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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10
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2answers
2k views

Is it possible to calculate a Lebesgue integral in Mathematica?

As the title says, I wonder if it is possible to calculate a Lebesgue integral in Mathematica, especially when the domain of integration is $\mathbb{R}^N$, or in other words multivatiate Lebesgue ...
1
vote
1answer
54 views

Drawing a tangent plane to the torus

I'm trying to draw a torus and a plane tangent to it. Here is my code so far: ...
0
votes
0answers
36 views

Numerical derivative step size

I am trying to perform a numerical derivative of some function using ND. I want to know if there is a way to set the step size of differentiation to the specific ...
0
votes
1answer
25 views

Multivariate integration of a compicated expression

I have an expression in multiple variables that is something like ...
5
votes
2answers
96 views

Inflection point and curvature

Through a coordinates list and then interpolating them I got a $f[x]$. ...
2
votes
0answers
84 views

Bug in the “PrincipalValue” method of NIntegrate?

Update As pointed by Emilio Pisanty, one should pay attention to EvaluationMonitor to monitor principal value integration. I update the codes, and find some more ...
-1
votes
1answer
197 views

Finding the maximum of a gradient vector

Im trying to find the maximum of my gradient vector G[x,y], I've tried several options including FindMaximumValue, FindMaximum etc. but i couldn't find it. The full function is shown below, any help ...
1
vote
1answer
38 views

Do double vertical bars in ConditionalExpression mean 'or'? [on hold]

I have a very simple interpretation question. I think that double vertical bars in a ConditionalExpression given by Mathematica as the result of an integration must mean 'or', but I am hoping to ...
0
votes
0answers
88 views

How to compute this integration? [closed]

I would like to compute the following integral: Where: a and k are reals, with the assumptions: 0 < a, and 0 > k > -4. y and p are reals too. I tried to write it as ...
-1
votes
1answer
29 views

Proof The union of an infinitely countable set and an element is countable [on hold]

If $A$ is countable infinite set and b $\notin$ $A$, then $A$ $\cup$ {b} is also countable. I thought of starting of saying lets assume $A$ $\cup$ {b} is countable and show that $A$ is a subset of it ...
1
vote
3answers
78 views

Evaluating $\int_0^\infty dx /{(x^2-a^2)^2+b^4} $ returns a negative number

I am trying to evaluate $\int_0^\infty dx /((x^2-a^2)^2+b^4)$. And I seem to have an error that I cannot detect. Using ...
0
votes
3answers
59 views

Derivative of a 2D Interpolating function

I cannot find a way to obtain the partial derivatives of a 2D Interpolating Function obtained with ListInterpolation. I didn't find anything useful on the web. Any ...
0
votes
1answer
252 views

Is it necessary to introduce ImplicitRegion and ParametricRegion in version 10?

In version 10.0, Mathematica introduced a new function about region plotting: ImplicitRegion. But I'm wondered that haven't this function been already existed in the older version? That is, ...
1
vote
1answer
62 views

Einstein summation and traditional form of partial derivative of implicit function

I am quite new in Mathematica (1 week new) and after many trials I did not manage to make my partial derivatives to look traditional way. There are few topics such as How to make traditional output ...
10
votes
2answers
2k 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 ...
0
votes
1answer
384 views

Integration of linked PDFs over probability simplex

I'm trying to integrate the following expression: f[a1,p1+p2+p3+p4]*f[a2,p2+p3+p4]*f[a3,p3+p4]*f[a4,p4] Where ...
3
votes
0answers
30 views

Differentiation and series expansion of dot product - inconsistent results

If I differentiate a dot product, I get the result I expect D[a.b[x], x] (* a.b'[x] *) However, a series expansion of the same expression gives a very different ...
0
votes
0answers
31 views

Coordinate transformation for gradients expressions

Let's suppose I have an expression in cartesian coordinates, involving gradients, such as: F[x,y]:=D[f[x,y],x]/D[f[x,y],y]+D[f[x,y],y,y] f[x,y] Now, I would like ...
4
votes
0answers
179 views

