13.2.0 or earlier. I am trying to integrate the following but it returns a Reduce::ivar ...
BabaYaga's user avatar
  • 1,876
11 votes
0 answers
284 views

MacDonald formula for Modified Bessel Functions

How can I make Mathematica understand these two integrals? $$\int_0^{\infty} e^{-x \cosh{\xi}} d\xi = K_0(x)$$ $$\int_0^{\infty} e^{-\frac{1}{2} \Big( \frac{x y}{u} + u \frac{x^2+y^2}{x y} \Big) } K_{...
Kevin Driscoll's user avatar
10 votes
0 answers
317 views

Possible Symbolic Integration Bug

Bug introduced between 5 and 8 and persisting through 12.0. I think I may have found a bug, and want to verify is this reproducible in other versions and platforms and not a mistake of mine. I ...
yyli's user avatar
  • 111
9 votes
0 answers
209 views

The most bizarre kernel crash ever. Kernel crash in Integrate under simple different setups. Unable to find cause

Reported to WRI. CASE:4330461 V 12 on windows 10 64 bit. Note: This problem do not show up in V 11.3. Only in V12. For the last 2 hrs, I've been trying to zoom into why V12 kernel crash when ...
Nasser's user avatar
  • 144k
9 votes
0 answers
187 views

A bug in Derivative?

Update This is a bug in v11.3 or earlier and is fixed in v12. Original Post Check this out: ...
luyuwuli's user avatar
  • 2,814
9 votes
0 answers
273 views

Incorrect evaluation for Thue-Morse signed harmonic series

I would like to evaluate $$s = 1 - \frac{1}{2} - \frac{1}{3} + \frac{1}{4} - \frac{1}{5} + \frac{1}{6}+\frac{1}{7}-\frac{1}{8} - ... + \frac{(-1)^{\textrm{binary digit sum}(n-1)}}{n} + ... $$ where ...
Douglas Zare's user avatar
8 votes
0 answers
152 views

Symbolic comparison of integrals

I'd like to prove algorithmically the fact that the integral of $e^{-x^2 - y^2}$ evaluated over a circular disk, ${\cal D}$, is greater than over a square, ${\cal S}$, of the same area regardless of ...
David G. Stork's user avatar
8 votes
0 answers
240 views

Two integrals that should not be equal

Bug introduced in 12.0 or earlier and persisting through 14.0 CASE:4539809 I think there is a bug here: ...
user avatar
8 votes
0 answers
128 views

Bug in integral related to beta distribution

I've encountered a problem, which I'm pretty sure is a bug. I've contacted Wolfram's support and they were less than helpful. I'd like to bring the issue up here to get either a confirmation that it ...
Meni Rosenfeld's user avatar
7 votes
1 answer
150 views

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 ...
smish's user avatar
  • 51
7 votes
0 answers
132 views

GreenFunction for Helmholtz equation in arbitrary Rectangle region doesn't evaluate

Bug persists through V13.0.0 or later Here is a basic example found in the documentation of GreenFunction: ...
Ulrich Neumann's user avatar
7 votes
0 answers
404 views

Integrating rational functions of several variables over $\mathbb{H}^4$

Let $W$ be a rational function of $8$ variables $a,b,c,d,e,f,g,h$ from this file, e.g.: ...
Ricardo Buring's user avatar
7 votes
0 answers
195 views

Convoluting inverse square root with Gaussian

Bug introduced in 9.0 and persisting through 11.0.1 or later I would like to convolute the inverse square root on the interval [0,inf] with a Gaussian function, ...
leopold.talirz's user avatar
6 votes
0 answers
105 views

Does IntegrateChangeVariables work with improper integrals?

Trying a new command of version 13.1 IntegrateChangeVariables, I obtain ...
user64494's user avatar
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6 votes
0 answers
6k views

What's the most difficult multidimensional integral that Mathematica has solved?

I am interested in benchmarking numerical integration methods and am trying to develop a wider set of difficult multivariate examples. For my particular methods, I only want to look at non-negative-...
ben18785's user avatar
  • 3,167
6 votes
0 answers
217 views

Mathematica gives wrong answer for a definite integral

I tried to compute the definite integral Integrate[Exp[Pi I t]/((-I + 1 + t) Cosh[Pi t]), {t, -Infinity, Infinity}] and obtained the answer (version 11.2.0.0) ...
juan's user avatar
  • 171
6 votes
0 answers
267 views

Strange Integrate behavior (a bug!)

