Questions related to the calculus and analysis branches of Mathematica, including, but not limited to, limits, derivatives, integrals, series, and residues.
25
votes
4answers
928 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 ...
21
votes
3answers
1k views
How to find the period of an arbitrary mathematical function?
Is there a function to find the period of an arbitrary (possibly complex) function in Mathematica?
20
votes
4answers
553 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:
$$
\frac{\left(h^2 +c^2 h^2 + \sqrt{4 h^2+\left(h^2+c^2 ...
17
votes
1answer
532 views
How can the {x,y,z} points that fall on the outer boundary of a set of values be selected and smoothly surfaced?
For a given set of x,y,z values, that may, or may not form a uniform shape, how can the center of the data cloud be found, and the surface points be located and a solid smooth surface created from ...
15
votes
1answer
289 views
Speeding up trigonometric integral
Context
On a possible non trivial toric topology for the Universe (nothing less!).
Problem
I would like to carry out the following integral for $\ell=2,4\cdots 20$.
$$\int _0^{\pi }\int _0^{2 \pi ...
15
votes
2answers
213 views
Inconsistent results from equivalent integrals
Why is Mathematica returning different values for these two integrals:
I am just being introduced to complex integration, so it's possible that I have a misunderstanding of how this works, but in ...
14
votes
4answers
465 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
$$\lim_{(x,y)\to (0,0)} ...
14
votes
3answers
564 views
Can we teach Mathematica about functional differentiation?
The key relation for functional differentiation is
$$\frac{\delta}{\delta f(y)}f(x)=\delta(x-y),
$$
where $\delta(x-y)$ is the Dirac delta function, and the usual properties of differentiation (e.g. ...
13
votes
5answers
626 views
Does $x>0$ imply that $x\in\mathbb{R}$?
Let’s assume I input
Assuming[x > 0, expression]
Is it assumed by Mathematica that $x$ is a real number? Or that the real part of $x$ is positive? Something ...
13
votes
3answers
607 views
How to evaluate the 0/0 type limit in Mathematica?
When I use Limit to evaluate the $k \to 0$ limit of
((k + 2) (α^2 - Sqrt[α^4 + k]) + k)/(α^2 - Sqrt[α^4 + k] + 2 k)
If ...
13
votes
1answer
341 views
What strategies can I use to evaluate a limit when Limit[] returns unevaluated
I'm trying to find the following limit using Mathematica:
$$\lim_{N\to\infty}\sum_{k=1}^N\left(\frac{k-1}{N}\right)^N$$
The problem is taken from here and is known to converge to ...
12
votes
2answers
453 views
Suppressing negative roots in Mathematica
Problem
Using Mathematica's Solve operator can sometimes lead to an output involving a positive and negative root (say when solving for a variable such as ...
12
votes
1answer
237 views
Extra factors appear when evaluating Euler integrals
When I perform the double integral in Mathematica,
Integrate[(x (1 - x))^z (y (1 - y))^z, {x, 0, 1}, {y, 0, 1}]
which should give
$$B(z+1,z+1)^2 = ...
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 ...
11
votes
4answers
2k views
How to find the domain and range of a function with Mathematica?
I'm studying calculus and in some exercises I am asked to find the domain and range of a function. Does Mathematica have already a built-in function for this?
I can imagine some ways of doing so, ...
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 ...
11
votes
1answer
670 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 ...
10
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 :
...
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 ...
10
votes
2answers
173 views
What kind of hypergeometric function is it?
I found a formula for an integral of a product of three Bessel functions at The Wolfram Functions Site:
I cannot understand what kind of hypergeometric function it is.
The Mathematica code given ...
10
votes
1answer
632 views
How to deal with complicated gaussian integrals in Mathematica?
As we know, for most Gaussian integrals, we can get the analytical
result. Now I have many Gaussian integrals to treat, which have the
following general form,
...
9
votes
2answers
204 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 ...
9
votes
2answers
285 views
Derivative from the left and right
I am currently trying to write a script that takes a curve $C$, a starting point $p$ and a direction vector $\vec{v}$ and calculates the path of a beam starting at $p$ in direction $\vec{v}$ when it ...
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
395 views
Maximizing a function with assumptions
Using
f[s_] := Log[(s/r)^α ((α - 2) n0 r^α + 2 π Pmax ρ r^2)
/((α - 2) n0 s^α + 2 π Pmax ρ s^2)]/s
When I run the following line:
...
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 ...
8
votes
4answers
436 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 ...
8
votes
4answers
2k views
Finding critical points of a function
Let's say we'd like to find the critical points of the function $f(x)=\sqrt{x-x^2}$. Finding out where the derivative is 0 is straightforward with Reduce:
...
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 ...
8
votes
2answers
369 views
Series expansion of an inverse
I have to find the series expansion of the inverse function of : $\arctan\left(\frac{\ln(1+x)}{1+x}\right)$
How do I find out the series expansion of any inverse ?
Note: The inverse of a function ...
8
votes
5answers
1k 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 ...
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 ...
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.
8
votes
2answers
660 views
8
votes
0answers
125 views
Why does Mathematica give an incorrect answer to this multiple integral?
This is not a new problem but I would like to understand why Mathematica gives the result that it does. (Volume of a hypersphere)
...
8
votes
0answers
102 views
7
votes
3answers
898 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 ...
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), ...
7
votes
2answers
384 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:
$$\int{\left(f(x) + ...
7
votes
2answers
362 views
Simplifying the derivative of $|x|$
Context
In[855]:= D[Abs[x], x] /. x -> 1
Out[855]= Derivative[1][Abs][1]
In[856]:= D[x, x] /. x -> 1
Out[856]= 1
Question
Why is ...
7
votes
2answers
901 views
Equation of a line that is tangent to a curve at point
A common problem in the derivative section of calculus texts is "find the equation of the line that is tangent to the curve $y = \ldots$ at the point $P$."
To find the line that is tangent to $y = 2 ...
7
votes
2answers
137 views
How to change coordinates of integration?
Is there some built-in routine, some easier method to change variables of integration, without/before solving the integral?
Say I have
...
7
votes
1answer
122 views
How do I get a general form for n-th derivative of a function? [duplicate]
I tried the following but it basically returns unevaluated:
...
7
votes
2answers
225 views
How to simplify the integration result of the following program?
In the following program, the fec[x] is a function depending on $x$ while q3[t] and q5[t] ...
7
votes
1answer
438 views
How to find the smallest root
I have a continuous, differentiable, monotonic, bounded function called F[t]. If t -> Infinity then ...
7
votes
1answer
247 views
7
votes
2answers
401 views
Integration with vector coefficients
I asked this same question in Mathematics, and it was suggested I might try here. I'm more comfortable with Maple, but if I can get Mathematica to do what I'm after, so much the better.
Basically ...
7
votes
1answer
138 views
Differentiate the product of some terms
How can I compute the following derivative,
$$
\frac{\partial}{\partial \lambda_j} \prod_{i=1}^k (1+\lambda_i)e^{\lambda_i} \quad \text{for }\; 1\le j \le k
$$
for some positive integer $k$ which is ...
7
votes
1answer
238 views
Symbolic Integration along contour: branch cut problem?
Context
Following this question on path integrals in the complex plane, having defined again a numerical and symbolic integrator along a path as
...
7
votes
0answers
280 views
Dual complex integral over implicit path using contour plot
Context
I am interested in doing double contour integral over paths which are defined implicitely.
For the sake of debugging, let's assume its
$$\oint_{\cal C}\oint_{\cal C} \frac{1}{u\, x} d u d x$$ ...
