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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6
votes
1answer
279 views
+50

How do I numerically evaluate and plot the Fabius function?

The Fabius function is a well-known example in analysis of a non-analytic function that is infinitely differentiable. I want to be able to numerically evaluate the function for any real argument, as ...
0
votes
1answer
37 views

Infinite Sum - Result not correct for all cases?

Evaluating the Sum Sum[a^i, {i, ∞}] yields -(a/(-1 + a)) which obviously only holds ...
0
votes
0answers
50 views
0
votes
0answers
34 views

What is wrong with my formulas for a mathematical model of a double pendulum? [on hold]

I wanted to create a computer simulation on Matlab, using a model for a pendulum from this study (A double pendulum model of tennis strokes. Rod Cross. Uni of Sydney, 2006) - Link I wanted to use the ...
0
votes
1answer
396 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 ...
0
votes
1answer
67 views

Can this expression be calculated symbolically?

I want to do definite integration of the following expression ...
2
votes
2answers
194 views

Curve fitting with this modified integrated log-normal function and extracting fitted?

The data to be fitted is for y vs x. We have y and x points. We do not have data for w. In fact, w variable is being varied during integration from 1e-9 to 100e-9. The fitting parameters are Ms,sigma,...
3
votes
1answer
140 views

Incorrect result when integrating

Mathematica 10.4.1.0 produces Integrate[ Log[1 + 2 a*Cos[2 x] + a^2]*Sin[x]^2, {x, 0, Pi/2}, Assumptions -> a > 1 ] ...
9
votes
4answers
3k views

Is it possible to “pretty print” my input integral in Mathematica like Wolfram|Alpha does?

With Wolfram|Alpha, if I enter: Integrate[r^3, {t, 0, 2Pi}, {r, 0, 2}, {z, r^2, 4}] Wolfram|Alpha will pretty print my input like so: Which is very readable, ...
5
votes
5answers
188 views

How to find the derivative value at $(\pi,0)$ for this implicit function $n$ times?

I am trying to take the implicit derivative at $\sin(x+y)+\sin(x)=y$ and substitute $x=\pi$ and $y=0$ at least 6-7 times since I need to find the Taylor series for this function. Since I barely ...
3
votes
1answer
67 views

Rule for derivatives

I have a differential equation of the form $$f'(x)=g(x,f(x))$$ (where $g$ is has a known explicit form, e.g. $x f(x)$) and I have an expression which contains several derivatives of $f$. I want to ...
0
votes
1answer
61 views

how can I get the right derivatives in this Sum?

I want to get the derivatives of $q_i$ of the following function Q in Mathematica: $$Q=\sum _{i=1}^N q_i$$, Using the command of $$\frac{\partial Q}{\partial q_i}$$, the software give me an answer of $...
0
votes
0answers
16 views

Seconde Derivatives of an integral [closed]

$$ g(x) = \int_{-\infty}^{x^2/2} e^{-(x^2+1)t^2}dt\,. $$ Trouble calculating the first and the second derivative of this function.
0
votes
0answers
64 views

How can I verify this limit? [closed]

Hello I have to verify this limit $$ \lim_{N \to \infty} \frac{2 \pi}{N+1}\sum_{n=0}^{N}\sin^2(q_n)\frac{1}{E_n}\left [ \delta(\omega -E_n)+\delta(\omega+E_n) \right ] = \mathrm{sgn}(\omega) \Theta(1-...
6
votes
3answers
418 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: $r=\sin 2\theta$, $\theta \in [0, \pi]$ which I believe is already in polar form. I ...
0
votes
0answers
46 views

Line Curvature and Gaussian Curvature [closed]

Use Mathematica to calculate the curvature of the line: Then calculate the Gaussian curvature of the plane Ax+By+Cz=D, in which A, B, C, and D are constants and C≠0 Here are some ...
1
vote
0answers
44 views

Does Mathematica interprets representations of Dirac's Delta? [duplicate]

