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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3 votes
1 answer
191 views

Quantum Field Regularization: Choice between ExpIntegralE and Gamma for Casimir Effect

For the Casimir effect with exponential regularization, we compute the vacuum energy between the plates (somewhat simplified) with: $$\omega = c\ \sqrt{q^2+k_z^2}$$ $$ E = \sum_{k_z=n\pi/a} 2 \int \...
3 votes
1 answer
106 views

How to use Mathematica functions ArcLength and ArcCurvature?

I want to use Mathematica to find the length of a curve and the sum of angles you would turn back and forth when travelling along that curve. To clarify, the angle I am looking for is the sum of all ...
3 votes
1 answer
91 views

Problem with calculation of Lyapunov exponent

The code given by Chris K for Lyapunov Exponent does not work and gives lot of errors for the dynamical system given in equation(2) of this paper The dynamical equations: ...
0 votes
1 answer
51 views

How do I differentiate this in Mathematica? [closed]

I have the following expression: $$v=-r\omega \left( \sin\theta + \frac{r\sin(2\theta)}{2l \sqrt{l^2-r^2\sin^2(\theta)}}\right).$$ I need to find $\frac{\rm dv}{\rm d t}$, given that $\frac{\rm d\...
5 votes
4 answers
510 views

Dealing with derivative of noisy experimental data

I have the following (time, position) dataset ...
0 votes
1 answer
154 views

Concept of tangent and residual in AceFem and automation

here trying to solve the pde: ut + ux = 0 in acefem. I have two questions regarding tangent and residual. Tangent shows the s$$ and residual shows ...
0 votes
0 answers
68 views

Problem with Integrate

I am trying to compute the Fourier expansion coefficients by Integrate. ...
2 votes
0 answers
161 views

How to plot the sum of this function series?

Let us consider the sum of a function series $$ \sum _{n=1}^{\infty } \frac{z^n}{\left(z^n+1\right)^n}.$$ This series absolutely converges if Abs[z] < 1 or ...
3 votes
0 answers
38 views

Cannot add a constant coefficient to LineIntegrate

I'm quite confused on how you're supposed to use constant coefficients in Mathematica integrals. The upvoted and accepted answer on Mathematics stackexchange (https://math.stackexchange.com/a/48021/...
5 votes
2 answers
313 views

Integrate has issues when integrating the general form of a trigonometric integral but not specific instances

I'm trying to solve with Mathematica the following integral for $m$ a positive integer $$s=\int_{0}^{1/2}\frac{\sin^4(\pi\nu m)}{m^2\sin^2(\pi\nu)}\mathrm{d}\nu$$ which should yield $1/(4m)$ (I ...
1 vote
0 answers
81 views

Differentiation of Parabolic Cylinder Functions and Hermite Polynomials w.r.t to Order

When I evaluated (ver. 12.3) parabolic functions below in the vanishing order limit: $u \rightarrow 0$, I got the following: ...
4 votes
0 answers
268 views

Legendre–Fenchel Convex Conjugate

What is the easiest way to compute the convex conjugagte of a real convex function $f: \mathbb{R} \to \mathbb{R}$, defined by $f^*(s) = \sup_{x} \{ s x - f(x) \}$ I know I can compute the derivative ...
3 votes
1 answer
114 views

Define a function whose variable is differential operator

I would like to act with a differential operator $D_x = x \partial_x^2+3\partial_x$ on a function $g(x)$ to compute something like $$ ((D_x)^2 +D_x) g(x) $$ Clearly, in this simple example, this does ...
0 votes
2 answers
179 views

Fitting data for an implicit function with two parameters

I have an implicit function of independent variable sB and dependent variable H with two parameters ...
0 votes
1 answer
51 views

Conformal Mapping Function and Scaling Factor

I am having problems determining a scaling factor for a conformal mapping function. The plan is to map the upper half plane into a rectangle (Schwarz Christoffel Transformation) to determine the ...
4 votes
1 answer
182 views

How to find a closed-form expression for this integral depending on parameter?

