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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0
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1answer
47 views

How to integrate with integer parameter in Mathematica?

I would like to integrate the following expression with N being Integer ...
5
votes
0answers
72 views

Wrong limit: Limit[(1 + (-1)^n/n)^n, n -> Infinity]=1 (Mathematica 10.4 and W-Alpha)

Since $(1-\frac{1}{n})^n\to 1/e$ and $(1+1/n)^n\to e$, the sequence $(1+\frac{(-1)^n}{n})^n$ has no limit as $n\to\infty$, but has limits for odds and even numbers. If $n$ were taken to be real, there ...
6
votes
1answer
176 views

Vastly incorrect answers obtained by increasing WorkingPrecision with modified Bessel functions

This is a follow-up to this question regarding numerical instabilities occurring with modified Bessel functions. In trying to explore J.M.'s answer to that question in greater depth by setting ...
0
votes
0answers
56 views

Tricky inverse Laplace transform

I'm trying to compute the inverse Laplace transform of $f(s) = s^c/(N + s^{ir} )$ where $c,N \in \mathbb{C}$ and $r \in \mathbb{R}^+$ using the Bromwich integral $$ F(t) = \frac{1}{2 \pi i} \int_{- ...
11
votes
3answers
252 views

Integrating a BesselJ integrand to obtain the same result as Maple 16

I would like to check the following integration: Integrate[y Integrate[1/x BesselJ[1, x Exp[I π/4]] BesselJ[1, x Exp[-I π/4]], {x, 0, y}], {y, 0, r}] ...
0
votes
0answers
49 views

Definite and indefinite integral give different results

I'm trying to find an expression that, given $x$, $y$, and $R$, gives the indefinite 2D integral of a function at the circle centered on $(x_0, y_0)$ with radius $R$. With $g$ as defined below, I'd ...
0
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0answers
3 views

Double Integral Range [migrated]

we want the Surface between the two equation: $r= 1 + cos\theta$ $r = 1$ (circle) we can use a duble integral to solve this: $$ S = \int_{-\pi/2}^{\pi/2} \int_1^{1+cos\theta} r \,dr\,d\theta $$ ...
2
votes
1answer
63 views

Why does NDSolve and NIntegrate not give the same result?

I have plotted solution of two equivalent equations one in Integral form (right chart) the other in Differential form (left chart) using NDSolve and NIntegrate but they give me completely different ...
3
votes
1answer
450 views

Gâteaux (directional) derivatives and higher order differentials of a functional

I would like to calculate the Gâteaux derivative of a functional (i.e. a function depending on functions). A simple example for the Dirichlet functional: $L(u(x))=\int_0^1 \frac{1}{2} (u'(x))^2 dx$ ...
5
votes
2answers
242 views

Strange phenomenon occurring in analytic integration result involving Bessel functions

For the following integral, Integrate[x^2 Exp[-a x^2 - b x^4], {x, -∞, ∞}, Assumptions -> {a > 0, b > 0}] Mathematica gives the following analytic ...
0
votes
1answer
226 views

Asymptotic forms of Bessel function

I want to replace Bessel functions by asymptotic forms, so the question is: can I find the best ones with help of Mathematica? And if it's possible, how can I do it? Update How can I get with ...
5
votes
0answers
109 views

MacDonald formula for Modified Bessel Functions

How can I make Mathematica understand these two integrals? $$\int_0^{\infty} e^{-x \cosh{\xi}} d\xi = K_0(x)$$ $$\int_0^{\infty} e^{-\frac{1}{2} \Big( \frac{x y}{u} + u \frac{x^2+y^2}{x y} \Big) } ...
0
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0answers
26 views

Compute score for a set of data [on hold]

So let's explain my problem. I have a set of items which have a score from 0 to 100. This set is dynamic which means that several values are expected to be added from moment to moment. In each item is ...
0
votes
0answers
33 views

Replace higher order derivatives given lower order derivative

I have an unknown function f[x1,x2] whose partial with respect to x1 is identically zero. I would like to make sure that ...
1
vote
3answers
109 views

How to calculate the double integral of a double integral?

