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.
763
questions with no upvoted or accepted answers
17
votes
0
answers
486
views
Strange behaviour of integrals with Cos, Sin, and Exp
Bug introduced in 8.0.4 or earlier and persisting through 13.2.0
During the study of the problem How to solve this integration? I have discovered a strange behaviour of some integrals.
I would ...
14
votes
0
answers
334
views
Is there a way to teach integrate new solutions?
I have an integral which I can solve, but integrate cannot:
...
- 181
12
votes
0
answers
227
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 ...
- 1,552
11
votes
0
answers
270
views
MacDonald formula for Modified Bessel Functions
How can I make Mathematica understand these two integrals?
$$\int_0^{\infty} e^{-x \cosh{\xi}} d\xi = K_0(x)$$
$$\int_0^{\infty} e^{-\frac{1}{2} \Big( \frac{x y}{u} + u \frac{x^2+y^2}{x y} \Big) } K_{...
- 659
10
votes
0
answers
317
views
Possible Symbolic Integration Bug
Bug introduced between 5 and 8 and persisting through 12.0.
I think I may have found a bug, and want to verify is this reproducible in other versions and platforms and not a mistake of mine.
I ...
- 111
9
votes
0
answers
206
views
The most bizarre kernel crash ever. Kernel crash in Integrate under simple different setups. Unable to find cause
Reported to WRI. CASE:4330461
V 12 on windows 10 64 bit. Note: This problem do not show up in V 11.3. Only in V12.
For the last 2 hrs, I've been trying to zoom into why V12 kernel crash when ...
- 135k
9
votes
0
answers
271
views
Incorrect evaluation for Thue-Morse signed harmonic series
I would like to evaluate
$$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 ...
- 191
8
votes
0
answers
150
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 ...
- 39.5k
8
votes
0
answers
224
views
Two integrals that should not be equal
Bug introduced in 12.0 or earlier and persisting through 13.2
CASE:4539809
I think there is a bug here:
...
user72309
8
votes
0
answers
176
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:
...
- 2,764
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 ...
- 311
7
votes
0
answers
126
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:
...
- 47.2k
7
votes
0
answers
400
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.:
...
- 171
7
votes
0
answers
190
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, ...
- 171
6
votes
0
answers
97
views
Does IntegrateChangeVariables work with improper integrals?
Trying a new command of version 13.1 IntegrateChangeVariables, I obtain
...
- 23.2k
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-...
- 3,167
6
votes
0
answers
216
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)
...
- 171
6
votes
0
answers
264
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.
...
- 6,418
5
votes
2
answers
158
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.
...
- 1,616
5
votes
1
answer
199
views
What is the volume of the intersection of four cylinders of equally radius equally spaced?
I saw that question at the certain forum and answered it with help of
Mathematica 13.1 in such a way. The angles between the unit vectors
...
- 23.2k
5
votes
0
answers
131
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))$$
?
I need more suitable and better code ...
- 183
5
votes
0
answers
229
views
Incorrect result by DSolve
For real $x$ consider the trivial equation
$$|y'(x)|=-|x|.$$
Since the left side is always positive and the right always negative, there is no solution.
Let's try
...
- 1,005
5
votes
0
answers
320
views
Help implementing Magnus Expansion
The Magnus expansion is a tool to approximate solutions to first-order linear differential equations (the Wikipedia page is quite instructive and concise) - it's particularly useful because all orders ...
- 831
5
votes
0
answers
228
views
Integrating a product of three Spherical Harmonics
The following command is returned unevaluated. The answer is well known to be related to Wigner's 3j Symbol which is also a defined function in Mathematica.
...
- 1,616
5
votes
0
answers
542
views
Convergence of approximate solutions to obstacle problem for the heat equation
Consider the problem
$$(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{...
user51360
5
votes
0
answers
85
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 $...
- 14.8k
5
votes
0
answers
231
views
Total variation integration of a discontinuous function
This question derives from this one, about mathematics and Maple.
Consider the following discontinuous function:
...
- 39.5k
5
votes
0
answers
351
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 $...
- 173
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
...
- 419
5
votes
0
answers
204
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}^...
- 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 ...
- 151
5
votes
0
answers
228
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):
...
- 1,471
4
votes
0
answers
149
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 ...
- 2,671
4
votes
0
answers
98
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:
...
- 26.1k
4
votes
0
answers
104
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 ...
- 81
4
votes
0
answers
73
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 ...
- 1,921
4
votes
0
answers
230
views
Finding Coefficients in a Perturbation Problem related to Chaotic Dynamics
I am trying to reproduce the results from this paper: Chaotic dynamics of a suspended string
in a gravitational background with magnetic field
I am currently stuck with finding the coefficients.
First ...
- 899
4
votes
0
answers
157
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
...
- 51
4
votes
0
answers
96
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 ...
- 153
4
votes
0
answers
150
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 $z$ ...
- 389
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 ...
- 49
4
votes
0
answers
159
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,...
- 1,670
4
votes
0
answers
94
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 $\...
- 14.1k
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
...
- 1,460
4
votes
0
answers
124
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}]]
...
- 11.9k
4
votes
1
answer
400
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 (...
- 2,315
4
votes
0
answers
179
views
Integral Representation of DiracDelta
I would like to get dirac delta from integral definition, of course if I ask
Integrate[Exp[I x t],{x,-Infinity,Infinity}]
I get that the integral does not ...
- 909
4
votes
0
answers
225
views
Symbolic second variation (quadratic form matrix)
Mathematica has a calculus of variations package that can compute the first variational derivative symbolically, rather nicely.
Does anyone know if there is a way to compute the quadratic form matrix ...
- 235
4
votes
0
answers
581
views
What kind of algorithm does Mathematica use to find limits?
Is there any information available about the implementation of the Limit command? In particular, I am interested in how Mathematica computes limits of real ...
- 161
4
votes
0
answers
107
views
Possible bug in Integrate
The integral I'm considering is:
$\int_0^{2\pi}\frac{e^{\pm t\theta}}{4\sinh^2\left(\frac{s+ i\theta}{2}\right)} = 2\pi e^{\mp s}\Theta(\pm s)$
for $s\neq 0$ (otherwise there some other subtleties). ...
- 141