In 14.0 under Windows 10 I try to find the improper integral $$\int\limits_{\sqrt{1-\sqrt{1-\psi }}}^{\sqrt{1+\sqrt{1-\psi }}} \frac{1}{r^2 \sqrt{-r^4+2 r^2-\psi }}\,dr$$ under the assumptions $\psi &...
2 votes
0 answers
169 views

How to perform the following deduction symbolically?

$$i \frac{d}{dt}a_n = (a_{n+1} + a_{n-1}) + |a_n|^2 (a_{n+1} + a_{n-1}), \; n \in \mathbb{Z}$$ In the above, $a_n := a_n(t)$. Consider $P = \sum\limits_{n}g(|a_n|^2)$ where $g$ is some function to be ...
4 votes
2 answers
182 views

How to improve the following code to get rid of the warning messages?

I have to solve the following initial Value problem $$y'=\frac{2xy}{y^2-1}, \quad y(2)=1$$ so, I wrote the following code to solve it ...
0 votes
0 answers
126 views

Unable to compute integral on implicit region

I am new to Mathematica. I need to deal with the following computation ...
1 vote
2 answers
100 views

Root of integer difference equation

I have a function f[k] where k is a nonnegative integer. My function f has a single point ...
8 votes
2 answers
186 views

Numerical integral does not agree with analytic integral

I am trying to evaluate a function, that I've now reduced to its minimum (not) working example. Unless I am doing something very wrong, it appears that NIntegrate ...
0 votes
1 answer
100 views

How to force Wolfram solve the ODE with respect to h[s]?

I have a problem with the DSolve operator. It just gives me the initial ODE as an answer, however, obviously, I need to find the answer h[s] as a function of f[s] and its powers (or whatever else). It ...
0 votes
1 answer
111 views

Help in solving the following bvp

...
4 votes
1 answer
85 views

Numerical Solution for a Non-Linear Functional Fractional Differential Equation (FFDE)

I tried solve non-linear Functional-Fractional Differential Equation (FFDE) with this method, but it works on only for range: $x\in \{0,1\}$. I what extend the solution range for example for general ...
1 vote
2 answers
110 views

Problem with evaluation of integral

My code is creating an error when I try to evaluate an integral. Could I ask you to reproduce this code and see what is wrong? Thank you. Here I simply define two variables: Find ...
4 votes
3 answers
255 views

Setting up a double integral over a polar region

Clear["Global`*"]; f[x_, y_] := 9 - x^2 - y^2; Integrate[f[x, y], {x, -1, 1}, {y, -Sqrt[1 - x^2], Sqrt[1 - x^2]}] Integrate[f[x, y], {x, y} ∈ Disk[]] ...
2 votes
2 answers
106 views

Interpreting the output of double integral over a Circle

Clear["Global`*"]; f[x_, y_] := 9 - x^2 - y^2; Integrate[f[x, y], {x, -1, 1}, {y, -Sqrt[1 - x^2], Sqrt[1 - x^2]}] Integrate[f[x, y], {x, y} ∈ Disk[]] ...
3 votes
2 answers
70 views

Using FindSequenceFunction to solve the integral of the product of Legendre polynomials and power functions

The integral of the product of Legendre polynomials and power functions: $I=\int_{-1}^1 x^n \mathrm{P}_l(x) \mathrm{d} x$ The calculation result from the textbook is: $$ \begin{aligned} I & =0 \...
1 vote
1 answer
107 views

How to correctly calculate the sum of this series with Mathematica?

The Mathematica code ...
1 vote
1 answer
83 views

How to integrate Legendre polynomials with parameters?

The orthogonality of Legendre polynomials: $\int_{-1}^1 \mathrm{P}_l(x) \mathrm{P}_k(x) \mathrm{d} x=0, \quad k \neq l$ But ...
4 votes
1 answer
61 views

How to compute the $2n^{\text{th}}$-order derivative of a function

Consider the function f[x_]=x^5. The first-order derivative of this function is given by D[f[x],{x,1}]. More generally, we can ...
1 vote
2 answers
112 views

How to impose a change of variable in a differential equation?

I have this differential equation in terms of the variable $y$ deq[y_]=y^2 U''[y] + (a y^2 + b y + c) U'[y] + (d y + f) U[y]=0 and I want to make the change of ...
1 vote
2 answers
49 views

Strange behavior when integrating vector valued piecewise function

The following straightforward integral should clearly equal $\{0,0\}$ for any $R\in\mathbb R$: ...
1 vote
1 answer
103 views

Find the range of Legendre polynomials

The range of Legendre polynomials in the Reals domain is [-1, 1]. How can we calculate it using FunctionRange or other MMA code? ...
0 votes
1 answer
100 views

Adding Assumptions in double integral written in "math input" mode

I've the following double integral and I'd like to write Assumptions->Re[c]>0, but if I try to do it, I get a syntax error. How can I add the above condition but without changing the ...
2 votes
2 answers
367 views

Why does this integral not converge?

