Questions related to the calculus and analysis branches of Mathematica, including, but not limited to, limits, derivatives, integrals, series, and residues.
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
...
1
vote
1answer
102 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 ...
8
votes
0answers
102 views
3
votes
1answer
266 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
357 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 $g$.
$$ f\left(\mathbf{x}\right) =
\left(
\begin{array}{c}
...
5
votes
3answers
485 views
Multivariable Taylor expansion does not work as expected
The basic multivariable Taylor expansion formula around a point is as follows:
$$
f(\mathbf r + \mathbf a) = f(\mathbf r) + (\mathbf a \cdot \nabla )f(\mathbf r) + \frac{1}{2!}(\mathbf a \cdot ...
0
votes
2answers
266 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 ...
1
vote
0answers
151 views
Function approaching incorrect limit in mathematica [closed]
I am interested in the complex function
$f(z)=\frac{1}{a}\log\left[2\sinh(a\sqrt{z})\right]$.
where $a > 0$ is a real parameter. Clearly for large $a$ this approaches $f(z) \to \sqrt{z}$.
But ...
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 ...
15
votes
1answer
289 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 $\ell=2,4\cdots 20$.
$$\int _0^{\pi }\int _0^{2 \pi ...
2
votes
1answer
241 views
How to find all roots and maxima and minima and graph them
I need to find all the roots of this function.
I know it is simple to find by hand, however, I wish to learn how to do it so I can apply it later. The problem is that it only gives me one root.
...
14
votes
3answers
564 views
Can we teach Mathematica about functional differentiation?
The key relation for functional differentiation is
$$\frac{\delta}{\delta f(y)}f(x)=\delta(x-y),
$$
where $\delta(x-y)$ is the Dirac delta function, and the usual properties of differentiation (e.g. ...
5
votes
0answers
162 views
Integral with HeavisideTheta takes too long to evaluate
I tried to compute $$\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)\,$$ where $\theta$ is Heaviside's step function, by using
...
2
votes
3answers
594 views
How can I differentiate Numerically?
Mathematica has two ways to integrate: Integrate and NIntegrate.
But what about D? ...
2
votes
0answers
66 views
How to expand terms within an integral? [duplicate]
Possible Duplicate:
Why aren’t these additions of integrals and summations equal?
I am trying to expand all the terms in a sum in order to apply assignment rules later, but can't figure out ...
3
votes
2answers
949 views
Simple ways to evaluate a derivative at a point?
The contrast in behavior between, say,
f[x_] = Sin[x^2];
f'[2]
vs.
u[x_, y_] = Cos[x + y^2];
has always bothered my ...
6
votes
1answer
103 views
Problem overloading D[] using UpValues
First I define an entry of UpValues overloading D[] for expressions with head ftest:
...
1
vote
0answers
255 views
Calculating the mean curvature of a surface - suggestions
I am attempting to calculate the mean curvature [1, 2] of a surface defined by a function of x and y.
My function is rather ...
0
votes
2answers
393 views
About generating power series
For an arbitrary function $f(x,y)$ I am defining functions LogMT1 and LogMT2 as follows,
...
7
votes
1answer
238 views
Symbolic Integration along contour: branch cut problem?
Context
Following this question on path integrals in the complex plane, having defined again a numerical and symbolic integrator along a path as
...
7
votes
0answers
279 views
Dual complex integral over implicit path using contour plot
Context
I am interested in doing double contour integral over paths which are defined implicitely.
For the sake of debugging, let's assume its
$$\oint_{\cal C}\oint_{\cal C} \frac{1}{u\, x} d u d x$$ ...
3
votes
1answer
332 views
Symbolic integration in the complex plane
Context
While answering this question, I defined (symbolic and numerical) path integrations as follows
...
2
votes
1answer
255 views
Finding residues of multi-dimensional complex functions
Say I have a function $f$ of $n$ complex variables, $\{ z_i \}_{i=1}^{i=Nc}$. And then I want to contour integrate the expression such that for each $z_i$ its an integration on an unit circle about ...
