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
votes
1answer
62 views

Module inside of Manipulate Example

Here is a very nice example from The Student's Introduction to MATHEMATICA. ...
0
votes
1answer
233 views

Integration of linked PDFs over probability simplex

I'm trying to integrate the following expression: f[a1,p1+p2+p3+p4]*f[a2,p2+p3+p4]*f[a3,p3+p4]*f[a4,p4] Where ...
2
votes
0answers
393 views

Simple contour integral with a parameter gone wrong

Bug introduced in 7.0 and fixed in 7.0.1 I run into the following problem, I tried to evaluate a very simple integral: ...
16
votes
0answers
2k 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
votes
0answers
27 views

Another Integration to prove [on hold]

We want to prove this integration $\displaystyle\int_{-\infty}^{\infty} \frac{x sin (x)}{1+x^2} = \frac{\pi}{e} $
2
votes
0answers
66 views

Minimax for conditioned UnitStep functions

I made a simplified version of my problem, and I have now: ...
5
votes
1answer
72 views

Integrate yielding a ConditionalExpression but I don't think the condition is necessary

Suppose I take the PDF of the LogNormal distribution with parameters m and s evaluated at x. I obviously get an expression involving m. I now want to integrate that expression not with respect to x ...
8
votes
1answer
232 views

Why don't products of Dirac deltas integrate correctly?

Bug introduced in 10.0.0 and fixed in 10.0.1 The integral $\int \int \ \delta(x) \delta(y) \ dx dy=1$ evaluates to 0 in Mathematica ...
1
vote
1answer
47 views

How to apply Simplify only to elements of list that cause specific Warning

I have a set of replacement rules I use to analytically compute nested integrals of very long symbolic expressions since Integrate literally takes forever. To do ...
3
votes
1answer
169 views

Derivatives of functions with arbitrary number of variables?

I am trying to define a function f[x1,x2,...,xn] with n integer but not specified. And then I would like this funktion to behave ...
1
vote
0answers
35 views

Discrepancy between laplace and fourier transform of gaussian

The function in question is Exp[x^2/2] for x>0 Laplace transform should be the same as the fourier transform since the function is absolutely integrable. Fourier: They are indeed the same ...
2
votes
3answers
349 views

How can I apply calculus to functions obtained from NDSolve?

Originally, I asked the question below, but the real underlying issue is as follows: When we solve an ODE numerically, I get the answer like this: ...
0
votes
1answer
51 views

Partial derivatives with respect to multiple variables

For any natural number $N$: $$_t k_{mn}^1 = \frac{1}{m!n!}\frac{\partial^{n+m} k_1 (0, 0)}{\partial x^m \partial t^n}, \qquad (m,n=0,1,\ldots, N)$$ where $k_1(x,t)$ is a known function, for example ...
9
votes
1answer
195 views

Why doesn't Mathematica evaluate this simple limit?

I want to evaluate $$\displaystyle\lim_{n\to\infty}\left(n-\sqrt{\sin(n)+10n+n^2}\right)^2$$ I used this code ...
0
votes
1answer
71 views

Strange behavior of `FourierSinCoefficient`

FourierSinCoefficient works normally like this and the result is checked with Integrate. ...
5
votes
1answer
173 views

Wrong Limit in Mathematica 10.0.0

Bug introduced in 9.0.0 and fixed in 10.0.1 Limit[(Log[(3+Sqrt[5])/2]/(2*Log[(1+Sqrt[5])/2]))^(-1-2*n), n -> Infinity] Mathematica (wrong) output: ...
4
votes
2answers
126 views

Finding the equation of a level curve

Is there a way in Mathematica to determine a parametric equation of a level curve in Mathematica. For example, consider: ...
0
votes
2answers
76 views

Easy way to add another parameter to Manipulate

Vector definitions: I'm sorry that I didn't include the Vector definitions. I forgot because I had them in initialization cells. Here they are: ...
8
votes
2answers
238 views

