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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9
votes
1answer
288 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
699 views
0
votes
1answer
115 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
377 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
1k 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
0answers
150 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 ...
4
votes
3answers
412 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
664 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
719 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
157 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
492 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
224 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
146 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
104 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
201 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, ...
16
votes
0answers
2k 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 ...
4
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
129 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
400 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 ...
2
votes
1answer
156 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
333 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
96 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
2k 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
488 views

How to integrate this cumbersome piece-wise function?

I have this systems of equation: ...
2
votes
1answer
99 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
209 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
167 views

Understanding output of multivariable integration

I'm new to Mathematica and I'm trying to integrate this function: ...
1
vote
0answers
224 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
369 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
88 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: ...
6
votes
3answers
315 views
1
vote
3answers
2k views

Find dy/dx given an equation

Here's a homework problem from the Coursera course Calculus: Single Variable by Robert Ghrist Find the derivative $\frac{dy}{dx}$ from the equation $x \tan y - y^2 \ln x=4$ I wanted to check my ...
19
votes
2answers
547 views

Have I found a bug in Integrate?

The following command gives 0 in Mathematica 9.0.1. ...
1
vote
2answers
320 views

Is this integration error from a misuse of Mathematica or is it my poor math skills?

I have input this: Integrate[1, f[t]*Exp[(v/V)*t]] That is $\large\int 1*d(f(t)*e^{\frac{v}{V}*t})$ I expected this output: $\large f(t)*e^{\frac{v}{V}*t} + C$ ...
2
votes
1answer
171 views

A question about derivatives (cubics)

Given an equation: f(z) = a (x + I y)^3 + b (x + I y)^2 + c (x + i y) + d Which can be re-written as: ...
1
vote
1answer
308 views

Difficulty with computing a limit

I meet some difficulties when trying to compute this limit, ...
3
votes
1answer
225 views

Using WhenEvent for derivative of discontinuous function

I have a discontinuous function ($u(t)$, a square wave) and I would like WhenEvent to trigger when the signal goes high/low, i.e. when the value of $u(t)$ changes. ...
1
vote
1answer
1k views

The underlying process of Integrate[] [duplicate]

Integrate[x^4 E^-x^2, {x, 0, +∞}] Output: (3 Sqrt[π])/8 Can someone explain to me the specific calculation process ?
8
votes
3answers
603 views

How can I calculate the perimeter of an equation-defined curve?

This is how the curve looks like: ...
6
votes
2answers
185 views

Series resulting in “No more memory available.”

Is there any way to force Mathematica to come up with the closed form to Sum[1/(i^18 (i^2 + j^2)), {i, 1, Infinity}, {j, 1, Infinity}] or ...
1
vote
1answer
406 views

integrating a Green's function for a damped harmonic oscillator

I have the following integrand: int = Sin[Sqrt[-g^2 + omega^2]*(t - tp)]*Exp[-g*(t - tp)]*A*Exp[-(tp - t0)^2/sigma^2]*Cos[Omega*tp]/Sqrt[-g^2 + omega^2] and am ...
8
votes
2answers
319 views

About an infinite product

I have a curiosity as regards the infinite product below. I wonder why Mathematica v.8.0. says the limit is $1$. This is not true. ...
1
vote
2answers
136 views

Integration leading to logarithms and chosing branch

I m doing this interal with Mathematica Integrate[(((1+b*t)^2)t)^(-1),t,Assumptions->b>0] what I get back is ...
4
votes
2answers
1k views

Can I define a function for vectors of arbitrary dimension?

Is it possible to do analytic calculations with Mathematica? For example, I want to compute: $$\partial \frac{\sum_{j=1}^n G_{j} \prod_{k=1}^{j-1} (1 - G_{k})}{\partial G_l}=-\prod_{k\neq l} ...
-4
votes
1answer
558 views

Symbolic Definite Integration

How do I perform Integrate[Sin[x]^2/(z^2 + R^2 - 2 zRCos[x])^(3/2), {x, 0, Pi}] where z and ...
9
votes
1answer
310 views

solve ODE with divergencies

The solutions of the second order differential equation $$\frac{1}{\eta}\frac{d}{d\eta}\left(\eta \frac{df}{d\eta}\right)+\left(1-\frac{s^2}{\eta^2}\right)f-f^3=0$$ is shown in Fig. 5.2 below, for ...
1
vote
2answers
622 views

Integration and Parallel Computing in Mathematica

In mathematica, how would I use parallel computing (or paralleltable) to compute the integrals of $x^2$, $x$, $3x$ from $x=0$ to $10$ just like in the first answer of ...
4
votes
1answer
257 views

The limit of a product to infinity

What is the difficulty with this question to Mathematica? Is it just a problem with Mathematica v.8.0 or all versions are in trouble with it? Or is something wrong with my input? ...
0
votes
1answer
264 views

Gram-Schmidt process with Hermite functions on [-1, 1]

Denote by $h_n$ the $n$-th Hermite function. $$ h_n(x) = \frac{(-1)^n }{\sqrt{2^n n! \sqrt{\pi}}} \mathrm{e}^{\frac{x^2}{2}} \frac{\mathrm{d}^n}{\mathrm{d} x^n} \mathrm{e}^{-x^2} $$ I am trying to ...