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.
4,584
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How to make Mathematica able to integrate $\int_{-\infty}^\infty \frac{e^{ikx}}{(x+ia)^3} dx = i\pi k^2 e^{ka} \theta(-k)$ for $a>0$?
I want to evaluate an integral which is essentially a sum of the form $\lim_{a\to 0^+}\int_{-\infty}^\infty \frac{e^{ikx}}{(x+ia)^m} dx$, where $k\in\mathbb R$ and $m\in\mathbb N$. By the residue ...
- 1,009
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How to use interpolating function for function composition?
The following codes give solutions of x(t),y(t), and v(t) as interpolating functions. How to transform y(t) to a regular function so that it can be used for function compostion of y(z(T))? Z(T) is a ...
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Post-process NDSolveValue result
For example, after solving the Poisson equation $-\nabla^2 T = 1$ on the unit circle using NDSolveValue and the following code
...
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47
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QFT Functional derivative
I am trying to calculate the correlation functions of scalar field theory using Mathematica. To do this, one needs to evaluate the expression.
The propagator ...
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2
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67
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Calculating Length of an Elliptical Arc [closed]
I am trying to calculate the arclength of a section of an ellipse, given the correspoding parametrization angle $\phi$. While doing so, I am running into problems with elliptic integrals. The ...
4
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2
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Symbolic integration in a region obtained using a boolean combination
Inertia tensors for non-typical rigid bodies
Calculate inertia tensors
Integrate over a region
I'm trying to calculate the moment of inertia for a complex figure depending on the $r$ parameter.
...
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Proving orthogonality of Hermite polynomials
The orthogonality relation for Hermite polynomials is:
$$\int\limits_{-\infty}^\infty e^{-x^2} H_n(x) H_m(x)\ dx = 2^n n! \sqrt{\pi} \delta_{nm}$$
where $\{ n, m \} \in \mathbb{Z}_{\geq 0}$, $H_n(x)$ ...
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I want to graph a heatmap for three parameter data
I want to plot a heatmap where $x$ varies from $0.1$ to $1$, $y$ varies $0.1$ to $1$, and a third parameter representing image generation times for different $(x,y)$ will be the input to a ...
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How to span a non-zero parameter space for integral?
https://github.com/adeelaafzal/Effective_degrees_of_freedom/find/main
I am working on the following code,
...
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87
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How to get the normal vector of a planar curve pointing "outside" (its circle of curvature)?
I have the following implicit function, for which I need to pick up a point $(x0, y0)$ on it and determine the expression of its normal vector pointing outside. "Outside" here means that the ...
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Evaluating a simple limit in Mathematica
Consider the function
$$M = \frac{\left[(f^{\prime})^2 - 2ff^{\prime \prime} - f^{\prime}_H~f^{\prime}\right]}{2f}$$
where $f \equiv f(r)$, and the primes are derivatives with respect to $r$. $f'_H$ ...
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Analytical Expression
I am trying to get an analytical expression for the following integral. But when I write this in mathematica, it just keeps on running and running. Do not give me any output. Can anyone please help me ...
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How to exactly calculate the volume?
Let us consider the intersection of four cylinders of the unit radius along the big diagonals of the cube $[-10,10]^3$ and the cylinder of the unit radius with the $z$-axis as its axis. More exactly,
<...
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Numerical integration of complicated multivariable expression with divergencies
I've been working on a weird case for some time now and I'm stumped. Any help would be appreciated.
The problem
I'm trying to calculate a 5-dimensional integral of a pretty complicated function (which ...
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A problem about the integration of blackbody radiation law using FormulaData. QuantityVariable limits
I am trying to integrate blackbody radiation law under 1500 Kelvin using the FormulaData command.
The integration seems not to work properly after I designate the ...
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How to improve the following codes for generating eigenvalues?
