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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32 views

Partial derivative with Dirac Delta as initial condition

I am trying to reproduce the Faynman-Kac results from this Wikipedia page: Faynman-Kac This is the equation that I am trying to solve: $$ \frac{\partial w(t,x)}{\partial t} = \frac{1}{2} \frac{\...
2
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4answers
147 views

Verify $\frac{a_{20}}{a_{20}+b_{20}}$ in Mathematica, May help me?

The solution of $\frac{a_{20}}{a_{20}+b_{20}}$ is $-39$ from the recursive system of equations : \begin{cases} a_{n+1}=-2a_n-4b_n \\ b_{n+1}=4a_n+6b_n\\ a_0=1,b_0=0 \end{cases} This is taken from $...
1
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0answers
45 views

Derivative of a function with indexed variables

I want to show that $$ \mathcal{U}(\mathbf x)=\prod_{i=1}^n U_i(x_i) $$ is a concave function, where $$ U_i(x_i)=\frac{c_i}{1+e^{b_i-a_i x_i}}+d_i $$ and $a_i$, $b_i$, $c_i$, and $d_i$ are positive ...
0
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1answer
51 views

Infinite Sum - Result not correct for all cases?

Evaluating the Sum Sum[a^i, {i, ∞}] yields -(a/(-1 + a)) which obviously only holds ...
1
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1answer
62 views

Make integration/ plot run faster? (takes hours)

I am using Mathematica 10.0.2.0. Here is the code I am working with: ...
0
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0answers
35 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 ...
3
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1answer
68 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 ...
3
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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 ] ...
0
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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
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1answer
62 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
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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-...
2
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2answers
198 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,...
1
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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
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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
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1answer
80 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
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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 ...
6
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4answers
193 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 ...
5
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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 ...
11
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0answers
155 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 ...
0
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1answer
64 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}(...
7
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1answer
313 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
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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. ...
1
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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 ...
0
votes
1answer
76 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 ...
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, ...
0
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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: ...
0
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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: ...
3
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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 $A$ ...
1
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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 ...
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
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1answer
76 views

Drawing a tangent plane to the torus

I'm trying to draw a torus and a plane tangent to it. Here is my code so far: ...
0
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0answers
42 views

Numerical derivative step size

I am trying to perform a numerical derivative of some function using ND. I want to know if there is a way to set the step size of differentiation to the specific ...
0
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2answers
85 views

Multivariate integration of a compicated expression

I have an expression in multiple variables that is something like ...
7
votes
2answers
134 views

Inflection point and curvature

Through a coordinates list and then interpolating them I got a $f[x]$. ...
1
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1answer
46 views

Do double vertical bars in ConditionalExpression mean 'or'? [closed]

I have a very simple interpretation question. I think that double vertical bars in a ConditionalExpression given by Mathematica as the result of an integration must mean 'or', but I am hoping to ...
2
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0answers
110 views

Bug in the “PrincipalValue” method of NIntegrate?

Update 2 The technical support suggests setting AccuracyGoal. As I quote here It does appear that the program is returning an incorrect result along with a ...
1
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3answers
83 views

Evaluating $\int_0^\infty dx /{(x^2-a^2)^2+b^4} $ returns a negative number

I am trying to evaluate $\int_0^\infty dx /((x^2-a^2)^2+b^4)$. And I seem to have an error that I cannot detect. Using ...
1
vote
3answers
72 views

Derivative of a 2D Interpolating function

I cannot find a way to obtain the partial derivatives of a 2D Interpolating Function obtained with ListInterpolation. I didn't find anything useful on the web. Any ...
2
votes
1answer
67 views

Einstein summation and traditional form of partial derivative of implicit function

I am quite new in Mathematica (1 week new) and after many trials I did not manage to make my partial derivatives to look traditional way. There are few topics such as How to make traditional output ...
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 ...
-1
votes
0answers
37 views

Coordinate transformation for gradients expressions [duplicate]

Let's suppose I have an expression in cartesian coordinates, involving gradients, such as: F[x,y]:=D[f[x,y],x]/D[f[x,y],y]+D[f[x,y],y,y] f[x,y] Now, I would like ...
1
vote
1answer
102 views

Computing integrals in Mathematica

I am trying to actually compute $\int \sqrt{\cosh{y}-\cos{x}}e^{inx} dx$ in mathematica. As an example i tried computing $\int \sqrt{1-\cos{x}}e^{inx} dx$ and got a result. But when i try $\int \...
0
votes
1answer
67 views

Can this expression be calculated symbolically?

I want to do definite integration of the following expression ...
1
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1answer
82 views

integrating with multiple indicator functions depending on each other

I need to compute the following integral with three indicator functions, $I(x)$, $I_1(x_1)$ and $I_2(x_2)$, $$ \int_0^1\int_0^1I(x)\left( \ I_1(x_1)\,x_{1} + I_2(x_2)\,x_{2}-1 \right) dx_1 dx_2 $$ ...
1
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1answer
69 views

Evaluating Bessel-Schläfli-type Integral with Infinite contour

I am attempting to evaluate the integral \begin{equation} \int_{\infty-i\pi}^{\infty+i\pi} dt \frac{e^{-\frac{l}{n}t}}{[(X'-X)^2+r^2+s^2- 2rs \cosh(t)]}, \end{equation} the only variable being $t$, ...