Questions related to the calculus and analysis branches of Mathematica, including, but not limited to, limits, derivatives, integrals, series, and residues.
11
votes
1answer
666 views
How does Mathematica integrate?
Basically, this question can be considered to be an extenstion to my other question.
What I wanted to do was this integral as homework (it is indefinite BTW so no approximations using Simpson's Rule ...
25
votes
4answers
926 views
Generating evenly spaced points on a curve
In the KnotData package a simple command such as
points = Table[KnotData[{3, 1}, "SpaceCurve"][t], {t, 0, 2 Pi, 0.1}];
will ...
3
votes
1answer
331 views
Symbolic integration in the complex plane
Context
While answering this question, I defined (symbolic and numerical) path integrations as follows
...
7
votes
3answers
884 views
Finding unit tangent, normal, and binormal vectors for a given r(t)
For my Calc III class, I need to find $T(t), N(t)$, and $B(t)$ for $t=1, 2$, and $-1$, given $r(t)=\{t,t^2,t^3\}$.
I've got Mathematica, but I've never used it before and I'm not sure how to coerce ...
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 ...
9
votes
2answers
1k views
How to specify assumptions before evaluation?
If I request mathematica evaluate an integral for me, I'll often get a more general ConditionalExpression than I want. Example :
...
8
votes
2answers
660 views
7
votes
2answers
225 views
How to simplify the integration result of the following program?
In the following program, the fec[x] is a function depending on $x$ while q3[t] and q5[t] ...
10
votes
5answers
1k views
How can you compute Itō Integrals with Mathematica?
How can you compute Itō Integrals with Mathematica? I tried searching through the documentations but I didn't find anything.
P.S. I was not at all sure how to tag this question. I had to put in at ...
9
votes
2answers
203 views
Find closed form expression for series expansion coefficients [duplicate]
Is there a built-in function that will find a general expression for the coefficient of the series expansion of a function?
Series will only give the explicit ...
4
votes
2answers
913 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 ...
1
vote
3answers
244 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
3answers
466 views
Using D to find a symbolic derivative
I need to do the following:
Define a function
Take the derivative of this function and have a look at the symoblic representation
Substitute in some values
With the bonus that I want to use the ...
8
votes
4answers
427 views
Plot a 2D vector path onto a surface
In my calculus 3 course, we're studying gradients and have a project that takes a combination of 3D Gaussian radial surfaces and a basic parametric path $r(t) = \{x(t),y(t)\}$ to see how the gradient ...
4
votes
5answers
927 views
Using implicit differentiation to find a line that is tangent to a curve at a point
A few days ago I asked about using differentiation to find a line that is tangent to a curve at a given point. J.M. provided a very elegant way to solve these kinds of problems in Mathematica.
Now ...
7
votes
3answers
1k views
Derivative of real functions including Re and Im
When deriving functions using Re, Im or Arg (and probably some other functions as well), ...
2
votes
3answers
594 views
How can I differentiate Numerically?
Mathematica has two ways to integrate: Integrate and NIntegrate.
But what about D? ...
13
votes
5answers
626 views
Does $x>0$ imply that $x\in\mathbb{R}$?
Let’s assume I input
Assuming[x > 0, expression]
Is it assumed by Mathematica that $x$ is a real number? Or that the real part of $x$ is positive? Something ...
3
votes
2answers
1k views
How to convert a system of parametric equations to a normal equation?
For example, I have a system of parametric equations (R is a constant number) :
...
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 ...
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 ...
7
votes
2answers
383 views
Why aren't these additions of integrals and summations equal?
I have the following code:
Simplify[Integrate[f[x] + g[x], x] == Integrate[f[x], x] + Integrate[g[x], x]]
To test:
$$\int{\left(f(x) + ...
6
votes
1answer
342 views
How to expand a function into a power series with negative powers?
Is there any way to expand this expression
a+b(1-Exp[-T/(b c)]/(z-Exp[-T/ (b c)])
(where a, ...
6
votes
5answers
561 views
Trying to prove that $x\sin(\frac{\pi}{x})\ge\pi \cos(\frac{\pi}{x})$ for $x\ge 1$
Consider the function
f[x_] := x Sin[Pi/x]
I want to prove that this function is increasing for $x\ge 1$. This can be done with the first derivative. We have to ...
6
votes
1answer
414 views
Hankel Transform integrals won't work in Mathematica
I'm trying to do this integral, which is shown on the Wikipedia page on the Hankel transformation:
$$\int_0^{2\pi}\mathrm d\varphi\;e^{\mathrm im\varphi}e^{\mathrm ikr\cos(\varphi)}$$
The answer is ...
