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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9
votes
1answer
136 views

Non-rectilinear integration of InterpolatingFunction

Bug introduced in 7 or earlier and persisting through 10.3.1 [CASE:3487737] I have a 2-d array that I would like to resample into a different coordinate system and integrate along one of the new ...
2
votes
2answers
130 views

A simple question: Input problem

I have a simple problem. First I define the value of variables, then I calculate the expression as the following: Code 1: ...
15
votes
3answers
2k views

How to evaluate the $0/0$ type limit in Mathematica?

When I use Limit to evaluate the limit $$\begin{align}\lim_{k \to 0} \frac{ (k+2)(\alpha^2 - \sqrt{\alpha^4 + k}) + k}{\alpha^2 - \sqrt{\alpha^4 + k} + 2 ...
0
votes
0answers
31 views

How do I numerically integrate the result of NDSolve if it depends on several variables?

I am trying to numerically integrate solution of NDSolve. I am trying to integrate my result of NDsolve in line 97 of my notebook, but am getting the error ...
4
votes
1answer
89 views

Pertubation series of given equations

I would like to define a function that does the following. If I have an equation, for example: $$f(x)+[1-f^2(x)]+f^{\prime\prime}(x)+f(x)\ f^\prime(x)=0$$ and I am given a value for $f$ to expand ...
0
votes
0answers
16 views

Integration with IntervalMemberQ

I define an interval and its associated index function. ...
1
vote
2answers
148 views

Is there a way to do a cyclic convolution analytically in Mathematica?

Let's say I want to convolve a Gaussian function Pulse[t_] = Exp[-(t - Mid)^2/(2 sig^2)]; with an exponential broadening function ...
0
votes
2answers
74 views

Expression which should be real-valued appears to be complex-valued [closed]

I want to plot the answer of the following integral: Λ = 10; Ω = 10; t = 1; Integrate[ω E^(-ω/Λ) Sin[(ω + Ω)t/2]^2/(ω + Ω)^2, {ω , 0, ∞}] The answer of ...
6
votes
2answers
299 views

An integral with a fractional part in 3 dimensions

The evaluation with Maple suggests the triple integral is around $1$, but Mathematica tells it's $0.0958758$. When using the code ...
-1
votes
2answers
89 views

Gradient of interpolated 3D data in mathematica 9

tt = Flatten[Table[{x, y, z, btot[x, y, z]}, {x, -1, 1, 0.1}, {y, -1, 1,0.1}, {z, -1, 1, 0.1}], 2]; ff = Interpolation[tt] Till here it is working fine as it is ...
4
votes
1answer
110 views

Products of Differential Operators

I have a differential equation defined as the product of operators which I want to expand out into a polynomial in powers of $z\frac{d}{dz}$ $\qquad \prod_{n=1}^p(z\frac{d}{dz}+a_n)$ However when I ...
3
votes
0answers
108 views

Why Can't Mathematica Integrate this? [closed]

I am trying to integrate Absolute-value functions from Wikipedia: ...
3
votes
3answers
208 views

Limit n->Infinity of recursive sequence

I have defined a recursive sequence a[0] := 1 a[n_] := Sqrt[3] + 1/2 a[n - 1] because I want to calculate the Limit for this ...
-1
votes
1answer
63 views

Taylor and Maclaurin series [closed]

I need to calculate limit of sin(x)*cot(tanx) at 0 using Maclaurin series I took the usual approach but did not get anything of help.
1
vote
1answer
121 views

DensityPlot showing artifacts [closed]

This is my first post on Mathematica.SE, and I'm still a bit new to Mathematica. I believe I'm probably doing something silly in my code, but I have not been able to find where. I'm trying to plot a ...
5
votes
2answers
155 views

Symbolic integration of SphericalBesselJ

Backslide introduced in v10 and persisting through v10.3.1. Consider the following integral ...
3
votes
0answers
89 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
546 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: ...
1
vote
1answer
173 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: ...
3
votes
0answers
94 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
98 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
1answer
93 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 ...
4
votes
0answers
106 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
418 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
271 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
303 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
157 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
85 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
81 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
59 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
104 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 ...
10
votes
2answers
221 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. ...
3
votes
1answer
82 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
118 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
3answers
270 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
828 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
124 views
5
votes
2answers
448 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
84 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
95 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
64 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
130 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
43 views
1
vote
1answer
104 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
230 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
129 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
129 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 ...