May I ask is there any wise way to calculate the following integration? $\int_{-1}^{1}\int_{-1}^{1}\int_{-1}^{1}\int_{-1}^{1}\frac{\left( 0.25 \left(-1+\eta _1\right) \eta _1 \left(-1+2 \eta ...
1
vote
1answer
51 views

Defining functions as derivatives

I'm working on a model that requires a sum over a function that uses various coefficients. In order to make the sum easier to perform, I am trying to keep things general through the use of a ...
3
votes
2answers
104 views

Problems with integrating over a region

Following the example of this answer: Find the volume of the region defined by $|x|+|y|+|z|<4$ I tried to compute the following integral over a region by using ImplicitRegion. However, when I ...
28
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1answer
3k views

How to visualize Riemann surfaces?

In WolframAlpha we can easily visualize Riemann surfaces of arbitrary functions, can we plot the Riemann surface of an arbitrary function using Mathematica and ...
1
vote
2answers
114 views

How to convolve the unit box function and the modified Bessel function of the second kind in 2D?

In 1D the convolution of the unit box function and the modified Bessel function of the second kind $K_0(x)$ works very well. ...
1
vote
2answers
223 views

How to compute this integral?

I'm trying to calculate this integral : $$I(z,k,a)= \displaystyle\int_{1}^{\infty} t^2\, \text{ArcTanh} \left(\sqrt{\frac{t^2-1}{t^2}} \dfrac{k}{z}\right)\, e^{-a\,t} \, dt$$ Where : ...
0
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0answers
27 views

NDSolve and Nintegrate give different plots of the same function? [closed]

So my problem is that I transformed a differential equation to an integral equation and the results from Nintegrate is different from what I get with NDSolve. Shouldn't it be the same? I was expecting ...
10
votes
1answer
1k views

How does a Pringle lose its curvature?

Nom! As part of a bigger project, I've was writing some code to calculate the scalar curvature of surfaces of the form $z = f(x,y)$. This uses a general calculation of the scalar curvature to produce ...
0
votes
1answer
82 views

Tedious integral

I'm new to Mathematica so I'm not familiar with its potential use to solve tedious symbolically integrals. I'm trying to solve the following one: ...
0
votes
1answer
50 views

integral involving square of exponential integral

I'm trying to compute the integral $$ \int_{a}^{\infty}\frac{e^x}{x}[\mathrm{Ei}(-x)]^2\,dx, $$ where $\mathrm{Ei}$ is the exponential integral, and $a>0$. The obvious ...
4
votes
1answer
1k views

How to take the derivative w.r.t. an arbitrary function?

EDIT: The answer posted by @Jens works for built-in functions only. However I'm starting to wonder if this is related to my Mathematica install somehow. I have version 7. Here is an example in the ...
1
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0answers
17 views
2
votes
1answer
144 views

How to linearize an expression involving derivatives?

I want to linearize the expression involving function $y(x,t)$ $$[y^3(y_x-y_{xxx})]_x$$ by assuming $y(x,t)=y_0(t)+y\prime(x,t)$, where $y\prime(x,t)$ is a small variation. In other words, I will ...
0
votes
1answer
43 views

Assumptions with Multiple Rules

What is the correct way to add more than one assumption while integrating? I am attempting to evaluate the following integral and Mathematica is just stalling, so I'm wondering if my assumptions are ...
6
votes
3answers
385 views

Integrate to calculate enclosed area

I am trying to 'use an integral in polar coordinates to calculate the area enclosed by this curve': The curve is: r=sin2θ, θ ∈ [0, π] which I believe is already in ...
2
votes
1answer
88 views

Dog chases his tail ! - “parametric differential/Integral equation”..?

I have the following situation where I am interested in the function $m(t)$ $$ \frac{dm}{dt}=4T(t)^{3}+T(t)^{2} $$ $$ T(\tau)=T_{0}-\int_{0}^{\tau}(\frac{dm}{dt})dt*Q_{S} $$ Is there a way to solve ...
1
vote
1answer
130 views

Differential Equations and Unit Notation

What do I get the Syntax::sntxi: Incomplete expression; more input is needed . error when I try to use the math palette for derivatives? What does the dot in ...
3
votes
2answers
110 views

How to plot a volume for triple enclosed by surfaces

My main issue (multivariate calculus), I want to plot the body made within the surfaces y + z = 4, y = 4 - x^2, ...
4
votes
1answer
184 views

Why can't the Bernstein Bears work normally?

