Questions tagged [calculus-and-analysis]
Questions related to the calculus and analysis branches of Mathematica, including, but not limited to, limits, derivatives, integrals, series, and residues.
0
votes
0answers
41 views
How to integrate a logsumexp function?
I am trying to integrate a logsumexp function which is a smooth function of another piecewise function.
Run
...
1
vote
2answers
54 views
How to output the level of the contour plot based on the desired area enclosed in the contour level?
I am a beginner using Mathematica. I am doing a physics experiment that involves this 3D surface; the function $ f(x, y) $. I need a program that can give me the contour level in the $ z $-axis which ...
0
votes
0answers
37 views
Laplace Transform of Complementary Error Function [migrated]
I need to apply one Laplace transform formula while I have no idea how to prove it:
$$\int_0^\infty e^{-st}e^{a k}e^{a^2 t}erfc(a \sqrt{t}+\frac{k}{2 \sqrt{t}})dt=\frac{e^{-k \sqrt{s}}}{\sqrt{s}(\sqrt{...
0
votes
3answers
55 views
Solution… 2nd Transcendental Equation with Graphical method [on hold]
If $ 2x^3 + \ln x = 5 $, then what is $ x $?
For beginning we started to find a solution for equation
$$
\begin{align}\label{eq:eq}
\ln x + cx = b \tag{1}
\end{align}
$$
We know from W function ...
0
votes
1answer
29 views
numerical integration with intermediate variables
I'm trying to perform the following integral, where the bounds of the inner integrals become the integration variable in the next integral.
disregarding the stuff in the exponential in the sample ...
1
vote
2answers
63 views
Define a partial derivative of a composite function
I would like to define the function $ \dfrac{\partial}{\partial y}\left(\dfrac{f(x,y)}{g(x,y)}\right)$, for example
...
2
votes
2answers
61 views
0
votes
1answer
53 views
gradient of Legendre polynomials
I need help to check my code if my way to generate the gradient of a function that contains Legendre polynomials is correct or not.
if f is a function in θ and κ ,I need the gradient of f with ...
1
vote
1answer
42 views
Solve for: When does the area bound by three curves == area desired
The program I've been making an attempt to write does the following:
Plot 2 lines, f(x) = –x and g(x,a) = e^(x – a) –a
Find area bound by the 2 lines and x-axis (shaded in red)
A "desired area" value ...
4
votes
2answers
178 views
How to check whether Laguerre polynomials are orthogonal?
I've got the problem with checking if Laguerre polynomials for n=1,...,10 are orthogonal.
I have to create the list of these polynomials, then create the matrix of integrals from 0 to infinity. ...
1
vote
1answer
41 views
Laplace transform giving incorrect result
In Mathematica:
LaplaceTransform[Exp[-Exp[-t]], t, s]
Out:= Gamma[s]-Gamma[s,1]
But performing InverseLaplaceTransform on the ...
11
votes
1answer
294 views
Symbolic integration of potential over a disc : branch cut problem?
Context
I am trying to explore the geometry of a crystal made of irregular bubbles.
See animation here.
very vaguely in the spirit of this post (it is in fact motivated by cosmology and galaxy ...
1
vote
1answer
49 views
Volume computation not returning expected result
I have tried to integrate the expression
(0.5*(3*Cos[t]*Cos[t]-1))*(0.5*(3*Cos[t]*Cos[t]-1))
where t is in [0, π], as a ...
0
votes
1answer
21 views
Function that takes derivative of another function (with any number of arguments) and evaluates it
I would like to supply a function as an argument to another function which then differentiates the function and evaluates it at a set of points.
Suppose I want to differentiate this function,
...
0
votes
1answer
69 views
Mathematica failing to compute function to calculate integral over a region bounded by straight line
Statement of the problem
Consider the following situation:
You have a model which employs a bivariate distribution with known parameters.
You have a random realization from the distribution ...
2
votes
1answer
134 views
What does “ Im' ” mean? [closed]
I am trying to take the curl/derivative of the function A. What does " Im' " mean here? Why does Mathematica attempt to take the derivative of the Im function here?
...
0
votes
1answer
59 views
How to use a previous numerical solution to solve a differential equation numerically?
