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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0answers
51 views

Reducing a long complex expression to a simplier one by using assumptions

so I have an ugly looking equation that I wish to solve under some specific circumstances. Using DSolve I write; ...
4
votes
1answer
165 views

How do I compute the entropy of the beta distribution?

I tried Expectation[-q*Log[q], q \[Distributed] BetaDistribution[a, b]] and got ...
1
vote
1answer
194 views

Gram Schmidt inner and outer products

I know the Gram-Schmidt orthogonalization generates an orthonormal basis from an arbitrary basis. I need help with: Write a program that inputs a list $\{b_1,\dotsc,b_n\}$ of linearly independent ...
0
votes
1answer
138 views

Plot[ D[Sin[x]] ] and Evaluate[] [duplicate]

Why does t = 2 Pi; Plot[D[Sin[x],x], {x,0,t}] (* Plotting the derivative of Sin[x] *) not work, but ...
2
votes
2answers
122 views

Limit of integral gives incorrect output

I'm trying to evaluate $\lim_{e\to 0} \, \frac{i}{e}\int_{\pi }^0 \frac{1-\exp (i e \exp (i \theta ))}{\exp (i \theta )} \, d\theta$ with Mathematica 9.0.1.0 on OS X. However, I get "Undefined" for ...
11
votes
3answers
187 views

How to efficiently find moments of a multinormal distribution?

I have a multinormal distribution with covariance matrix $\sigma$ and zero mean. I want to find moment $$ E[x_1^{r_1}x_2^{r_2}\cdots x_n^{r_n}] =\,? $$ Of course, there is a build-in function ...
14
votes
2answers
1k views

How to deal with complicated gaussian integrals in Mathematica?

As we know, for most Gaussian integrals, we can get the analytical result. Now I have many Gaussian integrals to treat, which have the following general form, ...
1
vote
3answers
172 views

plotting an Integration output

How to solve this integral by Mathematica even by numerical methods (plotting the solution) Integrate[(Cos[x] - a)/(1 + a^2 - 2*a*Cos[x])^1.5, {x, 0, 2*Pi}] It ...
6
votes
3answers
246 views

Where did I go wrong with my implementation of the trapezoidal rule?

One method for doing quadrature, called the trapezoidal rule, improves accuracy by connecting the points on the curve corresponding to the points of subdivision with line segments, forming trapezoidal ...
4
votes
2answers
225 views

How do I evaluate a symbolic integral involving Hermite polynomials?

I want to test a difficult integral : Integral on all reals of some complicated function involving the Hermitian polynomials, exponentials, squares, factorials, and being general considering any ...
0
votes
0answers
120 views

Approximating a stochastic integral with a Wiener process

I would like some assistance to solve the following: $\sum\limits_{i=0}^{n-1} e^{-k(n\Delta t -i\Delta t)}\Delta z_i$ where $z$ is a Wiener process, $\Delta z_i = z((i+1)\Delta t)-z(i\Delta t)$. ...
3
votes
3answers
177 views

What is the formula for this numerical series?

I'm developing a questions game. My goal is that the score for each correct answer will increase as the user answers more questions. Initially there are 15 points for each correct answer. Every 4 ...
10
votes
2answers
545 views

Derivative from the left and right

I am currently trying to write a script that takes a curve $C$, a starting point $p$ and a direction vector $\vec{v}$ and calculates the path of a beam starting at $p$ in direction $\vec{v}$ when it ...
9
votes
2answers
291 views

Recursive Integral for Volume of $n$-Ball

The volume of an $n$-ball (the $(n+1)$-dimensional analogue of a disk) of radius $r$ can be found by the following integral recurrence: $$V_0(r)=2r$$ ...
4
votes
3answers
189 views

Why does Integrate set the constant of integration to be one in this case?

Why does Integrate[(4 x)/(2 x + 1), x] give 1 + 2 x - Log[1 + 2 x] Notice the extra ...
13
votes
4answers
649 views

Mathematica gives wrong result while Wolfram|Alpha is correct

Why does Mathematica and Wolfram|Alpha give different results based upon the same code? I know Wolfram|Alpha's 7.85 is correct.
1
vote
3answers
401 views

How many iterations of Newton's method are needed to achieve a given precision?

Consider using Newton's method to solve the equation $arctan(x) = 0$. Using an initial guess of $x_0 = 1/2$ produces a sequence that converges rapidly. After $8$, iterations, $x_8$ is accurate to well ...
0
votes
1answer
128 views
1
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0answers
68 views

Help with Integrals (and conditional expectations) of the Beta distribution: Integrate[e^(az) z^a (1-z)^b, {z, 0, 1}]

I have a Beta distribution, and am interested in calculating expectations and conditional expectations. The domain on the distribution is $z \in [0,1]$ Ignoring constants of proportionality, the ...
1
vote
1answer
53 views

Am I missing something in this integral?

I am trying to solve this simple integral: ...
9
votes
1answer
253 views

Mathematica: computing a difficult integral

I am trying to compute the following integral: Integrate[Exp[Sum[-((cw λ - b[i])^2/(2 σ^2)), {i, 1, n}]], {cw, 0, 1}] And currently Mathematica outputs ...
5
votes
2answers
553 views

How to find the non-differentiable point(s) of a given continuous function?

For example, the non-differentiable point of the function $f(x)=|x|$ is at $x=0$. How to find the non-differentiable points of a continuous function that is defined numerically?
0
votes
1answer
997 views

Given x[u,v], y[u,v], and f[x,y] how do I find D[f,x] in terms of u,v?

