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.
4,869
questions
76
votes
5
answers
26k
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Multivariable Taylor expansion does not work as expected
The basic multivariable Taylor expansion formula around a point is as follows:
$$
f(\mathbf r + \mathbf a) = f(\mathbf r) + (\mathbf a \cdot \nabla )f(\mathbf r) + \frac{1}{2!}(\mathbf a \cdot \nabla)...
- 16.9k
58
votes
6
answers
6k
views
Generating evenly spaced points on a curve
In the KnotData package a simple command such as
points = Table[KnotData[{3, 1}, "SpaceCurve"][t], {t, 0, 2 Pi, 0.1}];
will ...
- 701
53
votes
1
answer
13k
views
How to visualize Riemann surfaces?
In WolframAlpha we can easily visualize Riemann surfaces of arbitrary functions,
can we plot the Riemann surface of an arbitrary function using Mathematica and ...
- 27.4k
45
votes
2
answers
23k
views
How to calculate contour integrals with Mathematica?
How to calculate the integral of $\frac{1}{\sqrt{4 z^2 + 4 z + 3}}$ over the unit circle counterclockwise for each branch of the integrand?
- 23.2k
42
votes
2
answers
1k
views
Why does Mathematica report that $\int_1^\infty\frac{\sin(\sqrt{x})}{\sqrt{x}}dx$ = $2\cos(1)$?
Bug introduced in 7.0 or earlier and fixed in 11.0
Mathematica 10 gives the following very odd result,
Integrate[Sin[Sqrt[x]]/Sqrt[x], {x, 1, ∞}]
(* 2 Cos[1] *)
...
- 1,147
40
votes
4
answers
2k
views
Some indefinite integrals evaluate in 11.2 but not in 11.3 - what can be done?
Bug introduced in 11.3 and fixed in 12.0.0
Reported to Wolfram: [CASE:4032137]
These integrals evaluate in version 11.2 on windows but
when I tried them under version 11.3 they returned unevaluated.,...
- 135k
40
votes
2
answers
2k
views
Bug in mathematica analytic integration?
Bug introduced in 9.0 or earlier and fixed in 13.0 or earlier
I found Mathematica provides me a wrong answer for a relatively simple analytically solvable integral:
...
- 503
39
votes
1
answer
998
views
How to augment the realm of functions Mathematica thinks it knows how to integrate symbolically
My question involves extending the functionality of Integrate over specific integrals in the most generic manner.
Specifically, is it possible to "hack into" ...
- 22.5k
38
votes
6
answers
2k
views
Finding length of intersection of two surfaces
I would like to know how we find the length of the intersection of two surfaces. For instance, in the following example,a surface intersects with a plane:
How do we find the length of intersection ...
- 483
36
votes
1
answer
2k
views
Mathematica 9 cannot solve this Integral. Mathematica 8 could. Is this a bug?
I was trying to (re)calculate a problem of an older Wolfram blog post (Problem 11457, by M. L. Glasser) with Mathematica 9.0.0.0 (on OS X 10.8.2).
...
35
votes
4
answers
10k
views
Usage of Assuming for Integration
For some reason, when I enter the following integration in Mathematica
Assuming[{k ∈ Integers}, Integrate[ Exp[ I k t], {t, -π, π}]]
the result turns out to be 0....
- 523
34
votes
2
answers
1k
views
Symbolic derivatives are being calculated numerically
Update: (1) By V11, not sure of the exact version, the derivative IntegerPart' has been given a symbolic definition. (2) The numeric derivative computed has changed ...
- 115k
33
votes
4
answers
26k
views
Finding Limits in several variables
Is there a way to find a limit of a multivariable function, like
$$\lim_{(x,y)\to (0,0)} f(x,y)$$
with Mathematica?
When $f$ is continuous, we can use
$$\lim_{(x,y)\to (0,0)} f(x,y)=\lim_{(x,0)\...
32
votes
4
answers
26k
views
Finding unit tangent, normal, and binormal vectors for a given r(t)
For my Calc III class, I need to find $T(t), N(t)$, and $B(t)$ for $t=1, 2$, and $-1$, given $r(t)=\{t,t^2,t^3\}$.
I've got Mathematica, but I've never used it before and I'm not sure how to coerce ...
- 423
32
votes
5
answers
14k
views
How can I differentiate numerically?
Mathematica has two ways to integrate: Integrate and NIntegrate.
But what about D? ...
- 1,599
32
votes
3
answers
6k
views
How to deal with complicated Gaussian integrals in Mathematica?
As we know, for most Gaussian integrals, we can get the analytical
result. Now I have many Gaussian integrals to treat, which have the
following general form,
...
- 321
31
votes
4
answers
7k
views
How to find the period of an arbitrary mathematical function?