When I solve this differential equation with NDSolve, I get a numerical solution for H[x]:
Code for the differential equation:
...
3
votes
1answer
639 views
Why can Mathematica solve integral(a)+integral(b), but not integral(a+b)?
I am new to Mathematica and came across the following problem. The integral at hand cannot be solved.
$$
\text{Integrate}\left[\frac{2 \left(3 t \epsilon \text{Li}_2(t)+3 \epsilon \text{Li}_2\left(...
0
votes
1answer
33 views
How to solve this differential equation? Error (NSolve :: ifun)
When I solve the differential equation I get an error:
error:
Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information.
...
3
votes
2answers
159 views
Simplify Derivative with Substitution
I try to evaluate:
$$ \frac{\partial}{\partial x} \log{u(x, y, z)}$$
Mathematica gives:
$$ \frac{1}{x+y+z}$$
I want to simplify the expression with my function:
$$ \frac{1}{u(x, y, z)}$$
How to ...
1
vote
0answers
34 views
Series doesn't seem to work on functions
When you try to use the Multivariable option of Series, it doesn't work as intended, since it keeps terms of higher order than intended.
Example: Expand a two-variable gaussian exponential to second ...
1
vote
1answer
55 views
Simplify sum of unnatural conditional functions generated by derivative
I try to use Mathematica to evaluate:
$$\frac{\partial}{\partial A_{abc}} \sum_{j=1}^{J}\sum_{k=1}^{K}\log\left(\sum_{l=1}^{L}A_{jkl}B_{jkl}\right)$$
I get:
$$\sum_{j=1}^J \sum_{k=1}^K \frac{\begin{...
1
vote
1answer
56 views
Derivative with respect to any element of a matrix
I try to evaluate
$$ \frac{\partial s}{\partial A_{ab}} $$
where
$$ s = \sum_{i} \sum_{j} A_{ij}$$
The result should be 1 regardless of $a$ and $b$. But the following code gives zero.
...
4
votes
2answers
150 views
Expansion of the Meijer G Function
I'm trying to do the integral
Integrate[ B^2*BesselK[0, ko*ρ]^2*2 π*ρ, {ρ, a, ∞}]
which I figured should be relatively simple as the integral of a Bessel ...
0
votes
1answer
35 views
Derivative of sum of elements of array [closed]
I try to get derivative of sum of an array with respect to an element of the array.
$$s = \sum_{i = 1}^{3}A_{i, 1}$$
Evaluate $\frac{\partial s}{\partial A_{1,1}}$
Below is the code.
...
0
votes
1answer
67 views
Computing an integral : Mathematica seems to say it's diverging or not defined and I think it is
I would like to compute :
Element[a, Integers]
Element[b, Integers]
Integrate[Exp[-a^2*(x (1 - x/b^2))^2], {x, 0, Infinity}]
which is obviously not diverging. ...
1
vote
0answers
38 views
0
votes
2answers
48 views
How to deal with the loss of significant digits in this expression with Fresnel integrals?
I needed to solve this integral:
$$\int_1^\infty \frac{dx}{\sqrt{x}} \cos \left(a x-\frac{\pi}{4} \right) \cos \left(b x-\frac{\pi}{4} \right) \cos \left(c x-\frac{\pi}{4} \right)$$
Coming from ...
0
votes
0answers
96 views
0
votes
2answers
100 views
Why won't Mathematica evaluate my integral?
I would like to integrate:
Integrate[
1/((0.64*E^(2*(a/d)^1.7)*Sqrt[Pi *a]))^3, {a, a0, ac},
Assumptions -> {d > 0, a > 0, ac > a0 > 0}
]
I ...
0
votes
0answers
50 views
Integral converges only if Mathematica kernel has just started?
After just starting Mathematica (or quitting the kernel) if I run:
...
0
votes
0answers
32 views
Solve nonlinear differential equation system - Plot
When I solve the system of differential equations I get two solutions for each variable (See the picture). My question is how I can neglect one of the solutions at the plot. Thanks.
Code:
...
1
vote
1answer
35 views
Output of Apart as a list?
I have a rational funtion of one variable that I know will always have a simple partial fraction decomposition of the form
$$\sum_{n=1}^N\frac{\alpha_n}{z- \beta_n},$$
but I do not know the values ...