I have a rather simple integral that I want to solve in Mathematica. Integrate[Sin[x*a/2]^4/x^1, {x, 0, Infinity}] where a is a ...
1 vote
2 answers
140 views

Mathematica cannot solve this complicated integration

first-time here. I am trying to use Mathematica to evaluate a solution from Duhamel's principle, the integration looks like $\int^t_0\frac{e^{-ks-\frac{r^2}{2(2Ds+\sigma^2+2D_pt)}}}{2Ds+\sigma^2+2D_pt}...
3 votes
2 answers
203 views

Can Mathematica simplify an ordinary differential equation (ODE) by assuming a power series solution and obtain the recurrence relation?

I have an ordinary differential equation and want to solve it using power series as ψ[x_] = Sum[Subscript[a, n] x^n, {n, 0, ∞}] to obtain the recurrence relation ...
0 votes
1 answer
82 views

Is it possible to get rid of `Root` and have an explicit closed-form solution for the given equation?

I want to solve the given equation for m>0 where all parameters are Reals and a<0. ...
0 votes
1 answer
95 views

How to find the maximum and minimum values of a function

I am given $\alpha = \sin \theta \cos \phi$ and $\beta =\sin \theta \sin \phi$. How should I proceed to find the maximum and minimum values for the below function subject to the constraint $x^2 +y^2 \...
2 votes
1 answer
144 views

Speeding up minimization problem related to a minimal surface

I am trying to find the minimum of the function $$f(x)=\int_0^1(1+t^x)\sqrt{1+x^2 t^{2(x-1)}}dt$$ which arises from trying to minimize the surface area of a function rotated around the $x$ axis. Here ...
0 votes
1 answer
118 views

How to expand the existing basis set so that it becomes more complete?

I have a set of anisotropic gaussian basis set which describes the ground state of the system with great accuracy. The Hamiltonian of the system has the following form: $$H=-\frac{1}{2}\Delta-\frac{1}{...
-1 votes
1 answer
106 views

Why are the plot curves bumpy rather than smooth?

I'm expecting plot curves to be smooth but the results are bumpy. Here is my code: ...
2 votes
1 answer
93 views

Maximize a function of $x$ and $y$ where $x^2 +y^2 \leq 1$

I have the following function: $f(x,y) = x(y^2-x^2)- \frac{(x^2 +y^2)^2}{2\rho}+\frac{3x^2(y^2-4x^2)}{\rho}$ where $\rho>0$ is a constant. My goal is to find the maximum value of this function ...
4 votes
1 answer
184 views

Two integrals equal to Euler-Mascheroni constant not evaluating symbolically

I'm trying to compute: Integrate[(x - 1)/((1 - x*y)* Log[x*y]), {x, 0, 1}, {y, 0, 1}] and ...
1 vote
1 answer
196 views

Accurate numerical values of a given functional

Consider the following functional : $$ I(x,s) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy, s) − F(x −\mathrm iy, s)}{\mathrm e^{2πy}-1}, $$ where $ F(z, s) = \dfrac{\sin^2[π\Gamma(z)/(2z)]}{z^s} $....
2 votes
3 answers
51 views

In the given code, how can I ask Mathematica to replace the obtained points in function $g$ and plot it?

I obtain a set of points which are the solutions of f[b, c, d] == 0 by this code ...
0 votes
1 answer
94 views

Riemann surface of the argument multivalued function

I tried to use Michael Trott's Mathematica Resource function RiemannSurfacePlot3D for plotting the Riemann surface of the Arg function without success. How can I plot this Riemann function with the ...
3 votes
1 answer
113 views

Strange analytical and numerical results during an integration of a complex integrand

Mma 13.2.1.0 Win 10 Pro I am solving the following integral: ...
3 votes
1 answer
144 views

Numeric integration and RegionDilation in higher dimensions

I would like to integrate over a dilationed region in higher dimensions. In 3 dimensions it works well. ...

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