Is there a function to find the period of an arbitrary (possibly complex) function in Mathematica?
- 3,473
31
votes
4
answers
1k
views
Is it possible to write a MMA version of LineInt like Maple?
When I asked this question I found that the available answers relied heavily on the official RegionConvert, but which is very weak at that time. Now, We have a more ...
- 26.1k
30
votes
4
answers
13k
views
Implementing discrete and continuous Hilbert transforms
What is an efficient and accurate Mathematica implementation of the Hilbert transform, for both continuous and especially discretely sampled functions?
This transform relates phase and amplitude in ...
- 269k
29
votes
2
answers
6k
views
How does Mathematica understand branchcuts of the complex logarithm?
Say I have the function $f(x) = x \tanh(\pi x) \log (x^2 +a^2)$ where $a$ is some positive real number. Then it seems to be me that Mathematica when given such a ...
- 1,151
29
votes
3
answers
5k
views
Can we teach Mathematica about functional differentiation?
The key relation for functional differentiation is
$$\frac{\delta}{\delta f(y)}f(x)=\delta(x-y),
$$
where $\delta(x-y)$ is the Dirac delta function, and the usual properties of differentiation (e.g. ...
- 1,197
29
votes
3
answers
28k
views
Quick Hessian matrix and gradient calculation?
I am absolutely new to Mathematica and I actually want to try implementing a little optimization method .
Long story short assuming I have a predefined two-variable function f(x,y) I want to ...
- 399
29
votes
1
answer
517
views
many indefinite integrals do not evaluate in 10.1, looking for the cause
Bug introduced in 10.1 and fixed in 10.2
Many integrals no longer evaluate in V 10.1 when they did in 10.0.2
Here are some 23 integrals as an example, that all produced results in V 10.0.2, but now ...
- 135k
28
votes
1
answer
2k
views
Undocumented use of Integrate: Integrating over regions
I have come across a few questions asking about integrating over regions. And while the answers are impressive there should be a better more consistent way. So my question is, are there ways, ...
- 32.9k
28
votes
3
answers
2k
views
Symbolic integration error
fixed in 10.1 (windows)
I'm running Mathematica 10.0.0 and encountered a disturbing error in the symbolic integration of a rather simple function
...
- 913
27
votes
4
answers
3k
views
Is it possible to get this 'nicer' solution for an integral from Mathematica?
On a recent CAS-enabled exam question a few weeks ago I was required to evaluate the following integral:
$$
\int_0^5\left(\sqrt[3]{125-x^3}\right)^2\,dx
$$
In Mathematica, using the ...
- 607
27
votes
4
answers
1k
views
An apparently "simple" limit?
Let $c$ and $h$ be real values.
I was using Mathematica to compute the limit $(h \rightarrow \infty)$ of the following expression:
$$
\frac{\left(h^2 +c^2 h^2 + \sqrt{4 h^2+\left(h^2+c^2 h^2\right)^2}...
- 11.9k
27
votes
3
answers
7k
views
Visualizing a Complex Vector Field near Poles
I've been playing around with a visualization technique for complex functions where one views the function $f: \mathbb{C} \rightarrow \mathbb{C}$ as the vector field $f: \mathbb{R^2} \rightarrow \...
- 655
27
votes
2
answers
486
views
Bug in ArcLength?
fixed in 10.1 (windows)
With Mathematica 10.0.2:
ArcLength[Line[{{0, 0}, {1, 0}, {2, 0}}]]
ArcLength[Line[{{0}, {1}, {2}}]]
(* 2 *)
(* 2 *)
However,
...
- 1,189
27
votes
1
answer
518
views
Are greek symbols causing different evaluation?
I've updated today to Mathematica 9.0.1.0 from version 8 and found something that absolutely confuses me.
Let us define a piecewise function:
...
- 371
25
votes
3
answers
583
views
Backslide of Limit
Backslide introduced in 9.0, and persisting through 12.0.
A friend of mine showed me this example:
Limit[Sum[Sin[Pi*k/n]/(n + 1/k), {k, 1, n}], n -> Infinity]
...
- 62.8k
24
votes
5
answers
934
views
Heuristics, tricks, and hacks in symbolic math
Mathematica sometimes fails to compute symbolic solutions when posed in the direct or obvious code, but succeeds when the same fundamental problem is posed in a slightly different way, or when ...
- 39.5k
23
votes
5
answers
6k
views
Why this real integral yields imaginary results?
This integral yields -1-4Iπ/3 in Mathematica:
Integrate[(y - y^2 + x - x^2 + 2*x*y)/(1 - x - y), {x,0,1}, {y, 0, 1}]
Since ...
- 685
23
votes
3
answers
2k
views
How to efficiently find moments of a multinormal distribution?