Version 8.0 integrates but Version 9.0.1 doesn't

I am trying to run the following integral in version 9.0, but it fails: ...
0
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0answers
23 views

Trouble evaluating an integral for certain parameters [closed]

I'm trying to evaluate the integral $$ \int_0^{2\pi}\pi(2(\Delta x-d+r'e^{ix})I_o(rd) - \sqrt{2b(b-\sqrt{b^2-1})-1}(e^{-b/2(r^2+4r'^2+8(\Delta x - d)r'cos(x))-2i\sqrt{b^2-1}drsin(x)}dx $$ using <...
1
vote
1answer
100 views

Computing integrals in Mathematica

I am trying to actually compute $\int \sqrt{\cosh{y}-\cos{x}}e^{inx} dx$ in mathematica. As an example i tried computing $\int \sqrt{1-\cos{x}}e^{inx} dx$ and got a result. But when i try $\int \...
0
votes
1answer
48 views

Can this expression be calculated symbolically?

I want to do definite integration of the following expression ...
1
vote
1answer
80 views

integrating with multiple indicator functions depending on each other

I need to compute the following integral with three indicator functions, $I(x)$, $I_1(x_1)$ and $I_2(x_2)$, $$ \int_0^1\int_0^1I(x)\left( \ I_1(x_1)\,x_{1} + I_2(x_2)\,x_{2}-1 \right) dx_1 dx_2 $$ ...
1
vote
1answer
67 views

Evaluating Bessel-Schläfli-type Integral with Infinite contour

I am attempting to evaluate the integral \begin{equation} \int_{\infty-i\pi}^{\infty+i\pi} dt \frac{e^{-\frac{l}{n}t}}{[(X'-X)^2+r^2+s^2- 2rs \cosh(t)]}, \end{equation} the only variable being $t$, ...
2
votes
2answers
118 views

Symbolic linear algebra gradients/matrix calculus

Can Mathematica generate symbolic expressions for gradients? For example, if $x_1$ and $x_2$ are two points, could I get Mathematica to generate expressions similar to the following? $\frac{\partial ...
0
votes
0answers
50 views

Doing vector calculus with element-wise defined formulas and arbitrary-sized vectors [duplicate]

Currently I am trying to make Mathematica handle vector expressions defined element-wise. For example, But I can find no way to do this. Every way I tried requires a fixed-sized vector and returns ...
3
votes
2answers
134 views

Symbolic Integrate fails after almost 1 hr calculation

Mathematica 9. I want this Integrate[( Cos[t ω]^2 Cos[α + t ω])/(-1 + A Cos[α + t ω])^2, {t, 0, (2 π)/ω}] Can Mathematica handle with it? I tied different ...
2
votes
3answers
129 views

NDSolve and differentiation of Abs

I have found several questions about the derivative of Abs and how it is not defined in the complex plane. What I have not found yet is a precise and simple ...
2
votes
1answer
130 views

Residue failed to reproduce this residue in a paper

This problem is a more clearly state of my original problem which is off topic, so I reconsider my problem carefully based on previous discussions, and gives the following valid code to reproduce my ...
1
vote
3answers
129 views

How to calculate the double integral of a double integral?

May I ask is there any wise way to calculate the following integration? $\int_{-1}^{1}\int_{-1}^{1}\int_{-1}^{1}\int_{-1}^{1}\frac{\left( 0.25 \left(-1+\eta _1\right) \eta _1 \left(-1+2 \eta _2\right)...
0
votes
1answer
56 views

Integral of $(\sin x)^3(\cos x)^3$ [closed]

For the integral of (sin x)^3(cos x)^3 I got 1/6(cos x)^6 - 1/4(cos x)^4 + C. This seemed to work but my book had 1/4(sin x)^4 - 1/6(sin x)^6 + C. This is similar to what I got but instead of cosine ...
2
votes
0answers
61 views

Why does Expectation function compute, while explicit integration does not?