I have two functions x = x(u,v) and y = y(u,v). The chain rule says that for a function f(x,y), df/dx = df/du du/dx + df/dv dv/dx, and similarly df/dy = df/du du/dy + df/dv dv/dy. How do I get ...
0
votes
1answer
445 views
0
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1answer
94 views

How do I generate arbitrarily many integration bounds? [duplicate]

I have a multiple integral of the following form: $$\int_{-\infty}^\infty \cdots \int_{-\infty}^\infty \exp(-\sum_{i,j=0}^n x_i A_{ij} x_j) dx_1 \cdots dx_n $$ Here, $A$ is a square matrix and $x_1 ...
3
votes
1answer
125 views

Double Integral and Assumptions

I'm afraid this is going to be a really stupid question. Evaluating the following definite integral Integrate[Sqrt[1 - (x^2 + y^2)], {x, -1, 1}, {y, -1, 1}] ...
7
votes
2answers
288 views

Find asymptotics of Sum[2^i*Binomial[n-i-1,2*n/3-1],{i,0,n/3}]

I have an expression 2^n / Sum[ 2^i Binomial[ n - i - 1, 2n/3 - 1], { i, 0, n/3}] ...
10
votes
5answers
492 views

Double series over primes

I'm very curious if the following double series over primes has a closed form: $$\sum_{k \in \mathcal{P}}\sum_{n \in \mathcal{P}}\frac{1}{k\;n(k+n)^2}$$ where $\mathcal{P}$ denotes the set of all ...
6
votes
1answer
436 views

Integral of the Sinc product

Let as consider the following integral $$ B_n = \int_0^\infty \prod_{k=1,3,5,\dots}^n\frac{\sin (x/k)}{x/k}dx $$ By definition, Sinc[x]=Sin[x]/x therefore ...
4
votes
3answers
285 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
vote
1answer
138 views

Minimum value with two factors

Approximate the minimum value of $2e^{x-2}-6x+xe^{x-2}$. Is my input correct? If not can you please correct it? ...
10
votes
1answer
350 views

Why does Mathematica return a Fourier transform for a function for which it is not defined?

The following function $$g(x) = (1 + x^{1/a} )^a $$ should NOT have a Fourier transform, as far as I am aware, for any real values of $a$ since $g(x)$ is not nice in the sense of decays quickly ...
3
votes
5answers
189 views

A double series with divisibility restrictions

How may I restrict $i,j$ such that they run over $\mathbb{N}$ excepting the numbers divisible by $2$ or $3$ or $7$? ...
1
vote
3answers
115 views

How to iteratively integrate a function and sum the result of each iteration

I want to repeatedly integrated a function and add the result of each iteration. Suppose I start a with a function $f(x) = x$ , I want to integrated it, it will be $x^2/2$, and again integrating, it ...
0
votes
2answers
84 views

How to add a condition to a formula so that it will not appear at the result? [duplicate]

Sometimes I have very long and complicated equations, when I evaulate the equations for example taking derivative or integral there will be several conditions with the answer. My question is that; is ...
1
vote
0answers
120 views

How to prevent simplification of hypergeometric functions resulting from integrations?

Definite integrals from 0 to Infinity over a product of two hypergeometric (including exponential, trigonometric, hyperbolic, ...
3
votes
2answers
249 views

Integrating special functions

I would like to integrate the following function with Legendre polynomial and Gamma function. I am open to suggestions. ...
1
vote
1answer
496 views

Definite integral using Riemann sums [closed]

I just started learning Mathematica. It seems like a good program to learn. I only know some basics though. I came across this problem online which I want to learn how to solve: Write a ...
3
votes
1answer
112 views

Making time differentials look like the textbook [duplicate]

I need to have time differentials to look like the 'textbook'. My code is Dt[x y^2] /. {Dt[x] -> dx/dt, Dt[y] -> dy/dt} which gives the output ...
8
votes
1answer
346 views

Integral with HeavisideTheta takes too long to evaluate

I tried to compute $$\int_{-1}^1 d x_1 \int_{-1}^1 d x_2 \int_{-1}^1 d y_1 \int_{-1}^1 d y_2 \theta(x_1 x_2 + y_1 y_2)\,$$ where $\theta$ is Heaviside's step function, by using ...
4
votes
1answer
880 views

Is it possible to impose “Assumptions” in DSolve or equivalent? (differential equation solving)

When DSolve tries to solve differential equations, it is sometimes not smart enough to generate conditions according to different possible value of parameters. ...
2
votes
2answers
76 views

ConditionalExpression Limiting Domain

I came across a problem when evaluating the following integral: $\int_0^t \sqrt{9 x^4+1} \, dx$ Now, when I evaluate that, I get ...
1
vote
1answer
412 views
2
votes
1answer
379 views

Volterra integral equation : 'literally match the independent variables' message

I want to solve an Volterra type Integral equation, and as a practice I entered the following code to my mathematica: ...
2
votes
1answer
94 views

Use Mathematica to determine the falling law

We have a one-variable equation $\rho(R)$ where ρ = (14656.4+277.526*R^2)/(45.9225+R^2)^{5/2} + 0.370036/(R*(0.25+R)^3) This equations describes the evolution of ...
17
votes
2answers
461 views

Have I found a bug in Integrate?

The following command gives 0 in Mathematica 9.0.1. ...
1
vote
0answers
152 views

Positive integrand giving negative answer

I'm integrating a positive function f(t) times sin(t) from 0 to pi/5 and get -38. Actually f is slightly negative for a short time (smallest value ~ -0.0005), but far from enough to explain this. ...
7
votes
1answer
220 views

Series expansion for small ratio of variables

I have a messy expression of variables that I would like to simplify under the assumption that certain ratios of the variables are small. For example, consider $\sqrt{x+y}\;$ expanded for small ...