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.
2,764
questions
1
vote
1answer
29 views
How to shorten the length of the tangent line
I have all the equations down to manipulate the tangent line on the graph... but the tangent line is too long, what would be the best way to shorten it?
...
0
votes
1answer
19 views
How can I find the minimum and maximum output of a function and specify the valid ranges and steping of each variable?
I am a software developer, and it has been far too long since the last time I've needed to do any math this complicated. I've forgotten a lot of the proper terminology, which has made finding answers ...
1
vote
0answers
30 views
Integrate Twice over Symbolic Interval [on hold]
I would like to solve:
$$
\int _0^1\int _y^11dxdy
$$
In one function call to Integrate[]. However, evaulating
Integrate[1, {x, y, 1}, {y, 0, 1}] (*1-y*)
Gives ...
0
votes
0answers
26 views
How to calculate electric field of a (Gaussian) charge density?
So I've been trying to reproduce a known result from electrostatic, namely, to write the electrostatic potential due to the ground state electron density of a hydrogen atom. So the problem is as ...
0
votes
0answers
41 views
something basic for ordinary differential equations [migrated]
Given$$\frac{\partial u}{\partial x}=M(x,y)$$
$$\frac{\partial u}{\partial y}=N(x,y)$$
$$\frac{d u}{dx} =M(x,y)+N(x,y)y^{'}(x)
$$
How could I get $u$?
Should it be
$$u = \int_{x_{0}}^xM(s,y)ds+g(y)
...
0
votes
0answers
38 views
How to obtain and generated and analytical formula from a Series
The scalar function in cylindrical coordinates which expresses a certain series, which physically this represent an electrostatic potential given by
\begin{align}
\phi(r,\theta) =
\sum _{n=0}^{\...
0
votes
1answer
92 views
Unexpected behavior from $\lim_{n\to \infty }a^{1/n}$
I am computing the limit: $\lim_{n\to \infty }a^{1/n}$
Using the code:
Assuming[Element[a, Reals] && a > 0, Limit[a^(1/n), n -> Infinity] ]
...
1
vote
0answers
27 views
Moment-generating function of the log-normal distribution. A question about integration
Consider the probability density function $f(x)$ of the log-normal distribution with parameters $-0.1$ and $0.01$:
...
0
votes
1answer
34 views
Integral for infinite resistor grid problem
I'm studying the infinite grid resistor problem, and I've noticed that at some point Mathematica solution diverges with the one that's on the theorical study
$$\frac{1}{i\pi}\int_{0}^{\pi}\frac{1-e^{...
5
votes
1answer
96 views
Computing the surface area and volume of molecules
Here is a somewhat difficult problem I've had for a long while.
Space-filling models of molecules can be easily generated by ChemicalData[], ...
-1
votes
0answers
30 views
Geometrical interpretation of (anti-)derivative [on hold]
Integration is anti-derivative and vice versa, analytically. But geometrically, integration is area under the curve where as derivative is slope of the curve at a particular point. How to understand ...
2
votes
1answer
56 views
Analytical integration of a composite Dirac delta
From the well-known properties of the Dirac delta function
$ \delta(f(x)) = \sum_k \frac{\delta(x-x_k)}{|f'(x_k)|}
\qquad x_k = \text{zeros of }f(x) $,
$ \delta(f(x))=0
\qquad \qquad\qquad \text{if ...
3
votes
0answers
47 views
Help implementing Magnus Expansion
The Magnus expansion is a tool to approximate solutions to first-order linear differential equations (the Wikipedia page is quite instructive and concise) - it's particularly useful because all orders ...
2
votes
1answer
73 views
Series applied to an infinite sum does not work
When I have
Sum[(-1)^n x^(n^2) y^n, {n, 0, ∞}]
and I try evaluating
...
-1
votes
1answer
108 views
Can Mathematica calculate the triple integral $\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \frac{dx dy dz}{(1+x^2+y^2+z^2)^2}$?
A recent post in Mathematics Stack Exchange claims that one can get from Mathematica the following result:
$$\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \frac{dx dy dz}{(1+x^2+y^2+z^2)^2}=\frac{\pi^2}{32}.
...
0
votes
0answers
28 views
Making a loop for an integral equation
I have a bunch of equations. (All the equations are a function of temperature now, but I will set a certain temperature for this calculation.) These will be constants in the last differential equation ...
3
votes
1answer
130 views
Analytical form of 2d integrals relevant to graphene
This question is continuation of my previous post.
Alex Trounev was very helpful in fixing a crucial typo in the analytic solution known from the article "Density Dependent Exchange Contribution to ∂𝜇...