Hello I'm trying to take the limit of this expression in Mathematica I know this is a representation of Dirac's Delta (more precisely pi times Delta), but when I do this, the answer is 0. So my ...
0
votes
2answers
183 views

Integrating SquareWave - don't understand result

Integrate[SquareWave[t], t] // InputForm Gives as output Piecewise[{{-t, Inequality[1/2, LessEqual, Mod[t, 1], Less, 1]}}, t] ...
0
votes
1answer
63 views

Numerical integration of a numerical integration [closed]

I have a complicated function $f(x)$ whose analytical form does not exist for the following integral $$g(x)=\int_0^x\mathrm{d}x'\,f(x'). $$ I finally need to evaluate the following integral $$\int_0^...
0
votes
1answer
78 views

Integration of harmonic function

I am trying to calculate the average distance of electron above the surface. However when I did the Integral, for the first couple of state(n from 1 to around 15) the results are pretty good. However, ...
0
votes
2answers
84 views

Defining the Hirota operator

The Hirota $D$-operator (derivative) is mathematically defined as follows: $$D_x^n f\cdot g=\left.\frac{\partial^n}{\partial s ^n} f(x+s)g(x-s)\right|_{s=0} $$ An example of this operator acting on ...
2
votes
2answers
159 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. ...
6
votes
4answers
192 views

Why does Mathematica return Indeterminate for this converging infinite sum?

Limit[Sum[k/(n^2 - k + 1), {k, 1, n}], n -> Infinity] This should converge to 1/2, but ...
11
votes
0answers
154 views

integrate giving wrong result

Bug introduced in 7.0 or earlier and persisting through 10.4.1 or later A question with the same title has been asked many times, but I can't find the solution of my problem in their answers. The ...
10
votes
0answers
161 views

Strange behaviour of integrals with Cos, Sin, and Exp

During the study of the problem How to solve this integration? I have discovered a strange behaviour of some integrals. I would consider it a bug. ...
-2
votes
1answer
365 views

Why doesn't Mathematica evaluate this limit? [closed]

I am trying to evaluate a few limits in Mathematica, and I can only get it to evaluate them in certain cases. I don't fully understand the evaluation order and when it substitutes numerical values in ...
5
votes
1answer
44 views

Modified Derivative Definition Does Not Thread Over Sum

I'm redefining how a derivative of a polynomial is taken. I have polynomials or posynomials where x is to a fractional power, and derivatives are of integer order. But if the order of the derivative ...
0
votes
1answer
69 views

Strange behavior of derivative when using Abs[] [duplicate]

This code where I try to visualize Newton's Method on the function $f(x)=\sqrt{|x|}$ works fine when I define $f(x)$ piecewise. Then why doesn't it work when I define ...
3
votes
2answers
241 views

NIntegrate and Integrate giving different results for ill-behaved function

I'm trying to integrate the following function with Mathematica 8: $$ I(a,b)= \int_0^1 \mathrm{d}x\int_0^1\mathrm{d}y \,\theta(1-x-y) \frac{1}{x a^2-y(1-y)b^2},$$ where $\theta$ is the Heaviside ...
2
votes
2answers
255 views

Incorrect numerical derivative of function that uses FindRoot

I am trying to plot the derivative of function g[x] below where g[x] is defined as the root of another equation. However, I am ...
0
votes
2answers
97 views

Integral of rational function is indeterminate

I need to integrate the rational function $$\frac{t^n}{bt+t^2+1}$$ where $n$ is a integer. Of course, it's no problem for Mathematica to do this. ...
8
votes
1answer
271 views

Incorrect evaluation of integral involving a DiracDelta, whose argument has infinitely many zeros

Let's say $X>0$ is a random variable with probability density $p_X(x)={\rm e}^{-x}$. Define the random variable $Y=\sin(X)$. From the transformation theorem for probabilities we know that its ...
3
votes
0answers
113 views