The following codes generates eigenvalues by using Cebyshev Collocation method of two matrix which consists of another matrices. The codes take more calculation time for larger collocation points, say ...
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Integral of the Fermi function
I am trying to integrate over $v_y$ and $v_z$ the following function in Mathematica 13.1
\begin{equation}
n(v_x, v_y, v_z)= \frac{1}{\exp\left[\frac{\frac{m}{2}\left(v_x^2 + v_y^2 + v_z^2 \right) - \...
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How to calculate the double integral with e and imaginary number? [closed]
How do I write the code for mathmatica to integrate the below function.
it contains e and imaginary i so it is very confusing to me
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Symbolic comparison of integrals
I'd like to prove algorithmically the fact that the integral of $e^{-x^2 - y^2}$ evaluated over a circular disk, ${\cal D}$, is greater than over a square, ${\cal S}$, of the same area regardless of ...
- 38.2k
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How to simplify an expression with simple symbolic integrations with the same boundaries?
I am new to Mathematica and I am trying to simplify an expression with symbolic integrals as below
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387
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Prove linearity of integration [duplicate]
I'd like to prove the linearity of integration over one real variable ($x$).
Integrate[f[x] + b g[x], x] == Integrate[f[x],x] + b Integrate[g[x],x]
which I was ...
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How to reproduce the data format in the picture into mathematica？
In order to study the relationship between an endemic disease in a certain place and the health habits of local residents (health habits are divided into good and not good enough), a medical team ...
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$\int_0^\infty -\frac{\log(|1-x|) dx}{x(2-x)} = \frac{3 \pi^2}{8}$ but Integrate gives $\frac{\pi^2}{4}$
I'm running Mathematica 13.0 Student Edition on Windows 10.
Consider the function $f(x) = -\frac{\log(|1-x|)}{x(2-x)}$. The function is well-defined at $0$ and $2$ and has a weak, integrate-able ...
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Solve Inequalities with Functional Arguments
I would like to solve the following inequality:
$\frac{(\rho +\tau -1) \left(n g'(n) \left((g(n)-c) F'(g(n))+F(g(n))-1\right)+(c-g(n)) (F(g(n))-1)\right)}{n^2}>0$
Under some assumptions (listed ...
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Rule for derivative of an unknown function
I am taking the derivative of a function with respect to another function, e.g.
D[g[f[x,y]],f[x,y]]
This outputs g'(f(x,y)), as ...
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Integration is not correct for the expression with a fraction?
What value Px2[16, 4] or N[Px22[16, 4]] is correct?
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IntegrateChangeVariables doesn't Evaluate at all
How do I perform the following variable change:
IntegrateChangeVariables[Inactive[Integrate][2x E^-x^2,{x,1,\[Infinity]}],u,u==E^-x^2]
Returns unevaluated. Am I ...
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How do I prevent, or get rid of the <code>Abs^\[Prime]</code> in the output of a partial of a Normalize[vector]? 12.0.0 for Microsoft Windows [duplicate]
12.0.0 for Microsoft Windows
How do I prevent, or get rid of the Abs^[Prime] in the output of a partial of a Normalize[vector]?
The code below shows the problematic ...
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Overflowing during computation in Improved Euler’s method
At my master with have an assignment and we study the Dynamic analysis and control of a new hyperchaotic finance system. For this reason we use Improved Euler's Method with the following code.
In ...
3
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Euler's Method for system of differential equation with more than two functions
According to what I have read, the first formula is the classical Euler method and the second is the improved Euler method for second-order equations.
Method A: accuracy of order h
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Error of boundary condition is not specified on a single edge
I am trying to solve a PDE, but Mathematica is showing a strange message which is saying" boundary condition is not specified on a single edge"
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Define regular function as difference of two piecewise functions
Consider a piecewise smooth function,
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Why does DSolve make implicit assumptions when handing sqrt roots? Compare to Reduce
We all know that for real $x,y$ that $\sqrt{x y} = \sqrt{x} \sqrt{y}$ only when $x,y$ are nonnegative. But Mathematica DSolve automatically does this when solving ...