1
vote
1answer
22 views
Visualizing a multiple parameter integration
I'm really new to Mathematica and all the other answers to similar questions don't seem to help me. I have an integral I'd like to solve, and I'm having problems. My integral looks like this:
$\int_{...
2
votes
1answer
67 views
Product from max to min [closed]
Product[f[i], {i, 1, 4}]
gives us
f[1] f[2] f[3] f[4]
Is there any way I can take the product it will give something ...
0
votes
1answer
99 views
Is this number $N$ a new mathematical constant? [closed]
Smallest real number $N$ such that the equation $2^x = Nx$ has solution in real number $x$. I have calculated $N$ up to 9 digit precision , $N= 1.88416938....$, I believe $N$ converges. Has this ...
4
votes
1answer
48 views
Inverse of vector-valued function
Having a differentiable function $f : \mathbb{R}^2 \longrightarrow \mathbb{R}^2$ such that $\det D(f)$ (Jacobi determinant) does not vanish.
How can we get the inverse function $f^{-1} : \mathbb{R}^2 \...
8
votes
1answer
596 views
Correct way of simplifying the result of an integral
Many times Mathematica gives enormous results to simple problems. One uses the program more for trouble than for not knowing how to solve the problem.
As an ejemplo I present this integral that ...
2
votes
0answers
74 views
Drawing a cyclogon
Is it possible to draw the path of a point on the boundary while the polygon is rolling, or writing a function of theta similar to the cycloid parametric equations:
$$
x = r (\theta - \sin(\theta)),\...
1
vote
0answers
30 views
How to compute Bott-Chern operator ddc
I wish I could make Mathematica compute the $dd^c=2i\partial\bar\partial$ of a function depending on several complex variables. The question involves complex or Wirtinger derivatives (https://...
0
votes
1answer
38 views
Calculation using Integrate & Plot take too long time & some error
I want to plot this equation.
mu, Eb, gamma, Eg are constant parameters and x is independent parameter.
...
12
votes
3answers
489 views
How to evaluate $ \int_0^1{\mathrm e^{\mathrm i\pi x}x^x(1-x)^{1-x}\mathrm dx} $ and get $ \frac{\mathrm i\pi \mathrm e}{2\cdot3\cdot4} $?
The following definite integral identity holds
$$
\int_0^1{\mathrm e^{\mathrm i\pi x}x^x(1-x)^{1-x}\mathrm dx} = \frac{\mathrm i\pi \mathrm e}{2\cdot3\cdot4}
$$
and is easy to verify numerically. ...
0
votes
1answer
79 views
What is the proper way to use the Prime typesetting in an integral? [closed]
I've got an integral I'm trying to calculate in Mathematica:
$$\tau(\eta)=\int_0^\eta n_e\sigma_T a\space d\eta\prime$$
My best attempt to represent that in Mathematica has been:
...
1
vote
1answer
36 views
Plotting a function on the different axis
I've solved the DE:
$$a\cdot x(t)+b\cdot\ln\left(1+c\cdot x(t)\right)=-p\cdot x'(t)\tag1$$
Where $x(0)=k$
And I got the following solution:
$$t=\int_{x(t)}^k\frac{p}{a\cdot z+b\cdot\ln\left(1+c\...
1
vote
1answer
47 views
visualizing branch cut of a complex function
I'm working in the plotting the branch cut of a complex function, namely:
$w(z) = (2+z) \ln(2+z) - 2(1+z) \ln(1+z) + z \ln z$.
To do so, I have tried this:
...
0
votes
0answers
24 views
Contracting the KroneckerDelta should bring it back
I would like to define a function
$$V[\phi] = \sum_{1\leq i,j,k,l\leq dim}\lambda_{ijkl} \phi_i\phi_j\phi_k\phi_l, $$
with $\lambda_{ijkl}$ defined by contractions of the type $\delta_{ij}\delta_{kl}$...
0
votes
0answers
30 views
Wrong derivative for a function that I try to define without explicitly giving number of variables
What I actually want is to define functions depending on the number of variables which will be specified during some computation. My (probably wrong) attempt is
...
0
votes
1answer
46 views
Get Inverse Function in a certain domain
I'm trying to make Mathematica giving me the inverse function
of $f$ (below) when restricted to the interval $[v,1]$.
Using ConditionalExpression or ...
0
votes
1answer
106 views
Verifying that an inequality has a finite limit
I am trying to verify the following inequality using Mathematica, but without any success.
$$
\lim_{a\rightarrow0^{+}}\frac{\intop_{0}^{1}e^{\frac{-30}{\sqrt[15]{a^{16}*\log\left(\frac{1}{x}\right)}}}...
