Questions related to the calculus and analysis branches of Mathematica, including, but not limited to, limits, derivatives, integrals, series, and residues.

learn more… | top users | synonyms (3)

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
256 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
765 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
313 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
120 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
138 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 ...
14
votes
0answers
243 views

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

Consider these integrals: ...
4
votes
1answer
890 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 ...
4
votes
1answer
1k 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} ...
15
votes
4answers
4k 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 ...
1
vote
2answers
567 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
372 views

Typeset unbalanced brackets

In LaTeX, you can typeset expressions containing unbalanced brackets, by balancing them with an invisible delimiter \left. or ...
16
votes
1answer
463 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
861 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. ...
22
votes
3answers
2k 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
508 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 ...
10
votes
4answers
4k views

How can I differentiate Numerically?

Mathematica has two ways to integrate: Integrate and NIntegrate. But what about D? ...
2
votes
0answers
69 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 ...
7
votes
2answers
6k 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
158 views

Problem overloading D[] using UpValues

First I define an entry of UpValues overloading D[] for expressions with head ftest: ...
1
vote
0answers
783 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
796 views

About generating power series

For an arbitrary function $f(x,y)$ I am defining functions LogMT1 and LogMT2 as follows, ...
10
votes
1answer
390 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 ...
17
votes
1answer
678 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 $$\oint_{\cal C}\oint_{\cal C} \frac{1}{u\, x} d u d x$$ ...
5
votes
1answer
1k views

Symbolic integration in the complex plane

Context While answering this question, I defined (symbolic and numerical) path integrations as follows ...
2
votes
1answer
845 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
1k 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
329 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
303 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. ...
3
votes
0answers
236 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
203 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
396 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
651 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 ...
9
votes
1answer
180 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
635 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
189 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
298 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
915 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. ...
1
vote
1answer
576 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
504 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 ...
5
votes
2answers
3k 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
352 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 ...
15
votes
5answers
761 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
186 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 ...
2
votes
3answers
283 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
354 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} + ...
5
votes
3answers
308 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
411 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
157 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
120 views

How to explain the results of the following integrals?

Why does the following program lead to different results of a21 and a211? ...