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.

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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? ...
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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 ...
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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 ...
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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 ...
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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) ...
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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}^{\...
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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] ] ...
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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 $0.01$: ...
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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^{...
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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[], ...
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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 ...
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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 ...
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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
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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 ...
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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}. ...
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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
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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
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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
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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
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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
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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 \...
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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
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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
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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://...
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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
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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. ...
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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: ...
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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
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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: ...
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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 $\delta_{ij}\delta_{kl}$...
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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 \...
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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
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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
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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
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0answers
38 views

Normalized eigenvectors are different depending on when the limit is calculated

I have this code: ...
0
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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
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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 ...
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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
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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
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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
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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 ...