The following two calculations should give the same result. After all, integration is a linear operation. I have pasted the code below in case you want to play with it. ...
Hector's user avatar
  • 6,438
5 votes
0 answers
118 views

Create custom definition for chain rule of partial derivative of a vector-valued function

I'm trying to calculate derivatives of expressions involving an unknown function $F: \mathbb{R}\times\mathbb{R}^n\times\mathbb{R}^n \rightarrow \mathbb{R}^n$, with respect to a parameter $h \in \...
D__'s user avatar
  • 191
5 votes
2 answers
177 views

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. ...
Quasar Supernova's user avatar
5 votes
0 answers
125 views

Derivation from a network

how can I represent the derivative of a network (e.g. NetGraph or NetChain) as a network? I have tried it with FunctionLayer[] and NetPortGradient, but without success. I would be very pleased to ...
haro21's user avatar
  • 91
5 votes
0 answers
134 views

Numerical verification of the estimate:

How to verify numerically with considerable accuracy in Mathematica the following : $$\int_2^x\dfrac{1}{z\Gamma(\sin^2[π\Gamma(z)/(2z)])}dz\sim\ln(\ln(x))$$ ? I need more suitable and better code ...
bambi's user avatar
  • 213
5 votes
0 answers
234 views

Incorrect result by DSolve

For real $x$ consider the trivial equation $$|y'(x)|=-|x|.$$ Since the left side is always positive and the right always negative, there is no solution. Let's try ...
JHT's user avatar
  • 1,005
5 votes
0 answers
386 views

Help implementing Magnus Expansion

The Magnus expansion is a tool to approximate solutions to first-order linear differential equations (the Wikipedia page is quite instructive and concise) - it's particularly useful because all orders ...
KHAAAAAAAAN's user avatar
5 votes
0 answers
244 views

Integrating a product of three Spherical Harmonics

The following command is returned unevaluated. The answer is well known to be related to Wigner's 3j Symbol which is also a defined function in Mathematica. ...
Quasar Supernova's user avatar
5 votes
0 answers
552 views

Convergence of approximate solutions to obstacle problem for the heat equation

Consider the problem $$(P) \qquad \begin{cases} \min\{\partial_t u - \Delta u, u -\varphi \} = 0 & \text{ in } (0,T)\times \mathbb{R}^N \\ u(0,\cdot) = \varphi(0,\cdot) & \text{ in } \mathbb{...
user avatar
5 votes
0 answers
89 views

Why does Mathematica recognize the conditional convergence of some integrals but not others?

Consider the following two polynomials: f1[t_] := t^3 - 1 f2[t_] := t^3 + 3t - 1 Both of these polynomials have a single real root: $f_1(t)$ at $t = 1$, and $...
Michael Seifert's user avatar
5 votes
0 answers
233 views

Total variation integration of a discontinuous function

This question derives from this one, about mathematics and Maple. Consider the following discontinuous function: ...
David G. Stork's user avatar
5 votes
0 answers
359 views

How to verify the convexity of a function?

I have an optimization problem with the following objective function in $(x,y)$ $$ A\log \left(\sum_{i=1}^n x_i\right)+\log\left(1-\frac{f}{n}\left(\sum_{i=1}^n\frac{x_i}{y_i}\right)\right) $$ where $...
user_lambda's user avatar
5 votes
0 answers
200 views

Integrate producing bad result

Bug introduced in 9.0 or earlier, and fixed in 10.2 I noticed a bug in Mathematica. It computes incorrectly a definite integral ...
user1765636's user avatar
5 votes
0 answers
208 views

Using NIntegrate and DiscretePlot to visualize pseudodifferential operators

In harmonic analysis, pseudodifferential operators are a way to generalize the notions of derivatives, through the use of Fourier transforms. The basic idea being, Let $u(x)\in\mathcal{S}(\mathbb{R}^...
Patch's user avatar
  • 201
5 votes
0 answers
1k views

Symbolic matrix calculus: What's new in Version 9

I have seen some of the related posts, which ask about doing matrix calculation on tensors unknown dimensions. One of the posts mentioned that version 9 has some capability, but it was concerned about ...
nonlinearism's user avatar
5 votes
0 answers
231 views

Calculating a limit with a result that is discontinuous in the parameters

The following limit is left unevaluated (Edit: added the assumption that $\epsilon$ is real thanks to the comment below): ...
Joe's user avatar
  • 1,471
4 votes
0 answers
81 views

How to obtain the boundary of a two-dimensional ParametricRegion surface in order to apply Stokes' theorem to compute surface integrals?