Make Sinc'[0] return 0 instead of Indeterminate

I want Sinc'[0] to return 0, but instead it returns Indeterminate. I've tried Unprotect[Sinc] Unprotect[Derivative] Derivative[1][Sinc][0] ^= 0 But it doesn't ...
14
votes
2answers
367 views

Symbolic area calculation for a parametric self-intersecting closed curve

The parametric equation of the curve is: $$\begin{cases} x &= -9 \sin (2 t)-5 \sin (3 t) \\[6pt] y & = 9 \cos (2 t)-5 \cos (3 t) \end{cases}\quad t\in[0,2\pi]$$ which can be easily ...
0
votes
1answer
63 views

Integrate with symmetric volume assumption [closed]

My problem is the following: I have the following matrix: S = {{0, -z, y}, {z, 0, -x}, {-y, 0, x}}} The integral of this matrix over any symmetric domain is 0. ...
3
votes
3answers
328 views

$L = \lim\limits_{z \to 1} \frac{1-z}{1-z^\ast}$ evaluates incorrectly

I'm about 90% certain this is a Mathematica issue, not me making a silly mistake. I'm trying to evaluate $$L = \lim_{z \to 1} \frac{1-z}{1-z^\ast}.$$ Naively entering ...
3
votes
1answer
95 views

Strange Integrate messages / $RecursionLimit being ignored

Bug introduced in 10.1.0 and fixed in 10.2.0 The following behavior was observed in Mathematica 10.1 (Windows 64 bit) When attempting to evaluate the following integral, Mathematica outputs ...
0
votes
0answers
54 views

Formally invert a $C^\infty$ diffeomorphism

I'm having a problem with a calculation. Starting with $f: \mathbb R^2 \to \mathbb R^2$ having an invertible gradient, I want to compute $\nabla^2(g \circ f^{-1})$. While I know $f$ and $\nabla f$ ...
-2
votes
0answers
49 views

order of convergence of iterative method [closed]

can anyone tell me whats the error in the code as it runs first time accurately and output the required result. but the second time the code started giving errors ...
9
votes
2answers
173 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}] Mathematica 9.0 is ...
2
votes
1answer
41 views

Limit problem calculating directional derivative

Given $f(x,y)=1-x^2-y^2$, find the directional derivative at the point $(x_0,y_0)$ in the direction of the unit vector $\vec u$. I am experiencing some strange behavior with the Limit function. I ...
4
votes
1answer
402 views

Strange result for the analytic integration leads to Hypergeometric2F1

The integration result for Integrate[1/(r^2 Sqrt[x/r^(4 - 2 \[Gamma]) + 1]), r] is: ...
21
votes
2answers
856 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 ...
4
votes
2answers
138 views

A Chain rule proof using Mathematica

If $z=f(x,y)$, where $x=r \cos\theta$ and $y=r\sin\theta$, how can I use Mathematica to prove that: $$\frac{\partial^2z}{\partial x^2}+\frac{\partial^2z}{\partial y^2}=\frac{\partial^2z}{\partial ...
0
votes
1answer
132 views

Taylor Series vs. Series Function [closed]

Could someone please explain why the these two functions give two different results ...
4
votes
0answers
115 views

DiracDelta and version 10.0.0 [duplicate]

Note: This issue seems to affect version 10.0.0 only and is fixed in 10.0.1, 10.0.2, 10.1 and 10.2 When evaluating the following two inputs: ...
2
votes
0answers
104 views

Function for the Second Derivative Test

I wrote the following function. It is based on Mathematica for Rogawski's Calculus, 2nd Ed, 2007, Based on Mathematica 7. See: http://users.rowan.edu/~hassen/Math_Rogawski_Calc.htm, Chapter 14. I made ...
2
votes
0answers
69 views

How can I get constant of integration? [closed]

When using Integrate, Mathematica sets a constant of integration automatically. But sometimes we are given that constant already and it should be used in order to ...
9
votes
2answers
673 views

Problem with numerical evaluation of analytically solved integral, solution way off

