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 13.0.0 During the study of the problem How to solve this integration? I have discovered a strange behaviour of some integrals. I would ...
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14 votes
0 answers
326 views

Is there a way to teach integrate new solutions?

I have an integral which I can solve, but integrate cannot: ...
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  • 181
11 votes
0 answers
254 views

Series with ArcTan gives wrong symbolic answer in Wolfram Language

Bug introduced after 9 and persisting through 13.0 Recently, I have found a very bad problem with Wolfram Language. It gives the wrong answer for a quite simple expression! When calculating ...
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  • 211
10 votes
0 answers
314 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 ...
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  • 111
9 votes
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235 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_{...
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8 votes
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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 ...
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8 votes
0 answers
172 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: ...
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8 votes
0 answers
119 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 ...
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8 votes
0 answers
558 views

Expansion for Modified Bessel Function Around Infinity

I'm somewhat new to Mathematica, and I don't understand why I'm getting inconsistent series expansions for the modified Bessel Function of first kind near $x=\infty$. First problem: I get different ...
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  • 81
8 votes
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260 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 ...
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7 votes
0 answers
193 views

Two integrals that should not be equal

Bug introduced in 12.0 or earlier and persisting through 12.1.0 CASE:4539809 I think there is a bug here: ...
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7 votes
0 answers
390 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.: ...
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6 votes
0 answers
112 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: ...
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6 votes
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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-...
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6 votes
1 answer
237 views

How can I inform Mathematica of an identity concerning Bessel functions?

I am doing some analytical work that includes the integral of $e^{i(n\, t - x \sin t)}$. I know the result of this integral is a Bessel function. $$J_n(x)=\frac1{2\pi}\int_{-\pi}^\pi e^{i(x\sin\tau-n\...
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6 votes
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209 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) ...
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  • 171
6 votes
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159 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, ...
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6 votes
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254 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. ...
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5 votes
0 answers
124 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 ...
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  • 183
5 votes
0 answers
200 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 ...
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  • 1,005
5 votes
0 answers
238 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 ...
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5 votes
0 answers
179 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. ...
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5 votes
0 answers
515 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{...
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5 votes
0 answers
76 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 $...
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5 votes
0 answers
200 views

Total variation integration of a discontinuous function

This question derives from this one, about mathematics and Maple. Consider the following discontinuous function: ...
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5 votes
0 answers
329 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 $...
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5 votes
0 answers
199 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 ...
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5 votes
0 answers
191 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}^...
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  • 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 ...
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5 votes
0 answers
154 views

Derivative of generating function (Example from documentation)

Bug introduced in 9.0 or earlier and persisting through 11.0.1 or later In the documentation for GeneratingFunction, the following example is given under Examples -...
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  • 1,325
5 votes
0 answers
221 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): ...
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  • 1,461
4 votes
0 answers
53 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 ...
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  • 1,365
4 votes
0 answers
98 views

Possible Mathematica bug

I believe the following two integrals should have the same value, since the Boole[] expression always evaluates to 1 in the ...
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4 votes
0 answers
222 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 ...
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  • 181
4 votes
0 answers
150 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 ...
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  • 51
4 votes
0 answers
113 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 $z$ ...
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  • 379
4 votes
0 answers
130 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 ...
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4 votes
0 answers
153 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,...
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  • 1,650
4 votes
0 answers
91 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 $\...
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  • 13.6k
4 votes
0 answers
126 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 ...
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  • 1,252
4 votes
0 answers
115 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}]] ...
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  • 11.5k
4 votes
1 answer
352 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 (...
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4 votes
0 answers
178 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 ...
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  • 899
4 votes
0 answers
199 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 ...
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4 votes
0 answers
496 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 ...
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4 votes
0 answers
106 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). ...
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4 votes
0 answers
94 views

Huge difference after changing a fraction to decimal

I have a limit to calculate. With[{p = 1/2, q = 0.1}, Limit[a^q Integrate[Sin[x]/x^p, {x, a, Infinity}], a -> Infinity]] gives correct result ...
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  • 390
4 votes
0 answers
165 views

Inconsistent results for symbolic trigonometric integral

I am trying to evaluate (on Mathematica v.9) the integral \begin{equation} \int_0^x [\sin (x) \sin (2 y)-\sin (2 x) \sin (y)]^t \,\mathrm{d}y, \end{equation} where $t$ is an even, positive integer. ...
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4 votes
0 answers
162 views

Exploring formal limit definition

I found a few demonstrations on the Wolfram Demonstration Project site that help users to explore the formal definition of a limit. That is, $$\lim_{x\to a}f(x)=L$$ if and only if for every $\epsilon&...
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  • 14.5k
4 votes
0 answers
709 views

Function for the Second Derivative Test

I wrote the following function. It is based on Mathematica for Rogawski's Calculus, 2nd Ed, 2007, Based on Mathematica 7. See: http://users.rowan.edu/~hassen/Math_Rogawski_Calc.htm, Chapter 14. I made ...
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