3
votes
2answers
117 views
Why am I getting a negative value from $\int_{0}^{\infty}x^{\frac{-9}{8}}*e^{-16*x^{\frac{-1}{8}}-\frac{2*x}{0.5^{9}}}dx$?
In Mathematica 9, I want to evaluate the following integral:
$\qquad \int_{0}^{\infty}x^{\frac{-9}{8}}*e^{-16*x^{\frac{-1}{8}}-\frac{2*x}{0.5^{9}}}dx$
The result should be positive. However, ...
4
votes
3answers
150 views
A discontinuity in an integral
I have a function which I'm trying to integrate, which is perfectly smooth and Mathematica sees it as well defined everywhere in the region over which I'm integrating it. The function is:
...
2
votes
1answer
154 views
Solving multi-field problems in MMA 11.3
Based on the PDE model proposed by @Schumacher Solving a second order coupled PDE system, the one dimensional multi-field Problem I would like to solve such benchmark test:
namely:
u(x): ...
0
votes
1answer
39 views
DSolve not satisfying such kind of initial conditions
I am trying to solve this equation in MMA 11.3
$2 \epsilon s^{\prime \prime}+\frac{1-s}{2 \epsilon}=0$;
BCs: $s^{\prime}( \pm 1)=0$ and $s\left(0\right)=0$;
The analytical solution is expressed as:$...
4
votes
2answers
124 views
Solve a logarithmic equation by NSolve
I'm trying to solve:
$$1 - |x| Log(\frac{|1+x|}{|x|}) = 0.84$$
by using NSolve:
NSolve[1 - Abs[x] Log[Abs[(1 + x)/x]] == 0.84, x]
Mathematica doesn't return ...
1
vote
1answer
78 views
Writing a function that can perform n-times fractional Caputo derivative of order alpha to a function u(x) inductively
I wrote this code, but it fails to give the intended answer which is D^(1/2) three times of x^2.
any suggestions to correct this code or write a new one.
0
votes
0answers
26 views
Define the partial derivatives of a multivariable function
I have some vector $K = [K_1, K_2,...K_n]$ which is a function of variables $(x,y)$ i.e. $K = K(x,y)$
I want to symbolically calculate the partial derivatives e.g. $\partial_x K$, $\partial_x \...
0
votes
0answers
107 views
Need help with symbolic integration
I have
f[x_] := g[x]/g'[x]
FullSimplify[Integrate[f[x], x]]
I'm trying to get F[x] in terms of ...
5
votes
1answer
119 views
Visualizing Riemann surface (two branches) of logarithm
I'm trying to plot two branches of the complex multi-valued function: (In a previous post - linked above-, Mathematica found the branch cut of the following function between -1 and 0)
$1-z\ln[(1+z)/z]...
4
votes
2answers
155 views
To plot branch cut of logarithm
I like to see the branch cut of the function:
$$1 - z \ln[(1+z)/z].$$
If I plot it in the complex plane:
...
0
votes
0answers
38 views
Normalized eigenvectors are different depending on when the limit is calculated
I have this code:
...
0
votes
0answers
56 views
Unexpected difference between integral and summation?
I am trying to integrating something like: Integrate[Exp[-i*(k*x+k*z)]*Exp[-(x^2+z^2)],{x,-largenumber,largenumber},{z,-largenumber,largenumber}]
My issue is that ...
1
vote
1answer
80 views
Is it possible to use mathematica to find the derivative of a function with respect to some “ax” where “a” is a constant?
Mathematica evaluates the derivative of most complicated functions within a fraction of second. But while doing analysis sometimes we might be in a situation to differentiate or integrate a function ...
0
votes
2answers
46 views
problem with constant variables
Variable x is supposed to be constant, but the program calculates for each variable x, 100 numbers of x. I want all my x variables to have the same definition, but the program constantly changes each ...
0
votes
0answers
35 views
How to obtain the gradient of a function wrt a vector?
I'm new to mathematica and I wanted to obtain the gradient of the following function:
where b_k(s) is the cubic B-spline basis function associated with control point
k.
I tried this code:
...
9
votes
3answers
1k views
How can I solve for the intersection points of two ellipses?
I need to find the points at which the two ellipses implicitly defined by
$\qquad y^2=4-4\,x^2 \quad[E1]$
and
$\qquad (1-(x/2))^2+(y-1)^2=1 \quad[E2]$
intersect.
So I isolated $y$ in E2 and ...
1
vote
1answer
79 views
How is the best way to integrate using loops
I'm using Mathematica, and I want to integrate a function f[wr] in wr using some method that works with a Table/Do/For in the variable wr.
I tried to use something like the Riemann's sum, to evaluate ...