The code: Expectation[1 + 1/y, {y \[Distributed] ExponentialDistribution[1]}] outputs 1 - EulerGamma While, ...
1
vote
2answers
134 views

How to convolve the unit box function and the modified Bessel function of the second kind in 2D?

In 1D the convolution of the unit box function and the modified Bessel function of the second kind $K_0(x)$ works very well. ...
1
vote
2answers
217 views

How to find a graphical solution for extreme values using Lagrange multipliers

I need to find the maxima and minima of a function. I managed this analytically with no problem, but now I need to show in a graphic the intersection of the functions. My function is: ...
11
votes
6answers
880 views

Integrate gives wrong results

Integrate[a/(Sin[t]^2 + a^2), {t, 0, 2 Pi}] $$\int_0^{2 \pi } \frac{a}{a^2+\sin ^2(t)} \, dt$$ gives $0$ This cannot be true. What is going on? If I insert a ...
7
votes
3answers
635 views

Any way I can evaluate this integral?

So, Mathematica can't evaluate it. Any workaround? $$\int_0^\pi \frac{\mathrm{d}\theta}{\sin\left(\frac{\theta}{2}\right)^\beta + 1}$$ The code is: ...
1
vote
1answer
103 views

Evaluate an integral with parameters

I already asked this on stack overflow but was redirected here. I am trying to solve the (highly non-linear) following system of equations : with and $g_0$, $\Lambda$ are positive constants ...
1
vote
1answer
66 views

Problem in limit of exponential function

I want to obtain Exp[-α/(1 - α)] /. {α -> 1} The output is Indeterminate. But I calculate without Mathematica and I ...
1
vote
1answer
64 views

Need help on simplifying the integral of a square root of a polynomial

I have the following expression a = 4 Cos[af / 2]^2; b = 1 - 5 Cos[af / 2]^2; c = Cos[af / 2]^2; s = Sqrt[a*x^4 + b*x^2 + c] and I need the integration of $s$ in ...
2
votes
2answers
97 views

Analytic differential of an inverse transcendental function

I have the following function: $$ y(x)=\frac{Ax}{Bx^{1/4}\sinh{(\frac{Bx^{1/4}}{2})}}\left(\cosh{(\frac{Bx^{1/4}}{2})}-1 \right), $$ where $A=112.941$ and $B=18.6588$. I want to find the differential ...
3
votes
1answer
99 views

Using Vector Operations In Mathematica

I would like to calculate $$ \mathbf T = (\mathbf B \bullet \nabla) \mathbf B$$ Where nabla (the upside down triangle) is the grad operator $(\partial/\partial x,\partial/\partial y,\partial/\...
0
votes
1answer
130 views

Integrating SquareWave - don't understand result

Integrate[SquareWave[t], t] // InputForm Gives as output Piecewise[{{-t, Inequality[1/2, LessEqual, Mod[t, 1], Less, 1]}}, t] ...
8
votes
2answers
155 views

Integrate and NIntegrate disagreeing over branch cut of Sqrt?

Bug introduced after v8.0.4 and before v9.0.1, and persisting through v10.4.1. In Mathematica 10.2 I'm trying to integrate this piecewise continuous function, but ...
3
votes
2answers
138 views

Integrate on a contour in the complex plane

I want to integrate the function f[z]= z + Conjugate[z] over a circle of radius 2 centered at the origin. For the sake of stating something that I have tried: <...
0
votes
1answer
83 views

Definite and indefinite integral give different results

I'm trying to find an expression that, given $x$, $y$, and $R$, gives the indefinite 2D integral of a function at the circle centered on $(x_0, y_0)$ with radius $R$. With $g$ as defined below, I'd ...
0
votes
2answers
92 views
3
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2answers
87 views

Application of # for the $n^{th}$ derivative

Let's say that I have x^2+x Is there a way to map $x$ to the first derivative of a function and $x^2$ to the second derivative of the same function? According to ...
10
votes
1answer
103 views

Slot number cannot be filled

I got a strange message with the derivative of a two-argument function defined by a definite integral. For example if "f" is defined as: ...