0
votes
1answer
71 views
Linearity of the Dirac delta function [closed]
It is known that the Dirac delta function is linear. I do not understand why Mathematica says that a DiracDelta[x] + b DiracDelta[x] is not equal to ...
1
vote
1answer
61 views
Making StreamPlot draw phase plane streams near saddle points
For example, I want to draw the phase portraits near the saddle points $(-\pi,0)$ and $(\pi,0)$, but the code that follows can not plot traces near those points, because $(y,\,-10 \sin x - y) = (0,\,0)...
4
votes
0answers
80 views
Integrate under assumptions
I am trying to solve the following integral:
$$\int_0^\infty u^{\sigma-1} \exp\left[-c u^\sigma\right] \mathrm{d}u$$
under the assumptions $\sigma \in (0,1)$ and $c>0$.
I know the result is
$$\...
3
votes
1answer
376 views
How to calculate this n dimensional integral?
The integral such as
It's easy to evaluate the first few items
...
0
votes
1answer
35 views
Total derivative after numerical solution of a system
I have a system of three equations in three unknowns, $k$, $\theta$ and $w$. I am interested in the behavior of the variables when one parameter changes. I first specify the system:
...
1
vote
1answer
42 views
Table of function derivative evaluation
I want to create a table with numerical values of a function derivative by using the following code:
...
0
votes
0answers
41 views
Convolve taking too long and producing error message Integrate::idiv:
The following functions do not produce an output."f1" is a convolution of Gaussian function and my model. It took too much time and gave an error message. I have attached that error message also. How ...
0
votes
2answers
47 views
Finding the 1st and 2nd derivatives of a function of two variables [closed]
I have an oscillatory function f(ф, ф'), and I want to differentiate it. function like this:
...
4
votes
2answers
258 views
Calculating curvature of a contour
I have an equation of a scalar field in the form
$$
f(x, y) = x^2 + y^2 + xy + c.
$$
I want to find the curvature of the contour of the curve at $ f_c = f(0.5, 0.5) $.
So I need to calculate the ...
1
vote
2answers
43 views
Integrate over matrix elements
I wish to perform the integral
$$
\int \prod_{i,j}dM_{ij}\exp\left(-Tr(M^2)\right)
$$
where we integrate over the matrix elements of $M$. I tried
...
1
vote
1answer
42 views
How to change the integral variable and integral upper/lower limit?
I am dealing with the symbolic operation of taking the antiderivative, e.g., Integrate[f, x] (where $f$ represents a complicated function which cannot return ...
0
votes
0answers
52 views
Discordance between the results of AsymptoticIntegrate and calculations
Mathematica 11.3 finds
r = Integrate[Cos[k*(x^4 - x)], {x, -Infinity, Infinity}, Assumptions -> k > 0]
...
0
votes
1answer
91 views
Problem with pendulum code cant get to plot Mathematica 11.3 [closed]
I copied from an example about a pendulum. (A Mathematica Primer for Physicists, p38). Can't get the LogPlot to work. Get just the axis. Using version 11.3 on ...
0
votes
0answers
35 views
How to differentiate with respect to sum of variables?
If f is a function of x,y,z. How do I differentiate f wrt x+y+z?
2
votes
1answer
39 views
How to define a function that can transform integrands to the unit hypercube?
We know most integrands can easily be transformed to the unit hypercube:
$$\int_{a_1}^{b_1}\cdots\int_{a_d}^{b_d}{\rm d}^{d}x\, f(\vec{x})=\int_0^1{\rm d}^dx\,f(\vec{x})\prod_{i=1}^{d}(b_i-a_i)\ ,$$
...
2
votes
1answer
77 views
Wrong limits of very simple problems in Mathematica running on Ubuntu
I tried to find the limit$$\lim_{x\rightarrow0} e^{-\frac{1}{x}}$$ with the assumption $x\in \mathbb{R}$ on my Macbook Air, and I got Indeterminate, but on my ...
0
votes
2answers
66 views
Why does my integral not evaluate?
I just started using Mathematica today, so I'm sure there is something I am doing incorrectly. I am trying to evaluate this integral:
...