On the help page of the SurfaceIntegrate, an example of Stokes' theorem is listed: Compute the Curl ...
lotus2019's user avatar
  • 2,131
4 votes
0 answers
92 views

Weird behavior of derivative D[] function in v11

Bug introduced after version 9.0.1, in or before version 11.0, fixed before 12.3. Recently I noticed that a code developed in v13 was giving unexpected errors about the formatting of matrices in v11 (...
Hans Olo's user avatar
  • 1,843
4 votes
0 answers
115 views

Calculate the integral of the Slater determinant

This is a Slater determinant: $$ s=\left|\begin{array}{ll} \psi_{1 s}\left(r_1\right) \alpha & \psi_{1 s}\left(r_1\right) \beta \\ \psi_{1 s}\left(r_2\right) \alpha & \psi_{1 s}\left(r_2\right)...
我心永恒's user avatar
  • 1,562
4 votes
0 answers
195 views

Differentiating with D vs. Derivative

I was tinkering with something and needed a high-order derivative of a function that, when differentiated, needs the product rule (and so, subsequent derivatives - without simplification - become ...
Kellen Myers's user avatar
  • 2,701
4 votes
0 answers
108 views

IntegrateChangeVariables give a wrong result?

Bug introduced in 13.1. RegionPlot[x^(1/(x + 1/x)) + y^(1/(y + 1/y)) >= E, {x, 0, 7}, {y, 0, 7}] We can calculate the area by integral: ...
yode's user avatar
  • 26.8k
4 votes
0 answers
83 views

Calculate an n-order determinant by FindSequenceFunction

Calculate an n-order determinant: $\left|\begin{array}{cccccc}1 & 2 & 3 & \cdots & n-1 & n \\ n & 1 & 2 & \cdots & n-2 & n-1 \\ n-1 & n & 1 & \cdots ...
lotus2019's user avatar
  • 2,131
4 votes
0 answers
159 views

Mathematica 12.0 returning a imaginary value for a real-valued improper integral

Bug introduced in 11.3 or earlier and persisting through 12.3.1 When I use Mathematica to evaluate this integral ...
Jazengm's user avatar
  • 51
4 votes
0 answers
99 views

Integrate providing incorrect result for complex exponentials

Consider the integral \[\int_{-\infty}^\infty\frac{(e^{iax}-1)(e^{ibx}-1)}{x^2}\quad a,b\in\mathbb R\] which can be manually computed to be $\pi|\text{sgn}(a)-\text{sgn}(b)|\min(|a|,|b|)$. However ...
Ariana's user avatar
  • 153
4 votes
0 answers
166 views

How to perform integration with Dirac Delta in Mathematica?

I am trying to calculate the integral $$ y(x)=\int^b_a dz\delta(x-w_z) $$ were $\delta$ is the dirac delta function, $a=-3$, $b=3$ and $w$ is a one-dimensional matrix such that for some values of $z$ ...
gbd's user avatar
  • 409
4 votes
0 answers
133 views

Integration involving DiracDelta

I tried the following integration Integrate[DiracDelta[Tan[x]], {x, -4, 4}] I got 1 as the result. However, between -4 and 4 ...
Dark Lord's user avatar
4 votes
0 answers
161 views

A bug Integrating Piecewise functions

I have found a very puzzling bug (version 12.0), where integrating the sum of a piecewise function with a DiracDelta with a variable integration limit causes errors,...
Alex Bogatskiy's user avatar
4 votes
0 answers
284 views

Legendre–Fenchel Convex Conjugate

What is the easiest way to compute the convex conjugagte of a real convex function $f: \mathbb{R} \to \mathbb{R}$, defined by $f^*(s) = \sup_{x} \{ s x - f(x) \}$ I know I can compute the derivative ...
Alex Shtoff's user avatar
4 votes
0 answers
96 views

Computing differentials on a manifold

Consider $\phi:SO(3) \to \mathbb{R}^3$, $R \mapsto (R^\top e_3)\times e_3$ where $R$ is a real $3\times 3$ orthogonal matrix and $e_3 = [0\ 0\ 1]^\top$. Can Mathematica compute the differential of $\...
anderstood's user avatar
  • 14.3k
4 votes
0 answers
129 views

Evaluate $\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx$

Saw this post and noted that it was posted back in 2013. Just tried V11.3 and got no analytic solution. Just wondering if there is a way for Mathematica to give the desired result? Machine VS Human ...
CasperYC's user avatar
  • 1,550
4 votes
0 answers
130 views

Trigonometric integral with assumptions fails

Consider the following integral Assuming[Element[{n, m}, Integers] && n >= 0, Integrate[Cos[ϕ]^n Exp[I ϕ m], {ϕ, 0, 2 π}]] ...
Kagaratsch's user avatar
4 votes
1 answer
420 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 (...
Paul B. Slater's user avatar

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