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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3
votes
0answers
80 views

Inconsistent results for symbolic trigonometric integral

I am trying to evaluate (on Mathematica v.9) the integral \begin{equation} \int_0^x [\sin (x) \sin (2 y)-\sin (2 x) \sin (y)]^t \,\mathrm{d}y, \end{equation} where $t$ is an even, positive integer. ...
5
votes
3answers
507 views

Plotting the direction field of a differential equation

I have to sketch the direction field for the following differential equation: $$\frac{dy}{dx}=\frac{-0.02 y +0.00002 xy}{0.08 x-0.001xy}$$ This is the code I used, which gives an incorrect plot: ...
0
votes
0answers
68 views

What is the function in Mathematica that can be used to find the derivative of a constant value function with respect to an arbitrary matrix?

I have a function that consists of some matrices. This function results in a constant value. I need to take the partial derivative of this function with respect to one of the matrices it depends on. ...
1
vote
1answer
147 views

Dealing with indicator functions

I am calculating the integral of two indicator functions with Mathematica. I'm having difficulty in imposing assumptions. What I have done so far: To define an indicator function: ...
4
votes
1answer
79 views

Somewhat Irreproducible Integrate Results

Backslide introduced in v10 and persisting through v10.3.1. In the course of considering question 102922, I encountered erratic results from a particular integration. It is illustrated as follows. ...
3
votes
0answers
84 views

Bug or feature? If statement inside double integrals

(taken from http://math.stackexchange.com/questions/1592901/bug-or-feature-if-statement-inside-double-integrals) Generally speaking, $$ \int_c^a f(x) dx = \int_c^{a-b} f(x) dx + \int_{a-b}^a f(x) ...
2
votes
1answer
96 views

Limit $\sqrt{3-x}$ as $x\to3^+$ [closed]

I'm trying: $$\lim_{x\to3^+}\sqrt{3-x}$$ Using real numbers, technically this limit does not exist. However, Limit[f[x], x -> 3, Direction -> -1] I also ...
13
votes
9answers
2k views

Finding maximum or minimum of implicit functions

is there any built in function that can be used to find maximum or minimum of implicit functions? For example, if we have the equation $$x^2 + y^2 = (2 x^2 + 2 y^2 - x)^2,$$ then we can visualize the ...
0
votes
0answers
64 views

Finding double inverse Laplace transform numerically

Edit: I asked Prof. Valko, he was so nice to point out where the mistake is. The problem is solved ! Thank you for your attention ! Recall the the double Laplace transform of a function $f(x,t)$ is ...
0
votes
1answer
86 views

Integrating the derivative of a function does not give expected result [closed]

I was trying to do the following integral: f[z_]:=Hypergeometric2F1[1,-a,1-a,1/(1-z)]/((1-z)^a-1) Integrate[D[f[z],z],z] I thought the result should be ...
5
votes
2answers
186 views

Can your Mathematica do the following integral?

Backslide introduced in v10 and persisting through v10.3.1. I was trying to do the following integral in Mathematica: ...
4
votes
0answers
102 views

Exploring formal limit definition

I found a few demonstrations on the Wolfram Demonstration Project site that help users to explore the formal definition of a limit. That is, $$\lim_{x\to a}f(x)=L$$ if and only if for every ...
10
votes
2answers
408 views

Symbolic integration fails while numerical integration succeeds

I am hoping to evaluate the following integral Integrate[((r^3 - 7)^(2/3)*(1 - (r^3 - 7)^(2/3)/r^2))/r^3, {r, 2, Infinity}] but Mathematica informs me that this ...
8
votes
1answer
247 views

Strange bug: Product[1/(1-1/2^(n-1)),{n,2,Infinity}] = 0?

This infinite product is correct: Product[1/(1 - 1/2^n), {n, 1, Infinity}] (* 1/QPochhammer[1/2, 1/2] *) Numerical value of this result is not equal to zero ...
8
votes
7answers
1k 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 ...
7
votes
3answers
298 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 ...
7
votes
1answer
156 views

Old fashioned region method?

Just stumbled across this idea in http://users.rowan.edu/~hassen/Mathematica/Volume%20III/Chapter%2015.pdf. ...
2
votes
0answers
82 views

Evaluating an integral

Here is the input: Integrate[E^(2 I*t)/(2*Pi*(E^(I*t) - z)), {t, 0, 2 Pi}] and here the output: ...
2
votes
1answer
80 views

Limits explanation in doc

Limit[expr,x->Subscript[x, 0],Direction->1] computes the limit as x approaches Subscript[x, 0] from smaller values. ...
-1
votes
2answers
55 views

computing a limit [closed]

I couldn't compute this limit: it should be -3. I couldn't find the analytic way to compute it.
4
votes
1answer
91 views

Parametric numerical integration [closed]

I would like to solve the integral: $$ \iiint\frac{e^{-\sqrt{x^2 + k^2}}x^2 \sin{\theta}}{\sqrt{x^2 + k^2}} dx d\theta d\phi$$ defined in $x \in [0,+\infty]$, $\theta \in [0,\pi]$, $\phi \in [0,2 ...
0
votes
0answers
75 views

Multiple Integral of equation including dx prime?

