Questions related to the calculus and analysis branches of Mathematica, including, but not limited to, limits, derivatives, integrals, series, and residues.

learn more… | top users | synonyms (3)

1
vote
3answers
328 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 ...
7
votes
3answers
366 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
389 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 ...
3
votes
3answers
227 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
1answer
339 views

Limit of sequence of functions behaving strange

I'm trying to determine the limit of the sequence of functions $$f_n(x)=\left(\frac{1}{\pi}\arctan(n x) + 1/2\right)^n. $$ I define ...
23
votes
2answers
10k 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?
10
votes
2answers
449 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: $$V_0(r)=2r$$ $$V_n(r)=\int_{-r}^rV_{n-1}\left(\sqrt{r^2-x^2}\...
4
votes
3answers
421 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 1...
19
votes
5answers
1k 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.
5
votes
1answer
174 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
782 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
162 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 ...
16
votes
1answer
578 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: $$\theta(x)=\...
1
vote
0answers
106 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
170 views

Better way to evaluate integral containing Boolean function

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

Am I missing something in this integral?

I am trying to solve this simple integral: ...
9
votes
1answer
305 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
943 views
0
votes
1answer
129 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 \...
11
votes
3answers
410 views

Find asymptotics of $\sum\limits_{i=0}^{n/3} 2^i \binom{n-i-1}{\frac{2n}{3}-1}$

I have an expression 2^n / Sum[ 2^i Binomial[ n - i - 1, 2n/3 - 1], { i, 0, n/3}] ...
6
votes
2answers
2k 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?
2
votes
1answer
272 views

step by step interactive indefinite integral

WolframAlpha can give step by step solution for indefinite integral. There seems to be similar question but for derivatives. Is there a way that I can generate my own step by step solution for ...
5
votes
3answers
562 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: <...
8
votes
1answer
875 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 ...
12
votes
5answers
814 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
160 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? ...
11
votes
1answer
573 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
238 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 $7$? ...
2
votes
3answers
197 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 ...
3
votes
1answer
2k views

Convolution of two distribution functions

I have two functions; f[x_] = (1/k) Exp[-x/k] ; g[x_] = (1/p) Exp[-x/p] ; How I can convolve them? In Mathematica for convolving two functions we have this ...
0
votes
2answers
112 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 ...
2
votes
0answers
282 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, ...
28
votes
1answer
3k 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 ...
5
votes
4answers
2k views

Numerical differentiation methods

Is it possible to write code in Mathematica that implements various differentiation methods (like forward, central, extrapolated, etc.)?
3
votes
1answer
138 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 ...
5
votes
2answers
568 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 $H[H[f(x)]]...
2
votes
1answer
164 views

Integral of GeneratingFunction

I know that GeneratingFunction can be used to compute the generating function $\sum_{n=0}^\infty a_n x^n$ of a sequence $(a_n)_n$ via GeneratingFunction[a[n],n,x] ...
3
votes
2answers
382 views

Integrating special functions

I would like to integrate the following function with Legendre polynomial and Gamma function. I am open to suggestions. ...
2
votes
2answers
100 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 ...
8
votes
2answers
3k views

How do I find line integrals?

For example, how can I calculate $$\int_{\left | z \right |=1}\frac{dz}{z}$$ I know that the answer is $2\pi i$ but how do I do it using Mathematica?
1
vote
1answer
509 views

How to integrate this cumbersome piece-wise function?

I have this systems of equation: ...
2
votes
1answer
101 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 ...
1
vote
2answers
249 views

Symbolic integral with distributions

Why is Integrate[ HeavisideTheta[1 - x^2] DiracDelta[1 - x^2] , {x, 0, 1} ] (*0*) It should be HeavisideTheta[0]?
0
votes
1answer
196 views

Understanding output of multivariable integration

I'm new to Mathematica and I'm trying to integrate this function: ...
2
votes
3answers
6k views

Step-by-step definite integration

I am interested in obtaining the intermediate steps for rather straightforward symbolic definite integrations (e.g., from an elementary calculus text). As an example, for $\int_0^1 x\,dx$ I'd like ...
2
votes
0answers
292 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. ...
0
votes
1answer
469 views

Pull Constants outside of integrals

I would like Mathematica to pull constants outside of an integral: e.g., $\int_0^t f[t] g[s] dt \to g[s] \int_0^t f[t] dt$ This has previously been discussed at replacement rule to pull independent ...
1
vote
2answers
89 views

Obtaining a particular form of solution for an integral

When I do: Integrate [E^(-(t/\[Tau])) Cos[\[Omega] t], {t, 0, \[Infinity]}] How could I get the solution in the form: ...
7
votes
3answers
345 views