1
vote
1answer
429 views
How to solve this integral
I'm trying to deal with this article, but I don't understand how equations 5.9 and 5.10 were obtained. I tried to use Mathematica 8 to solve it, but the answer was different.
I have a problem with ...
14
votes
4answers
461 views
Finding Limits in several variables
Is there a way to find a limit of a multivariable function, like
$$\lim_{(x,y)\to (0,0)} f(x,y)$$
with Mathematica?
When $f$ is continuous, we can use
$$\lim_{(x,y)\to (0,0)} ...
7
votes
2answers
898 views
Equation of a line that is tangent to a curve at point
A common problem in the derivative section of calculus texts is "find the equation of the line that is tangent to the curve $y = \ldots$ at the point $P$."
To find the line that is tangent to $y = 2 ...
12
votes
2answers
453 views
Suppressing negative roots in Mathematica
Problem
Using Mathematica's Solve operator can sometimes lead to an output involving a positive and negative root (say when solving for a variable such as ...
11
votes
4answers
355 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 ...
8
votes
5answers
377 views
simplifying $\frac{\log x^a}{a} = \log x$
If one makes the assumptions $x>0,a>0$, then $\frac{1}{a}\log x^a = \log x$. Thus, in Mathematica, Simplify[1/a*Log[x^a], {a > 0, x > 0}] returns ...
7
votes
1answer
237 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
...
6
votes
3answers
530 views
4
votes
4answers
547 views
Mathematica gives wrong answer for integral
When I execute the following,
Integrate[Exp[-w^2 + I w^3], {w, -\[Infinity], \[Infinity]}]
I get
...
3
votes
3answers
403 views
Definite integral takes a very long time
I'm trying to integrate the following function:
$$
r_e=\int_0^z\frac{dz}{\sqrt{\Omega_m(1+z)^3+\Omega_\Lambda}}
$$
where $\Omega_\Lambda=1-\Omega_M$ and both $\Omega_\Lambda$ and $\Omega_M$ are real ...
1
vote
2answers
247 views
3N-dimensional integral
I'm trying to compute a multidimensional integral with a variable number of dimensions.
The integral is as follows:
$$
\int d^{3N}\!p~e^{-\frac{\beta}{2m}\vec p^2}.
$$
I have tried this
...
0
votes
2answers
193 views
Using Mathematica to find poles of Gamma functions [closed]
I am concerned about the expression on the RHS of equation A.5 (page 19) in this paper:
$$\int\frac{d^d ...
7
votes
2answers
362 views
Simplifying the derivative of $|x|$
Context
In[855]:= D[Abs[x], x] /. x -> 1
Out[855]= Derivative[1][Abs][1]
In[856]:= D[x, x] /. x -> 1
Out[856]= 1
Question
Why is ...
6
votes
2answers
147 views
Why does Integrate declare a convergent integral divergent?
When I try this command
Integrate[1/Sqrt[(s^2 - u)^2 - 1], {s, m, Infinity}, Assumptions -> u > 2 && m > 10]
Mathematica declares that the ...
5
votes
3answers
151 views
Integration gives ComplexInfinity when it shouldn't
A simple example of my problem is that
Assuming[k ∈ Reals, Integrate[Exp[l]/((1 + Exp[l])*Exp[k*l]), l]] /. {k -> 2}
gives ...
5
votes
3answers
485 views
Multivariable Taylor expansion does not work as expected
The basic multivariable Taylor expansion formula around a point is as follows:
$$
f(\mathbf r + \mathbf a) = f(\mathbf r) + (\mathbf a \cdot \nabla )f(\mathbf r) + \frac{1}{2!}(\mathbf a \cdot ...
5
votes
1answer
217 views
Check homework integration in Mathematica
Given in my homework I have to compute (by hand)
$$\iint\limits_{x^2+y^2\leq 1}(x^2+y^2)\,\mathrm dx\,\,\mathrm dy.$$
My solution so far:
Let $f(x,y)=x^2+y^2$ and $K=\{(x,y)\in\mathbb{R}^2:x^2+y^2\leq ...
3
votes
1answer
92 views
Why does Assuming for Integrate not work as expected?
I'm trying to perform the following integral with Mathematica 7:
...
3
votes
2answers
948 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 ...
3
votes
2answers
216 views
1
vote
1answer
101 views
how to reduce the time to calculate triple integral
i want to evaluate the following integral, but it takes 15 min to do it. i must reduce the time to calculate it, i have to do a table and create a graph, but takes too long.
anyone have an idea ?
...
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 + ...
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 ...
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,
...