9
votes
1answer
262 views
Plot showing discontinuity where it shouldn't
I was trying to integrate a continuous function with a kink and I did it two ways and both ways the plot of the result shows a discontinuity. I also later want to differentiate the Integrated ...
2
votes
1answer
161 views
About high dimensional integrals
I want to be able to do high dimensional integrals like,
(..naively I wrote it as this..)
...
3
votes
3answers
219 views
Proving a recurrence in Mathematica
I have
$$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)\, ?$$
I keep getting a recurring integration by parts and I can't simplify it.
...
2
votes
0answers
163 views
Numerical-Symbolical Integration (Calculus)
I created a simple numeric-symbolic integration. Here you can use symbolical and numerical techniques at the same time. You can also interpolate numerical integrals.
The problem with my function is ...
3
votes
1answer
165 views
One Integral — Two Results
I compared Mathematica's default integral evaluation to what you get by computing the indefinite integral and then plugging in the limits.
(I did this because the latter approach was much faster for ...
4
votes
2answers
175 views
Define product derivative
How do I define the $n$th product derivative of a function in Mathematica?
The first product derivative $f^\ast$ of a function $f$ is
$$
f^\ast(x)=\exp\left(\frac{f^\prime(x)}{f(x)}\right)
$$
The ...
5
votes
2answers
328 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 ...
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 ...
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]
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 ...
4
votes
2answers
342 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. ...
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 ...
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 ...
4
votes
2answers
914 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 ...
8
votes
1answer
234 views
ReplaceAll[] and Limit[] don't give correct results for this expression under extreme variables [duplicate]
Possible Duplicate:
Funny behaviour when plotting a polynomial of high degree and large coefficients
1/x^2 + (3 + x)/(6 (1 - Exp[x] + x))
——This is a ...
11
votes
4answers
356 views
How to make a Line[] with no end?
I'm trying to do this:
In this graph, the secant points are aproximated in order to become the tangent, it seems I need some kind of function which plots a line based on two points and it's points ...
1
vote
0answers
116 views
How to move differentation before the integral?
I have quite a lot expressions that need to be integrated after switching the integral sign with differentation operator. The additional problem is that the bounds are dependent to one of the ...
1
vote
1answer
169 views
Need help evaluating definite integral to a function of Y
Suppose
$Y = \sqrt{2T}\cos(U)$, $ 0 \le u \le \pi $, and $ 0 \le \cos^{-1}(\frac{y}{\sqrt {2t}}) \le \pi ) $, so $ -1 \le \frac{y}{\sqrt{2t}} \le 1 $, with all $ \mathbb{R}$. The iterated integral ...
5
votes
3answers
246 views
ComplexInfinity/Indeterminate error when evaluating derivative
Motivation: I want to find the coefficients for the polynomial that is obtained when one adds together the first $n$ natural numbers to the power of $a$; that is, when you consider $1^{a} + 2^{a} + ...
6
votes
3answers
217 views
How to force Dt[ ] to recognize dependencies
I want to differentiate a function f[] for which I don't have a specific expression.
f[] depends on ...
13
votes
1answer
341 views
What strategies can I use to evaluate a limit when Limit[] returns unevaluated
I'm trying to find the following limit using Mathematica:
$$\lim_{N\to\infty}\sum_{k=1}^N\left(\frac{k-1}{N}\right)^N$$
The problem is taken from here and is known to converge to ...
1
vote
1answer
138 views
How to collect q[t] from the following integration
As shown in the following program, the q[t] in a can be collected from the integration by defining the integration of ...
1
vote
1answer
103 views
How to explain the results of the following integrals?
Why does the following program lead to different results of a21 and a211?
...
2
votes
0answers
284 views
Simple contour integral with a parameter gone wrong
I run into the following problem, I tried to evaluate a very simple integral:
Assuming[ a > 0 , Integrate[Sin[a*s]/(s - I)^2, {s, -Infinity, Infinity}]]
...
