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.

418 questions with no upvoted or accepted answers
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16
votes
0answers
339 views

Strange behaviour of integrals with Cos, Sin, and Exp

Bug introduced in 8.0.4 or earlier and persisting through 11.0.1 During the study of the problem How to solve this integration? I have discovered a strange behaviour of some integrals. I would ...
13
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0answers
305 views

Is there a way to teach integrate new solutions?

I have an integral which I can solve, but integrate cannot: ...
10
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0answers
170 views

Workaround for DSolve in V 12 when it gives undefined as solution to 1D heat PDE?

Reported to WRI CASE: 4278450 Comparing the following, all done from clean kernel The strange thing is that V 12 can solve this same PDE without the assumptions ...
10
votes
0answers
281 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 ...
9
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0answers
114 views

Finding simplifying substitutions for an integral involving limits and integrand

[The following is based on a William Lowell Putnam Mathematical Competition problem.] Consider the definite integral: $I = \int\limits_2^4 \frac{\sqrt{\log (9-x)}}{\sqrt{\log (9-x)}+\sqrt{\log (x+3)}...
8
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0answers
407 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 ...
8
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0answers
207 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_{...
8
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0answers
244 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 ...
7
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0answers
134 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 ...
7
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0answers
130 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: ...
7
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0answers
102 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 ...
6
votes
0answers
175 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) ...
6
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358 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.: ...
6
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0answers
141 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, ...
6
votes
0answers
242 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. ...
5
votes
0answers
128 views

Integrate under assumptions

I am trying to solve the following integral: $$\int_0^\infty u^{\sigma-1} \exp\left[-c u^\sigma\right] \mathrm{d}u$$ under the assumptions $\sigma \in (0,1)$ and $c>0$. I know the result is $$\...
5
votes
0answers
432 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{...
5
votes
0answers
68 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 $...
5
votes
0answers
188 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 ...
5
votes
0answers
165 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}^...
5
votes
0answers
143 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 -...
5
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0answers
195 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): ...
4
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0answers
60 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 ...
4
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0answers
71 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 $\...
4
votes
0answers
71 views

How to calculate specific area on surface of sphere?

I am trying to calculate the solid angle subtended by arbitrary-shaped loops on a sphere's surface. First, I parametrize circular loops by: $$\theta(t,k_{x0},r) = k_{x_0} + r \cos(t);$$ $$\phi(t,k_{...
4
votes
0answers
101 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}]] ...
4
votes
0answers
116 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. ...
4
votes
3answers
283 views

Evaluate a certain two-dimensional integral involving an inverse hyperbolic tangent

Evaluate the integral of ...
4
votes
1answer
276 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 (...
4
votes
0answers
125 views

Total variation integration of a discontinuous function

This question derives from this one, about mathematics and Maple. Consider the following discontinuous function: ...
4
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0answers
312 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 ...
4
votes
0answers
104 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). ...
4
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0answers
159 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 $...
4
votes
0answers
88 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 ...
4
votes
0answers
152 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&...
4
votes
1answer
258 views

Integral of DiracDelta giving an unusual answer

I have been getting a number of seemingly inconsistent solutions to integrals of Dirac delta functions in which the integrand evaluates to DiracDelta[0] at one of ...
4
votes
0answers
313 views

Strange behaviour of MMA in derivatives of some standard functions

There are some peculiar things to be discovered in derivatives of some standard functions in MMA: Strange behaviour Example 1: Abs We have ...
4
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0answers
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 ...
4
votes
0answers
241 views

Linearised Einstein equations

I need to compute the linearised Einstein Equations around a fixed metric $$g_{μν}=\text{Minkowski metric} + h_{μν}$$ which is not the flat metric. Does anyone know a Mathematica package or a ...
4
votes
0answers
549 views

Spherical harmonic derivative

Consider the following substitution Derivative[2, 0][S][th, ph] /. S -> Function[{th, ph}, SphericalHarmonicY[3, 0, th, ph]] which gives correct answer. While ...
3
votes
1answer
98 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\...
3
votes
0answers
63 views

Derivative of eigenvalues

I work with 4x4 Hermitian matrices (r). I want to calculate a derivative of a function f[t,r] (ff[t_,r_]=1/2*D[f[t,r],t]), where the function f depends on the absolute value of the eigenvalues of r. ...
3
votes
0answers
110 views

How to test whether an improper integral converges or diverges?

To verify the improper integral $\int_{1}^{\infty}x\cdot\left | \sin (x^4)\cdot\sin x \right |dx$ converges or diverges,I code ...
3
votes
0answers
113 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 ...
3
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0answers
87 views

Analytic Integration Bug of Hypergeometric function in Version 11.3?

I got this integral ...
3
votes
0answers
85 views

Is Mathematica doing this integral of a wave form approximation correctly?

I am trying to integrate a square wave approximation I found in this answer by ybeltukov. Nothing wrong with the answer but when I try to integrate it I run into trouble. I modified the code from ...
3
votes
0answers
151 views

How can I solve a nonlinear fractional order integro-differential equation?

How do I solve the nonlinear fractional order integro-differential equation shown above in Mathematica?
3
votes
0answers
56 views

Definite integals that can be expressed in terms of itself

I came across the integral $\int_0^\infty \frac{1}{\left(x^2+1\right) \left(x^{2015}+1\right)}\,dx$ in a Youtube video. It apparently appeared in the MIT Integration Bee of 2015. In the video, it was ...
3
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0answers
77 views

Extract explicit region (integration bounds) from ImplicitRegion

Using an ImplicitRegion in NIntegrate by far best performance is obtained by using ...
3
votes
0answers
261 views

Integrating the Associated Legendre Polynomials

I know the following identity: $\qquad \int_{-1}^1 P_l^m(t)^2dt=\frac{2(m+n)!}{(2n+1)(n-m)!}$ I would like to verify this result using Mathematica. This is what I entered: ...