The following command in Version 9.0.1: N[Integrate[x^50*Sin[x], {x, 0, 1}]] gives $1.4615\times 10^{48}$ which is way off from the correct solution which is ...
7
votes
1answer
211 views

Suspected bug in Integrate

Bug introduced in 9.0.1 or earlier and fixed in 10.1.0 In version 10.0: ...
1
vote
1answer
80 views

How to simplify and evaluate Integral with lots of DiracDelta dists

I have a question about an Integrals which I can do by hand but I want to implement this Integral in Mathematica. Actually it is the definition of a 2 Particle Phase Space. The Expression is the ...
-1
votes
1answer
64 views

How can I prove the following equality [closed]

I have the following equality : $$\int_0^\pi \frac{\cos(2t)}{a^2\sin^2(t)+b^2\cos^2(t)}dt=\frac{a}{a+b}-\frac{1}{2}$$ where $0 <b \leq a$. I used the residues but I could not prove this equality
1
vote
1answer
73 views

Multivariable Calculus Chain Rule

I am getting ready to write a notebook for the chain rule in Multivariable Calculus for my students. I do know: ...
10
votes
1answer
164 views

Dirichlet coefficients as limits: wrong

Perhaps I should have included the word "bug" in my question. Here we go There is a bug in this Limit Limit[3^s (-1 - 2^-s + Zeta[s]), s -> ∞] (* 0 *) which ...
0
votes
1answer
45 views

Setting trigonometric polynomial terms to zero

Consider an expansion $$\eta = a\cos (kx-\omega t) +\frac{1}{2} a^2 k \cos 2 (kx-\omega t).$$ I would like to compute expansions like $$ V = \frac{1}{2\pi} \int _0^{2\pi} \eta^2 \ d\theta ,$$ ...
1
vote
0answers
44 views

Numerical Derivative after numerical integration [closed]

I am trying to find the numerical derivative of a function whose argument defines the bounds of a numerical integral. ...
1
vote
1answer
71 views

Manipulate to determine $\delta$ given $\epsilon$ in continuity question

This is a continuation of a question I posed at: Examining the function $f(x,y)=xy(x^2-y^2)/(x^2+y^2)$. The quest is to analyze the partial derivative $$ f_x(x,y)=\begin{cases} ...
9
votes
2answers
393 views

Why do I get a different value when I change the order of integration?

I think the following two-dimensional integrals should be equal, since they both integrate the function over the half plane defined by $t>\tau$. $$\int_{-\infty}^\infty \mathrm{d}t ...
8
votes
1answer
133 views

LogisticSigmoid residue bug?

Bug introduced in 10.0 and fixed in 10.2.0 On version 10.0 for arm: Residue[LogisticSigmoid[z],{z,I Pi}] gives 0 not 1. ...
2
votes
2answers
115 views

Dealing with Norm, Complex situation not desired

I am working on a notebook for my calculus students and am dealing frequently with the norm of vector valued functions. I always seem to run into this kind of situation. ...
5
votes
1answer
75 views

Examining the function $f(x,y)=xy(x^2-y^2)/(x^2+y^2)$

Consider the function $$f(x,y)=\begin{cases} xy\dfrac{x^2-y^2}{x^2+y^2},&(x,y)\ne(0,0)\\ 0,&(x,y)=(0,0) \end{cases}$$ I can show that $f_x(0,0)=f_y(0,0)=0$. ...
2
votes
1answer
144 views

Evaluating a surface integral

I'm trying to compute the integral $$\int_S x^2+z^2\,{\rm d}S,$$where $S$ is the surface $$S\colon~ \frac{x^2}{2} + \frac{y^2}{3} + \frac{z^2}{2} = 1, \quad y \geq 0.$$ One possible parametrization ...
5
votes
2answers
375 views

How can I numerically solve for fractional functions and fractional derivatives?

I would like to plot fractional functions. Say, $f(x)=\sin^{(1/2)}(x)$. By that, I mean that $f(f(x)) = \sin(x)$. Similarly, I can define a half-derivative to be an operator $H$ such that ...