I'm hoping someone can verify I've written this function properly since I'm a little new to Mathematica. What I'm attempting to do is code a function that describes a gravitational field of an ...
10
votes
2answers
215 views

Demonstrating the behavior of a function as its independent variable approaches zero

I have several questions regarding the function $$f(x)=\frac{\sqrt{x^2+9}-3}{x^2}$$ that I would like to help my students with in the upcoming semester. Now, the limit as $x\to 0$ is 1/6. ...
5
votes
2answers
145 views

RegionPlot of ImplicitRegion is incorrect

Why does RegionPlot of the ImplicitRegion give an incorrect plot? Also, will how does this affect calculations with ...
3
votes
1answer
77 views

Newbie Problem with integral

when I type Integrate[Sign[x], x] to Mathematica, the answer is ∫Sign[x] dx so nothing new. When I type the same to Wolfram ...
1
vote
0answers
109 views

How do I calculate this integral along a complex line (not a contour) in mathematica?

Given the function: $$f(z) = \frac{\pi ^2 \cot(\pi \sqrt z) \cot(\pi \sqrt {z-a})}{4\sqrt{z^2 - az}}$$ Where a is a positive constant and: $$z = \frac{i}{b} + t $$ $$t>a$$ Where b is also a ...
3
votes
1answer
79 views

Correct behavior of Integrate involving logarithmic cut, or possible bug?

I just discovered that Integrate with simple poles in its integrand sometimes gives different results depending on number type of its limit (Exact vs Inexact). ...
3
votes
4answers
238 views

Computing the area between two curves

I want to find the area in between the two curves whenever they overlap each other. I've tried multiple different formulas but none of them have been working. I have to find it using the ...
6
votes
3answers
747 views

Why does Mathematica spit this back out?

I'm struggling with the integral, $$\iint\sqrt{4a^2-x^2-y^2}dxdy$$ taken over the upper half disk of radius a centered at (a, 0). When I type it into Mathematica (10.2), Mathematica spits it back ...
5
votes
3answers
106 views
5
votes
2answers
350 views

Replace rule with function? Derivatives don't evaluate

Say I have an expression (call it expr) involving a function, f[x]. I'd like to be able to evaluate that for a particular choice of ...
0
votes
0answers
74 views

Mathematica: Derivative appears to be wrong

I'm first getting f(x) by computing an implicit equation. Then I want to take the derivative. Later on, when I plot both f(x) ...
0
votes
0answers
93 views

Integrate dealing with Sqrt gives wrong answers

I often find when I integrate something with a Sqrt[x] term in it, Mathematica gives an incorrect result. I assume this must be due to some kind of incorrect ...
2
votes
1answer
61 views

Using a grid table to estimate a limit

Consider finding the limit of $f(x)=(x^2-4)/(x-2)$ as $x\to 2$. f[x_] := (x^2 - 4)/(x - 2); I'm looking for some easy ways for my Calc I students (next semester) ...
0
votes
0answers
35 views

Error Message: Invalid integration variable or limit(s) in 1

I write the line: Integrate[1/{1 + Sin[2 x]}, x] And gives the error message: Invalid integration variable or limit(s) in 1. No solutions I'm trying work, and other integration problems are working ...
1
vote
0answers
40 views
1
vote
1answer
78 views

Evaluate a double integral over a region [duplicate]

How can I evaluate a double integral over a region in Mathematica? For example $\qquad \iint(x^4+y^2)\,dx\,dy$ for the bounded region $y=x^2$ and $y=x^3$
6
votes
1answer
221 views

Confirming conservation laws for complex valued functions

Consider the nonlinear Schrödinger equation (I would like to do this for a more complicated set of equations, but to gain understanding I'll consider this simplified case) $$A_t+iA_{xx}+i|A|^2A =0,$$ ...
1
vote
1answer
92 views

Integrating differential solid angle over the unit hemisphere

When reading about rendering topics I commonly run into integrals over a hemisphere similar to this one: $\int_\Omega n \cdot l \, d\omega$ If I want to put that in Mathematica then I can ...
2
votes
1answer
121 views

Triple/Double integral over outer side of a cylinder

$$\underset{S}{\iint} (yz\text{dx}\text{dy}+xy\text{dxdz}+xz\text{dydz})$$ Where S is the outer side of a part of a cylinder: $ x^2+y^2 = r^2 ; x\le0,\ y\ge0, \ 0\le z\le H$ The problem about that ...
2
votes
1answer
86 views
2
votes
1answer
148 views

Why the strange limit?

When we take the limit, we get an Interval. Why? What does it say about the function? I know that when $0$ or $\pi$ are substituted we get exact values, otherwise ...
1
vote
1answer
66 views

Integrating sinc function over discontinuity

I would like to analytically integrate the sinc function. First of all, if I just perform the integration the following way, everything is as expected: ...
5
votes
1answer
193 views

A triple sum related question

I'm trying to compute the triple sum Sum[ 1/(i! j! k! ), {i, 1, Infinity}, {j, i + 1, Infinity}, {k, j + 1, Infinity}] but Mathematica doesn't return any value. ...
2
votes
5answers
302 views

Simplify an integral involving trigonometric functions

We were doing variation of parameters in differential equations tonight and had to do the following integral, the result given by hand calculations. $$\int \cos t\tan^2 t\,dt=\ln|\sec t+\tan t|-\sin ...
2
votes
0answers
211 views

About calculating Integrals

I am trying to Integrate the following Integral (all of the variables are reals) $$\int Exp(-(\cos^{-1}\left[\frac{\text{n}_0 (\text{v}_0-\text{x}_0)+\text{n}_1 (\text{v}_1-\text{x}_1)+\text{n}_2 (u ...
2
votes
1answer
96 views

Two different results for same symbolic integral

I am trying to compute the following integral $\int_0^\infty dk \frac{e^{i k a}}{k^2}[\sin(kb)-kb\cos(kb)]$ When computing it as-is, i.e, with the code ...