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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2
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1answer
95 views

Integrate and ConditionalExpression Output

I am integrating some difficult functions and using the conditional output to place bounds on parameters to ensure integrability, but it seems that not all of the conditions are being returned. The ...
0
votes
0answers
57 views

limit calculation step by step [duplicate]

I calculate such limit with Mathematica: ...
-1
votes
3answers
1k views

How do I determine the maximum value for a polynomial, given a range of x values?

I need to determine the maximum value for y = a x^2 + b x + c, where I know the coefficients and the upper and lower x values. ...
0
votes
1answer
97 views

How can I find the derivative of a combinatorial expression? [closed]

I want to differentiate the following equation: n = (nchoosek)*p*(1 - p)^(n - k) After differentiating my result is: ...
5
votes
0answers
174 views

Strange Integrate behavior (a bug!)

The following two calculations should give the same result. After all, integration is a linear operation. I have pasted the code below in case you want to play with it. ...
3
votes
2answers
532 views

Algorithm for parts integration

Sorry if this is a duplicate, I've searched how to do this to no avail. What I'd like to do is a function that integrates by parts $n$ times, i.e $$ \int u(x) v(x) dx = u ...
3
votes
1answer
506 views

Integral of Lorentzian yields different results depending on when parameter assigments are made

I'm evaluating the integral of a Lorentzian, which I know equals one. First I define the function and evaluate the integral in two slightly different ways. Surprisingly, I do not get the right answer ...
4
votes
2answers
1k views

Paths integrals in the complex plane

I can't find how to calculate path integrals of complex functions in the complex plane. For example: $$\oint_{\mid z \mid =2}\frac{1-e^z+z}{z^3 (z-1)^2}dz$$
12
votes
2answers
1k views

How can I implement the method of Lagrange multipliers to find constrained extrema?

I want to form the function $h=f-\lambda_{1}g_{1}-\lambda_{2}g_{2}$ where $f$ is the function to optimize subject to the constraints $g_{1}=0$ and $g_{2}=0$ so that I can solve the first partial ...
2
votes
2answers
215 views

Calculate information entropy integral in infinite square well problem

In the context of information theory, entropy is a measure of uncertainty of a random variable. In quantum mechanics, the uncertainty principle states that $\Delta x\Delta k \ge 1/2$. The same can be ...
5
votes
3answers
224 views

Calculate integral for arbitrary parameter n in infinite square well problem

I'm continuing[1,2] the study of an infinite square well in the context of quantum mechanics. Ultimate goal is to calculate the product $\Delta x\Delta k$, for various eigenstates, that is for ...
4
votes
1answer
217 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
235 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
153 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 ...
18
votes
1answer
451 views

Incorrect result from Integrate

I attempted to calculate the following integral: ...
2
votes
2answers
140 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
219 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 ...
1
vote
3answers
204 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
271 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
270 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
148 views
0
votes
0answers
160 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
195 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 ...
13
votes
2answers
4k views

How to calculate contour integrals with Mathematica?

How to calculate the integral of $\frac{1}{\sqrt{4 z^2 + 4 z + 3}}$ over the unit circle counterclockwise for each branch of the integrand?
9
votes
2answers
343 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
234 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
830 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.
3
votes
1answer
107 views

Integration over a convex combination of a region: $\int_{\Omega} (w_1 z_1 + w_2 z_2)^{1-\sigma} d (z_1, z_2)$ where $\Omega = \{ z_1 + z_2 = 1\}$

Take $w,z\in R^{n}$. I am interested in integrating (as generically if possible) $$\int_{\Omega}(w \cdot z)^{1-\sigma} d z$$ Where the domain of $\Omega$ is $1$ dimension, and includes the convex ...
1
vote
3answers
506 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 ...
1
vote
0answers
104 views

Simple Jacobians, Gradients, etc. with arbitrary length vectors/matrices?

Is there any way (or a package built for it) can do simple operations with vectors and matricies of arbitrary size, but conforming extents? For the simplest example to test, given an arbitrary vector ...
15
votes
1answer
385 views

How to represent a continuous monotonic phase of Airy functions?

Note: In this question I am concerned only with real-valued variables and functions. DLMF, §9.8 Airy Functions, Modulus and Phase, formula $9.8.4$ defines the phase of Airy functions: ...
1
vote
0answers
86 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 ...
0
votes
1answer
137 views

Better way to evaluate integral containing Boolean function

I am trying to compute the integral ...
1
vote
1answer
54 views

Am I missing something in this integral?

I am trying to solve this simple integral: ...
9
votes
1answer
279 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 ...
0
votes
1answer
567 views
0
votes
1answer
102 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 ...
7
votes
2answers
301 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}] ...
5
votes
2answers
938 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?
4
votes
3answers
352 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: ...
7
votes
1answer
524 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 ...
10
votes
5answers
561 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 ...
1
vote
1answer
144 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
428 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
212 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
127 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
93 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
151 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, ...