Questions related to the calculus and analysis branches of Mathematica, including, but not limited to, limits, derivatives, integrals, series, and residues.
0
votes
1answer
122 views
Different expressions for an integral if I split the region of integration
Consider the assumptions
...
1
vote
0answers
123 views
Help for double parametrized integration: algebraic vs Mathematica solutions
I have to solve a double integral with parametrized functions, that is (written as a latex equation)
$$
\int_{-\infty}^\infty \int_0^\infty u(t)u(t-\Delta)\frac{1+e^{-2\int_{t-\tau-\Delta}^{t-\tau} ...
0
votes
1answer
145 views
Evaluate multivariable function [closed]
I'm trying to use Mathematica to evaluate many values of $f(x,y) $, like $f(-\pi,\pi) $ for
$$f(x,y) = \left( \frac{-1}{4}\sin \left( \frac{x-y}{2} \right)\sin y \right)\left( \frac{-5}{4}\sin ...
1
vote
1answer
176 views
2
votes
2answers
192 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
184 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 ...
1
vote
1answer
430 views
How to solve this integral
I'm trying to deal with this article, but I don't understand how equations 5.9 and 5.10 were obtained. I tried to use Mathematica 8 to solve it, but the answer was different.
I have a problem with ...
7
votes
1answer
238 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
...
5
votes
4answers
520 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 ...
0
votes
0answers
160 views
How can I calculate an integral, perhaps using residues?
Could you advise me on how to calculate the following integral, perhaps by using Residue?
...
2
votes
4answers
176 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.
$$\frac{1}{8 \sqrt{2}}\left(-2 i ...
6
votes
3answers
212 views
How to calculate this sum?
I want to find the sum
$$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
...
0
votes
1answer
72 views
Why does “Pattern(1,0)[w,_]” show up in the resulting expression of a differentiation? [closed]
I would like to differentiate the function s, and be able to evaluate it for different w afterwards. I tried the following:
...
8
votes
2answers
660 views
5
votes
2answers
149 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
98 views
Simplifying an Integral over Two DiracDeltas
Integrate[DiracDelta[a + k] DiracDelta[-b + k], {k, -\[Infinity], \[Infinity]}]
(*DiracDelta[a + b]*)
This works fine.
But
...
14
votes
3answers
564 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
0answers
102 views
3
votes
1answer
267 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
358 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 $g$.
$$ f\left(\mathbf{x}\right) =
\left(
\begin{array}{c}
...
5
votes
3answers
486 views
Multivariable Taylor expansion does not work as expected
The basic multivariable Taylor expansion formula around a point is as follows:
$$
f(\mathbf r + \mathbf a) = f(\mathbf r) + (\mathbf a \cdot \nabla )f(\mathbf r) + \frac{1}{2!}(\mathbf a \cdot ...
0
votes
2answers
267 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 ...
9
votes
3answers
182 views
Typeset unbalanced brackets
In LaTeX, you can typeset expressions containing unbalanced brackets, by balancing them with an invisible delimiter \left. or ...
2
votes
1answer
242 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.
...
2
votes
3answers
594 views
How can I differentiate Numerically?
Mathematica has two ways to integrate: Integrate and NIntegrate.
But what about D? ...
5
votes
3answers
444 views
recursive integration
I am trying to do multiple integrations recursively. For instance, I would like to do the following equation for arbitrary integer $n$:
$\displaystyle R_n(t) = \int_0^t \mathrm dt' R_0(t-t') ...
6
votes
1answer
103 views
Problem overloading D[] using UpValues
First I define an entry of UpValues overloading D[] for expressions with head ftest:
...
3
votes
2answers
951 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 ...
5
votes
0answers
163 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
...
2
votes
0answers
66 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 ...
1
vote
0answers
255 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 ...
8
votes
1answer
360 views
Is there any automatic differentiation package?
I'm wondering if an automatic differentiation package exists for Mathematica.
This is what I mean by automatic differentiation.
3
votes
1answer
333 views
Symbolic integration in the complex plane
Context
While answering this question, I defined (symbolic and numerical) path integrations as follows
...
2
votes
1answer
255 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 ...
2
votes
1answer
161 views
About high dimensional integrals
I want to be able to do high dimensional integrals like,
(..naively I wrote it as this..)
...
0
votes
2answers
318 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]
9
votes
1answer
262 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
0answers
163 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
165 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 ...
5
votes
1answer
110 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 ...
4
votes
2answers
175 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
329 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 ...
7
votes
1answer
138 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
0answers
197 views
About doing an integral
For a given integer $N$ and some function $f(x,y)$ I want to be able to do an integration of the kind,
$\int_{0} ^{\infty} \prod_ {i=1}^N da_i e^{[\sum_{n=1}^{\infty} \frac{1}{n} \left[ N + ...
2
votes
0answers
147 views
How to plot function & integral with vector variable?
I want to plot functions & integrals with vector variable the present in this paper.
For example the equation (5) at p. 3: I want to plot a chart like figure 5, i.e. the $x$-axis is the angle ...
1
vote
0answers
133 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 ...
0
votes
1answer
193 views
How to define a derivative/Variation operation in Mathematica from scratch
I am looking for a method in Mathematica 8 to define my own derivative/variation (independent of built-in operations). I do not want to re-define built-in objects since I fear that this could cause ...
4
votes
2answers
344 views
Sum of series with log in each term [closed]
I was solving recurrence relation of Introduction to Algorithms by CLRS, 3rd. edition. Problem 4-3 (i)
$$
T(n) = T(n-2) + \frac{1}{lg \; n}
$$
I tried few ways, like expending with iteration method. ...
4
votes
2answers
915 views
How to find (numerical) value of a derivative at point?
I have the following function:
f[0, 0] = 0
f[x_, y_] := Exp[-(x^2 + y^2)^(-1)]
How do I find its partial derivatives at any given point, including $(0,0)$? This ...
11
votes
1answer
436 views
Incorrect results for elementary integrals when using Integrate
There is a rather simple integral ($K_0$ is the 0-th order MacDonald function)
$$\int_0^\infty e^{-x \cosh\xi}\, d\xi = K_0(x)$$
which mathematica cannot solve. This even though the documentation ...
