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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1
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0answers
88 views

Product of two Meijer's Function

I want to evaluate an integral $I_1$ defined in $Eq.(1)$ as \begin{align} I_1=\int_{0}^{\infty}\frac{x\exp(-\beta x)K_1(\alpha x)}{1+x}dx\tag{1} \end{align} Where $\alpha\geq0$, $\beta\geq0$, and ...
0
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1answer
79 views

How to evaluate the integral of exponentials

I want to evaluate the following integral wrt $\phi$ from 0 to $2 \pi$: ...
0
votes
1answer
59 views

Plotting a function based on complicated integral

I have this function : f[Lambda_] := K Integrate[x Exp[- x^2-Lambda x] HypergeometricU[-Lambda,1/2,(x+ Lambda/2+2)^2],{x,0,Infinity}]; where ...
0
votes
1answer
62 views

Improper Integral Difficulty [closed]

Trying to do the problem $$\int_0^{\infty}e^{-sx}\cosh(ax)\,dx.$$ where we have the assumption that $s>|a|$ and everything is a real number. We tried: ...
1
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1answer
62 views

NIntegrate::inumri: error [closed]

I am getting the following error: NIntegrate::inumri: The integrand c (1-0.6 (1-(1+Times[<<2>>])^0.5)-0.4 (1- (1+Times[<<2>>])^0.5)) (1-0.6 (1-(1+Times[<<2>>])^0.5)-0.4 (1- ...
1
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1answer
45 views

numerical integration errors

I have plotted the following integrals with numerical integration nicely without any error: ...
-3
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1answer
104 views

How to symbolically integrate 2D exponential function? [duplicate]

Is it possible to symbolically solve a 2D integral of the following form $$\int d \vec{r}e^{-\frac{\left| \vec{r} - \vec{r}' \right|^2}{2c}}e^{-i \vec{k} \cdot \vec{r}} $$ where $\vec{r} = (x,y)$ and ...
2
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1answer
229 views

Existence of an analytical form for integral

Can the following integration be performed with an analytical output? ...
2
votes
2answers
164 views

Symbolic vs numerical integration

This is related to a question that I asked before but didn't resolve quite well. I want to obtain the solution of the following expression analytically: ...
1
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0answers
30 views

Hidden assumption in Mathematica integration [duplicate]

If I do Integrate[Exp[I*omega*t],t], it gives me $\frac{1}{\omega} e^{i \omega t}$, as expected in most of the cases, except when $\omega$ is 0. Mathematica seems ...
3
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1answer
87 views

Euler-Maclaurin summation

I want to compute asymptotic approximations to partial sum of harmonic series in Mathematica, using Euler-Maclaurin summation formula. ...
2
votes
1answer
52 views

Why do Wolfram Alpha and Mathematica give different answers? [closed]

I've just got my hands on Wolfram Mathematica and tried to see if I can figure out how to use it by myself. I've plugged in an integral $\int \frac{dx}{(1-x^2)^{2/3}}$, knowing that the result is ...
2
votes
1answer
109 views

Problem with Volume [closed]

A bug probably. This should give the volume of the unit sphere: vol = Volume[{1. r, th, ph}, {r, 0, 1}, {th, 0, Pi}, {ph, -Pi, Pi}, "Spherical"] but it gives: ...
17
votes
3answers
375 views

How to efficiently find moments of a multinormal distribution?

Update: Starting from V10.0 the build-in Moment is fast enough for practical use. I have a multinormal distribution with covariance matrix $\sigma$ and zero ...
2
votes
2answers
136 views

Confirmation of a nice closed form

Can Mathematica 10 confirm a nice closed form for ...
1
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2answers
116 views

How to get the $0$ result for this limit?

Consider the following series: $$\sigma (\text{x$\_$})\text{:=}\sqrt{\sum _{k=0}^{x-1} \frac{(x! x!) (-1)^{x-k}}{(x-k) (k! (2 x-k)!)}+\frac{\pi ^2}{12}}$$ From the following codes, it seems that the ...
4
votes
1answer
101 views

Finding Euler spiral convergence point for powers other than 2

I'm playing around with Euler spirals, using a variable power to see how the plot changes, $ \left\{\begin{matrix} x = \int_0^s \cos \left ( s^n \right ) ds \\ y = \int_0^s \sin \left ( s^n \right ) ...
1
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1answer
65 views

Wrong Limit[…] for inexact expressions

I noticed that Limit can return nonsense when using inexact parameters. In the following code, a is my (exact/inexact) ...
9
votes
1answer
135 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
129 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
30 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
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0answers
15 views

Integration with IntervalMemberQ

I define an interval and its associated index function. ...
1
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2answers
147 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
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2answers
295 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
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2answers
82 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
104 views

Why Can't Mathematica Integrate this? [closed]

I am trying to integrate Absolute-value functions from Wikipedia: ...
3
votes
3answers
184 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
62 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
119 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
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2answers
153 views

Symbolic integration of SphericalBesselJ

Backslide introduced in v10 and persisting through v10.3.1. Consider the following integral ...
3
votes
0answers
88 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
537 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
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1answer
170 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
97 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
105 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
416 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
268 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
302 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
84 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
57 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.