Tagged Questions

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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4
votes
1answer
178 views

Why does Assuming for Integrate not work as expected?

I'm trying to perform the following integral with Mathematica 7: ...
2
votes
0answers
165 views

About calculating Integrals

I am trying to Integrate the following Integral (all of the variables are reals) $$\int Exp(-(\cos^{-1}\left[\frac{\text{n}_0 (\text{v}_0-\text{x}_0)+\text{n}_1 (\text{v}_1-\text{x}_1)+\text{n}_2 (u ...
20
votes
4answers
802 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 ...
3
votes
0answers
139 views

A triple sum related question

I'm trying to compute the triple sum Sum[1/(i! j! k! ), {i, 1, Infinity}, {j, i + 1, Infinity}, {k, j + 1, Infinity}] but Mathematica doesn't return any value. ...
1
vote
0answers
49 views

Interested in knowing what substitutions were made to perform an Integrate[]? [duplicate]

Possible Duplicate: How does Mathematica integrate? Mathematica can perform the following integral with apparent ease (returning very fast with the result) ...
2
votes
1answer
189 views

Computing derivatives of a moment generating function

Dear Mathematica users, I'm trying to compute higher order derivatives of a moment generating function and then evalutate them in 0 (in order to get some moment conditions for a GMM estimation). ...
0
votes
1answer
187 views
6
votes
1answer
834 views

How to expand a function into a power series with negative powers?

Is there any way to expand this expression a+b(1-Exp[-T/(b c)]/(z-Exp[-T/ (b c)]) (where a, ...
1
vote
1answer
318 views

How to find this convolution?

How can I use Mathematica to find the convolution $f*f$ for ...
4
votes
1answer
474 views

Order of integration changes output of indefinite multiple integral in Mathematica 7

I'm trying to integrate a form-factor used in the calculation of radiation between two rectangles in perpendicular planes. While the integral is usually done over fixed limits, I am trying to do the ...
3
votes
1answer
302 views

Turn list of edges into a polygon function

I have a list of coordinates that define the edges of a polygon and I would like to get a function defining the area Inside out if it (The polygon is convex and the points are in order) So that for ...
11
votes
6answers
4k views

Finding the centroid of the area between two curves

When I have an area bounded by curves, is there a built-in way to find the center of the area? Or do I have to plot it first and then use ComponentMeasurements on ...
2
votes
4answers
196 views

How to evaluate this indefinite integrate $\csc(4x)\sin(x)$

I tried to integrate the following integral using Integrate[Sin[x]Csc[4x],x] and I am getting a strange result. $$\frac{1}{8 \sqrt{2}}\left(-2 i ...
5
votes
4answers
1k views

How to find the sum all even numbers of this sequence?

I have a sequence $(u_{n})$ $$u_1= 1, \quad u_2 = 2, \quad u_3 = 3, \quad u_{n}= -u_{n-3} + 3u _{n-2} +2 u_{n-1}, \quad \forall n \geqslant 4.$$ I want to list the first $20$ terms of this sequence ...
6
votes
3answers
253 views

How to calculate this sum?

I want to find the sum $$S=f\left(\dfrac{1}{2012} \right) +f\left(\dfrac{2}{2012} \right) +\cdots + f\left(\dfrac{2011}{2012} \right), $$ where $$f(x) = \dfrac{4^x}{4^x + 2}.$$ I tried ...
1
vote
3answers
522 views

How do I evaluate several n-th derivatives of a function at one point?

I have a question where I have to compute a Table containing $f^{(n)}(0)$ for n = 1, ..., 5, where $f^{(n)}$ denotes the $n$th ...
6
votes
2answers
250 views

Force integration to be linear over sum?

Is there an obvious way to force Mathematica to separately integrate the terms in a sum? According to the docs, When part of a sum cannot be integrated explicitly, the whole sum will stay ...
1
vote
1answer
118 views

Simplifying an Integral over Two DiracDeltas

Integrate[DiracDelta[a + k] DiracDelta[-b + k], {k, -\[Infinity], \[Infinity]}] (*DiracDelta[a + b]*) This works fine. But ...
1
vote
1answer
129 views

Assumptions are not being applied to integral

I am trying to use assumptions to simplify an integral, but I can't seem to get the assumptions to apply themselves. Here is the code I am using: Note the expressions in the assumption can be found ...
12
votes
0answers
193 views

Why does Mathematica choose branches as it does in this situation?

Consider these integrals: ...
4
votes
1answer
626 views

Computation of parametric integral

I am trying to compute the integral Integrate[(g^(u^(g - 1)))/(1 + u^g), {u, 0, t}] but as an answer I get my input expression. There must be something wrong ...
2
votes
1answer
872 views

Lie-Bracket of two vector fields

I'm new to Mathematica (installed couple of hours ago) and I need to compute a few Lie brackets between two vector fields $f$ and $g$. $$ f\left(\mathbf{x}\right) = \left( \begin{array}{c} ...
12
votes
3answers
3k views

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 ...
0
votes
2answers
468 views

Why Can't Mathematica Integrate this?

