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

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4
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4answers
797 views

Mathematica gives wrong answer for integral

When I execute the following, Integrate[Exp[-w^2 + I w^3], {w, -\[Infinity], \[Infinity]}] I get ...
11
votes
4answers
4k 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
167 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 ...
3
votes
2answers
254 views
6
votes
1answer
523 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 ...
5
votes
3answers
654 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 ...
5
votes
0answers
112 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): ...
14
votes
2answers
954 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, ...
3
votes
2answers
985 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 ...
12
votes
1answer
1k 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 ...
1
vote
2answers
321 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 ...
6
votes
1answer
286 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 ...
10
votes
2answers
527 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 ...
5
votes
1answer
253 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 ...
22
votes
3answers
2k 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?
7
votes
1answer
336 views

Limit not evaluating

Why is the input Limit[IntegerPart[Sin[x]/x], x -> 0] not being evaluated?
7
votes
3answers
982 views

how to simplify symbolic integration

The following is the code: ...
6
votes
3answers
980 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 ...
7
votes
1answer
615 views

How to find the smallest root

I have a continuous, differentiable, monotonic, bounded function called F[t]. If t -> Infinity then ...
11
votes
1answer
498 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 ...
10
votes
2answers
747 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
2answers
561 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 ...
4
votes
3answers
448 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)\, ...
8
votes
3answers
2k views

Derivative of real functions including Re and Im

When deriving functions using Re, Im or Arg (and probably some other functions as well), ...
13
votes
3answers
773 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 ...
6
votes
5answers
715 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 ...
8
votes
2answers
772 views

Bug in Integrate for Mathematica

Consider the following: ...
13
votes
5answers
705 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 ...
8
votes
5answers
471 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 ...
2
votes
1answer
189 views

Checking if first derivative is zero at a given point

I have a function V[a_, h_, tau_] := (Sqrt[3]/2) h (Sqrt[3] a + h/3 tau)^2 now I want to check, whether its first derivative is zero in a given point. Say for ...
2
votes
1answer
242 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. ...
11
votes
5answers
2k 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 ...
16
votes
1answer
656 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 ...
3
votes
1answer
255 views

n-fold symbolic integral in Mathematica

I am trying to compute symbolically a n-fold integral (n is a parameter of a function) over, say, the cube [0,a]^n. My code looks like this ...
2
votes
1answer
431 views

Speeding up a slow indefinite integral

I have found an alternative way to express an old solution: ...
9
votes
4answers
4k 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: ...
11
votes
2answers
3k 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 : ...
7
votes
3answers
482 views

Inverting a function in a certain region

InverseFunction works well for globally invertible functions, like f = 2*# + 2 &; InverseFunction[f] ...
3
votes
1answer
537 views

Definite and Indefinite integral give different results for piecewise function

I have the following function: $$ f(q,y)= \begin{cases} \tfrac{11720+p}{37791360} & -11720<p<-7720 \\ 0 & \text{True} \end{cases} $$ where $p = 443\ y-777600\ \sin^{-1}\left(\frac{q ...
5
votes
3answers
637 views

recursive integration

I am trying to do multiple integrations recursively. For instance, I would like to do the following equation for arbitrary integer $n$: $\displaystyle R_n(t) = \int_0^t \mathrm dt' R_0(t-t') ...
7
votes
1answer
550 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 ...
4
votes
5answers
2k 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
1answer
485 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.
7
votes
2answers
2k 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 ...
13
votes
2answers
695 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 ...
8
votes
2answers
546 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 ...
14
votes
2answers
137 views

Symbolic derivatives are being calculated numerically

Just found the following while debugging a problem. Mathematica is calculating the derivative of IntegerPart[x] in some odd way: ...