Update: Starting from V10.0 the build-in Moment is fast enough for practical use.
I have a multinormal distribution with covariance matrix $\sigma$ and zero mean. ...
- 43.5k
23
votes
1
answer
1k
views
Implement fractional Laplacian
What is a way to implement the Fractional Laplacian with Mathematica?
How can we apply such implementation to numerically solve the problem
$$(-\Delta)^su = 1 \text{ in } B_1(0), \\
u = 0 \text{ in ...
user52420
23
votes
2
answers
4k
views
Differentiating functions of vectors/matrices?
I'm dealing with derivatives of scalar functions of matrices and wondering if Mathematica can help me here.
The standard approach of expanding it in terms of components is cumbersome. As an ...
- 8,059
23
votes
2
answers
1k
views
How can the {x,y,z} points that fall on the outer boundary of a set of values be selected and smoothly surfaced?
For a given set of x,y,z values, that may, or may not form a uniform shape, how can the center of the data cloud be found, and the surface points be located and a solid smooth surface created from ...
- 4,478
23
votes
1
answer
791
views
Teaching Mathematica more about DiracDelta and KroneckerDelta
As the documentation and some experimentation indicates, Mathematica contains little information about representations of the DiracDelta and ...
- 832
23
votes
2
answers
910
views
Integrate returns imaginary answer for smooth, real function
Bug introduced in 7 or earlier and persisting through 12.0.0.0 or later
I'm trying to evaluate the integral:
$$\int_0^{\infty} \frac{1}{4 b \sqrt{\pi} r} e^{-(b-r)^2}(e^{4 b r} - 1) \mathrm{d}r$$
...
- 659
22
votes
2
answers
2k
views
Incorrect results for elementary integrals when using Integrate
Bug introduced in 8.0 or earlier and persisting through 13.2 or later
There is a rather simple integral ($K_0$ is the 0-th order MacDonald function)
$$\int_0^\infty e^{-x \cosh\xi}\, d\xi = K_0(x)$$
...
- 1,424
22
votes
2
answers
854
views
Derivative of a pure function with SlotSequence
I can live with this but I can't figure out why the following is 0:
Derivative[1][f[##] &][x]
0
From documentation for ...
Kuba♦
- 136k
22
votes
2
answers
2k
views
Why does Mathematica give the wrong answer when integrating?
Bug introduced in 8.0 or earlier and fixed in 9.0.0
I integrate
Integrate[Exp[I Cos[b - c]] Cos[b], {b, 0, 2 Pi}]
Mathematica gives:
...
- 983
21
votes
5
answers
15k
views
How to do implicit differentiation in Mathematica?
I have an equation which is
K^(1/2)*L^(1/2)-(K+L) = 24
but I don't know how to do implicit differentiation to find dK/dL because I only know how to do the normal ...
- 211
21
votes
5
answers
15k
views
How to find the domain and range of a function with Mathematica?
I'm studying calculus and in some exercises I am asked to find the domain and range of a function. Does Mathematica have already a built-in function for this?
I can imagine some ways of doing so, ...
- 5,121
21
votes
3
answers
10k
views
How can I implement the method of Lagrange multipliers to find constrained extrema?
I want to form the function $h=f-\lambda_{1}g_{1}-\lambda_{2}g_{2}$ where $f$ is the function to optimize subject to the constraints $g_{1}=0$ and $g_{2}=0$ so that I can solve the first partial ...
- 517
21
votes
4
answers
515
views
Negative probability?
Bug introduced in 9.0.1 and fixed in 10.0.2
I am trying to get the sum of the squares of seven random variables, all uniformly distributed. This is what I tried.
...
- 2,618
21
votes
1
answer
6k
views
How does Mathematica integrate?
Basically, this question can be considered to be an extenstion to my other question.
What I wanted to do was this integral as homework (it is indefinite BTW so no approximations using Simpson's Rule ...
- 1,089
21
votes
1
answer
1k
views
Solver for COVID-19 epidemic model with the Caputo fractional derivatives
As it is known in biological system with memory it would be suitable to use fractional derivatives to describe evolution of the system.
In a current version of Mathematica 12.1 there is no special ...
- 41.1k
21
votes
1
answer
1k
views
Dual complex integral over implicit path using contour plot
Context
I am interested in doing double contour integral over paths which are defined implicitly.
For the sake of debugging, let's assume its
$$\oint_{\cal C}\oint_{\cal C} \frac{1}{u\, x} d u d x$$
...
- 22.5k
20
votes
2
answers
7k
views
How to make traditional output for derivatives
Hello again after some pause. i have a problem how to present partial derivatives in traditional form, not as Mathematica gave it to me in its notation. So I want to present this
...
- 1,099