Questions related to the calculus and analysis branches of Mathematica, including, but not limited to, limits, derivatives, integrals, series, and residues.
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 ...
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 ...
7
votes
2answers
902 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 ...
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
218 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
342 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 ...
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}]]
...
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?
...
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] ...
1
vote
2answers
264 views
How to draw a tangent to a surface in a given direction?
It seems quite embarrassing for me, but somehow I can't draw a tangent line to a surface in a given direction. Assume that I want to visualize directional derivative of the function, say, ...
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 :
...
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 ...
1
vote
2answers
575 views
Find the dimensions of a cylindrical can that will minimize the cost of the material
A cylindrical can is to be made to hold 1 L of oil. Find the dimensions
that will minimize the cost of the metal to manufacture the can.
This is a standard calculus problem.
The volume of the ...
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) :
...
4
votes
2answers
386 views
Chain rule while differentiating
I am trying to find the derivative of a function defined in polar coordinates with respect to $x$ and $y$. My function is defined as follows:
$ v_x(r, \theta ) = v_r \cos (\theta ) - v_{\theta }\sin ...
17
votes
1answer
533 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 ...
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
$$B(z+1,z+1)^2 = ...
7
votes
2answers
384 views
Why aren't these additions of integrals and summations equal?
I have the following code:
Simplify[Integrate[f[x] + g[x], x] == Integrate[f[x], x] + Integrate[g[x], x]]
To test:
$$\int{\left(f(x) + ...
2
votes
1answer
340 views
Speeding up a slow indefinite integral
I have found an alternative way to express an old solution:
...
4
votes
4answers
548 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
2answers
216 views
5
votes
3answers
437 views
Can't compute definite integral
Consider a scalar field (in polar co-ordinates), $f(r) = l-r$.
Now I want to evaluate the field integral over a circular region of radius $b$, centered at a distance of $x$ from the origin. According ...
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, ...
2
votes
1answer
103 views
Assumptions with D
I want to give assumptions for the D function. Say I want to calculate $\frac{d|x|}{dx}$ for $x>0$, which is 1.
I write ...
5
votes
1answer
380 views
Getting an InterpolatingFunction from a ContourPlot
I have a function, say
minimizeme[Ω_][β_][ϵ_] = ϵ^2 Ω - Log[2 (Cosh[2 β] + Cosh[2 β ϵ])]/(2 β);
I want to find its critical points in $\epsilon$ for a given ...
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 ...
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 ...
7
votes
1answer
247 views
5
votes
0answers
88 views
Calculating a limit with a result that is discontinuous in the parameters
The following limit is left unevaluated (Edit: added the assumption that $\epsilon$ is real thanks to the comment below):
...
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,
...
11
votes
1answer
671 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 ...
3
votes
2answers
569 views
How to substitute numeric values in a symbolic Jacobian matrix?
I have a multi-variate function from $\mathbb{R}^n\to\mathbb{R}^n$. Choosing any desired initial vector, we can produce the corresponding function value, which is a vector as follows. The main problem ...
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:
...
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 ...
1
vote
2answers
180 views
Quickly differentiate and evaluate a function of several variables
How can I differentiate a function with respect to several variables and evaluate it at the same time ? I want to specify also the variable index that I want to differentiate and the number of times I ...
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 ...
5
votes
1answer
137 views
Finding ranges of a parameter for which a function is always positive
I have a complex function of a single variable expressed in analytical form, which also depends on a parameter. I would like to have Mathematica show me for which values of the parameter the real part ...
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 ...
6
votes
3answers
535 views
4
votes
3answers
308 views
Integrating with multiple indicator functions
I am trying to calculate an integral involving multiple indicator functions, such as:
$$ h(u,v,w) = -\int_0^1 J^{\prime\prime}(s) (I_{(0,s]}(u) - s)(I_{(0,s]}(v) - s)(I_{(0,s]}(w) - s)\, ...
6
votes
3answers
469 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 ...
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:
...
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
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 ...
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 ...
2
votes
1answer
185 views
Integral not simplifying
I specify a function in terms of an integral and then try to evaluate it with Simplify. However, the answer is not really simplified to what it should be.
...