Bug introduced in 7.0 and fixed in 9.0 I want to use the built-in BernsteinBasis[] to learn about Bezier curves. I tried the following code: ...
2
votes
3answers
196 views

Mathematica Integration Problem

I am new user of Mathematica. Can anybody help me to solve this integration? I really needs your help. Integrate[(1 - (1/(1 + A x^(-ao)))) x, {x, 0, r}] Can not ...
14
votes
2answers
1k 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 ...
4
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0answers
138 views

Integrate producing bad result

I noticed a bug in Mathematica. It computes incorrectly a definite integral ...
3
votes
2answers
72 views

Integral invoving product of Whittaker functions

I'm trying to evaluate the integral $$ \int W_{0,a}(x)\,W_{1,b}(x)\,\frac{dx}{x}, $$ where $W$ denotes the Whittaker $W$ function, and $a,b\in[0,1/2]$. Using the Whittaker differential equations that ...
0
votes
1answer
44 views

I would like to be able to get this integral evaluation to show up in terms of variables

this is very much a beginner question; I was hoping that someone may be able to take a look at this code and tell me why it outputs all of these crazy numbers instead of just the variables specified ...
2
votes
0answers
60 views

Cauchy Principal Value

Cauchy principal value of the following integration gives nothing in Mathematica. $\int\limits_{-\infty}^\infty\frac{\text{dx}}{x-2}$ I tried some hand calculation which is ...
-1
votes
1answer
38 views

Maclaurin Series Help [closed]

My problem is to numerically approximate the series (1 - Cos[x])/x over the interval [0,1]. I typed it into Mathematica as so: ...
0
votes
1answer
70 views

Faster code for calculat in Integral include derivative

I saw this question How I can get faster I tried help that. But I can not understand. I want to write a faster code for: ...
3
votes
2answers
138 views

Definite integration in mathematica

I tried to integrate the following in Mathematica. $\int\limits_{-\infty}^\infty\cfrac{\text{e}^{\text{i}ax}}{x^2-b^2}\text{d}x$ where $a,b>0$ The analytical result is well known. ...
5
votes
3answers
252 views

Complex limit gives wrong answer

I'm evaluating $$\lim_{z\to e^{i \pi/2 n}} \frac{z-e^{i \pi/2 n}}{z^{2 n}+1}$$ with Mathematica 9.0.1: ...
0
votes
0answers
57 views

Learning Mathematica for Math Majors [duplicate]

I just downloaded Mathematica and I'm taking a course in Multivariable Calculus. I realize there is a lot of complexity to Mathematica and there are Many types of learning resources out there. I was ...
5
votes
0answers
65 views

Huge difference after changing a fraction to decimal

I have a limit to calculate. With[{p = 1/2, q = 0.1}, Limit[a^q Integrate[Sin[x]/x^p, {x, a, Infinity}], a -> Infinity]] gives correct result ...
3
votes
0answers
75 views

Wrong limit involving sine?

Using Mathematica 9.0.0.0, I was trying to calculate the sum $$ B_n = \sum_{i=1}^n \sin^4\left(\frac{(i-1)}{(n-1)}\cdot\frac{\pi}{2}\right). $$ where I thought it would be $B_n = \frac{3(n-1) + ...
0
votes
1answer
67 views

How to perform this nth derivative operation in MMA?

I have to solve this following equation where I need to differentiate a function $n$ times. $$A=\sum_{n=0}^{m-1}\frac{s^n}{n!}(-1)^n\frac{d^nf(s)}{ds^n}$$ How to express the above equation in ...
0
votes
0answers
36 views

Recycle boundary condition

I'm having the following issue that can not find a solution , by using this boundary condition u[0, t] == 2 u[1, t] that Mathematica does not understand, someone ...