Obtaining a Previously Constructed Function as Desired Output

I was playing around with some Generalized Trigonometric Functions defined as follows; ...
0
votes
1answer
63 views

Dealing with numerical integration singularities

I'm trying to perform the following integral (numerically) (1) And their behaviour is where PV denotes principal Value. In principle I don't know the functions $\rm{Im}(\alpha)$ and $\rm{Re}(...
3
votes
1answer
55 views

Differentiation and series expansion of dot product - inconsistent results

If I differentiate a dot product, I get the result I expect D[a.b[x], x] (* a.b'[x] *) However, a series expansion of the same expression gives a very different ...
0
votes
1answer
75 views

Cauchy integration [closed]

I am evaluating Integrate[1/(1 + 3*cosin(x)*cosin(x)), {x, -Pi, Pi}] in Mathematica, which should give me Pi, but instead I ...
1
vote
0answers
38 views

Setting derivative of function equal to 0 at x=0 using UpValues

I'm trying to set the derivative of a function to zero at a specified point. This will act as an initial condition for an algorithm I'm writing: so I'm specifying the value of the function at x=0, and ...
0
votes
1answer
51 views

Constructing a root solution diagram (not to tell bifurcation)

Suppose I have a third order equation depending of one parameter say .5 4 x (1 - x) (x - .5) - .5 m x^2 where m the parameter. Obviously there are 3 roots for ...
2
votes
2answers
77 views

Integrate over a piecewise function [closed]

I want to calculate the indefinite Integral of $$f(x)=\begin{cases} 2x\cos(\frac{1}{x})& \text{ if } x\ne 0 \\ 0& \text{ if } x=0 \end{cases}.$$ I use the following code: ...
-1
votes
1answer
42 views

How can I solve this integration numerically? [closed]

how can I integrate this function numerically w.r.t r from 1 to 8, the function is as, ...
1
vote
2answers
124 views
0
votes
0answers
39 views

NIntegrate and Integrate seem to over-estimate an integral involving a sharp Lorentzian (Cauchy) distribution

I am trying to do a numerical integral with NIntegrate, but it seems that I get a bigger number than I should. The integral is fairly simple: ...
20
votes
4answers
12k views

Finding Limits in several variables

Is there a way to find a limit of a multivariable function, like $$\lim_{(x,y)\to (0,0)} f(x,y)$$ with Mathematica? When $f$ is continuous, we can use $$\lim_{(x,y)\to (0,0)} f(x,y)=\lim_{(x,0)\...
0
votes
0answers
34 views

NSolve on a function with NIntegrate

I have a function defined with an NIntegrate. The function is of x,y,z, integrated across t: ...
1
vote
1answer
64 views

Collect partial derivatives and rewrite them with delta/nabla

Ultimately, I wish to get the higher generalized product rule for Laplacian. That is, we have $\Delta(fg)=f\Delta g+2\nabla f\cdot\nabla g+g\Delta f$ but what is $\Delta^n(fg)$? I am tying to guess ...
3
votes
2answers
123 views

Using `Limit` with the Option `Assumptions`

I want to calculate the following limit $$L=\lim_{r \to 0} \left( 1 + r \frac{El^{'}(r)}{El(r)} \right)$$ by letting Mathematica to know $$\lim_{r \to 0} \frac{El^{'}(r)}{El(r)} = A $$ where $A$ ...
0
votes
1answer
87 views

How is Integrate Implemented? [duplicate]

At the end of this 90 minute lecture by Stephen Wolfram on the history of the development of Mathematica, Wolfram takes questions from the audience. One of those questions asked how the ...
1
vote
1answer
462 views

Plotting derivatives of B-splines

I am trying to solve an electrodynamics problem numerically by using B-splines a basis function. Please don't ask me why, but if you really wanna know, here is the assignment. Using the following ...
0
votes
2answers
84 views

Multivariate integration of a compicated expression

I have an expression in multiple variables that is something like ...