I have the following problem from a textbook I am trying to integrate: So, following the directions in text, I am required to integrate each function. However, I cannot get Mathematica to integrate ...
10
votes
3answers
319 views

Typeset unbalanced brackets

In LaTeX, you can typeset expressions containing unbalanced brackets, by balancing them with an invisible delimiter \left. or ...
15
votes
1answer
433 views

Speeding up trigonometric integral

Context On a possible non trivial toric topology for the Universe (nothing less!). Problem I would like to carry out the following integral for $\ell=2,4\cdots 20$. $$\int _0^{\pi }\int _0^{2 \pi ...
2
votes
1answer
652 views

How to find all roots and maxima and minima and graph them

I need to find all the roots of this function. I know it is simple to find by hand, however, I wish to learn how to do it so I can apply it later. The problem is that it only gives me one root. ...
20
votes
3answers
1k 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. ...
8
votes
1answer
377 views

Integral with HeavisideTheta takes too long to evaluate

I tried to compute $$\int_{-1}^1 d x_1 \int_{-1}^1 d x_2 \int_{-1}^1 d y_1 \int_{-1}^1 d y_2 \theta(x_1 x_2 + y_1 y_2)\,$$ where $\theta$ is Heaviside's step function, by using ...
9
votes
3answers
2k views

How can I differentiate Numerically?

Mathematica has two ways to integrate: Integrate and NIntegrate. But what about D? ...
2
votes
0answers
68 views

How to expand terms within an integral? [duplicate]

Possible Duplicate: Why aren’t these additions of integrals and summations equal? I am trying to expand all the terms in a sum in order to apply assignment rules later, but can't figure out ...
5
votes
2answers
4k views

Simple ways to evaluate a derivative at a point?

The contrast in behavior between, say, f[x_] = Sin[x^2]; f'[2] vs. u[x_, y_] = Cos[x + y^2]; has always bothered my ...
6
votes
1answer
143 views

Problem overloading D[] using UpValues

First I define an entry of UpValues overloading D[] for expressions with head ftest: ...
1
vote
0answers
601 views

Calculating the mean curvature of a surface - suggestions

I am attempting to calculate the mean curvature [1, 2] of a surface defined by a function of x and y. My function is rather ...
2
votes
2answers
724 views

About generating power series

For an arbitrary function $f(x,y)$ I am defining functions LogMT1 and LogMT2 as follows, ...
8
votes
1answer
334 views

Symbolic Integration along contour: branch cut problem?

Context Following this question on path integrals in the complex plane, having defined again a numerical and symbolic integrator along a path as ...
9
votes
0answers
526 views

Dual complex integral over implicit path using contour plot

Context I am interested in doing double contour integral over paths which are defined implicitely. For the sake of debugging, let's assume its $$\oint_{\cal C}\oint_{\cal C} \frac{1}{u\, x} d u d x$$ ...
4
votes
1answer
780 views

Symbolic integration in the complex plane

Context While answering this question, I defined (symbolic and numerical) path integrations as follows ...
2
votes
1answer
649 views

Finding residues of multi-dimensional complex functions

Say I have a function $f$ of $n$ complex variables, $\{ z_i \}_{i=1}^{i=Nc}$. And then I want to contour integrate the expression such that for each $z_i$ its an integration on an unit circle about ...
13
votes
1answer
859 views

Plot showing discontinuity where it shouldn't

I was trying to integrate a continuous function with a kink and I did it two ways and both ways the plot of the result shows a discontinuity. I also later want to differentiate the Integrated ...
2
votes
1answer
279 views

About high dimensional integrals

I want to be able to do high dimensional integrals like, (..naively I wrote it as this..) ...
3
votes
3answers
291 views

Proving a recurrence in Mathematica

I have $$j_n=\int_0^1 x^{2n} \sin(\pi x)dx.$$ How do I show that $$j_{n+1}= \frac{1}{\pi^2}(\pi- (2n+1)(2n+2)j_n)\, ?$$ I keep getting a recurring integration by parts and I can't simplify it. ...
2
votes
0answers
222 views

Numerical-Symbolical Integration (Calculus)

I created a simple numeric-symbolic integration. Here you can use symbolical and numerical techniques at the same time. You can also interpolate numerical integrals. The problem with my function is ...
3
votes
1answer
197 views

One Integral — Two Results

I compared Mathematica's default integral evaluation to what you get by computing the indefinite integral and then plugging in the limits. (I did this because the latter approach was much faster for ...
4
votes
2answers
332 views

Define product derivative

How do I define the $n$th product derivative of a function in Mathematica? The first product derivative $f^\ast$ of a function $f$ is $$ f^\ast(x)=\exp\left(\frac{f^\prime(x)}{f(x)}\right) $$ The ...
5
votes
2answers
515 views

How to receive the result of the integral $ \int\frac{1}{x}dx$ is $\ln|x|$?

When I input \[Integral]1/x \[DifferentialD]x in Mathematica, I get Log[x]. How can I make the result of the integral ...
6
votes
1answer
143 views

What exactly does GenerateConditions do?

Consider for example this strange behavior: Integrate[1/x, {x, 0, Infinity}, GenerateConditions -> False] (*0*) I'd also like to know the difference between ...
0
votes
2answers
541 views

Integral with the derivative of Dirac delta does not evaluate correctly?

Integrate[DiracDelta'[y - z] DiracDelta[x - z], {z, -Infinity, Infinity}, Assumptions -> x \[Element] Reals && x \[Element] Reals]
7
votes
1answer
180 views

Differentiate the product of some terms

How can I compute the following derivative, $$ \frac{\partial}{\partial \lambda_j} \prod_{i=1}^k (1+\lambda_i)e^{\lambda_i} \quad \text{for }\; 1\le j \le k $$ for some positive integer $k$ which is ...
1
vote
1answer
280 views

Asymptotic Rate of Growth

How can I calculate the Asymptotic Rate of growth of a function, for instance like: $X^3 - X^2 - X -1$ EDIT: For instance, as you can see in this graph, after the 1200 the function approximates to ...