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Is there an option to automatically make solve return the principal value of inverse trig functions?
I am porting some of my Maple packages to Mathematica. In Maple, solve(sin(y)=x,y) returns arcsin(x) and I'd like to get same ...
- 127k
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5
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Extracting the major and minor axes values from the elliptic equation
If the equation of ellipse is:
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$$
then how can the major and minor axes values in the denominators be extracted from eq?
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How to improve this plot? Behaviour of the derivative
I wanted to plot the function
$$f(x) = \frac{x^2}{2} - 3\ln(\vert x\vert +2)$$
together with its derivative
$$f'(x) = x - \frac{3x}{\vert x \vert(\vert x\vert +2)}$$
which exhibites a cusp point at $x ...
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How to define such a complex region and NIntegrate over it?
This may be in my urgent, I'm currently dealing with an integration problem over the support of high dimensional [0, 1] cubic. The integration function itself is not that complicated, you can assume ...
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Interrupted plot lines when multiple branches of an ODE solution are plotted
I am plotting the following:
...
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How can I get KroneckerDelta to contract an index?
I have a sum like
f = Sum[x[i]^2, {i,1,Infinity}]
and I want to take a derivative of this with respect to one of the variables, e.g.
...
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How to compute the rational approximant of a polynomial?
given the chebyshev series expansion up to the fourth order
$$f(z)= \sum_{0}^4 c_k T_k(z) $$
with some fixed $c_k$
i need to find the coefficients of the rational approximant (2,1)
$$ R_{2,1}= \frac{...
- 2,549
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How to calculate the values at the points of a 10-point partition (Fourier Series)?
We have the following periodic function.
$$f(x)=\begin{cases}
x+2,& -2<x<0 \newline
2-2x, & 0<x<2
\end{cases}$$
We define the function and we make the plot
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Understanding the calculation of the periodic function $f(x)=x-x^3, \quad -1<x<1$
It is given the following periodic function
$$f(x)=x-x^3, \quad -1<x<1$$
This is an odd function, so the coefficients $a_n,a_0$ of the Fourier Series are zero.
Usually, I face function with two ...
2
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1
answer
76
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Using Mathematica Confirm Bessel's Inequality
I am working on the following Fourier series problem
f[x_] = Which[-1 < x < 0, 0, 0 < x < 1, x^2]
Plot[f[x], {x, -1, 1}]
...
3
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How to find the derivative of one matrix with respect to another?
I have an expression from a continuum model,which looks somewhat like this excluding a few constants.
Here epsilon is the 3x3 strain tensor and sigma is 3x3 stress tensor. I wish to compute the ...
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How to calculate the integral only in real domain?
I have the following integral:
Integrate[1/(Sqrt[2] + Cos[4 X] + Sin[4 X]), X,
Assumptions -> Element[X, Reals]] // Simplify
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AsymptoticIntegrate of a difficult integral [closed]
I would like to use AsymptoticIntegrate to address this problem: to get an asymptotic expression for:
$$\int\limits_0^\infty \frac{\sin (\sin x)}{\Gamma (x+1)} \ln ...
- 38.2k
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Use Mathematica to solve equation derive the probability distributions using the Principle of Maximum Entropy
I am reading the article Deriving probability distributions using the Principle of Maximum Entropy and trying to derive some of the equations in it automatically using Mathematica.
I want to solve the ...
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How to increase the precision of numerical derivatives?
I need to make sure about precision of numerical derivatives in my calculations. I use finite element method to obtain a wave function, then I take derivatives of this numerical wave function to get ...
- 1,228
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2
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196
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Stop Mathematica from expanding Dirichlet beta and eta functions
It seems that Mathematica treats DirichletBeta[s] and DirichletEta[s] merely as symbols for ...
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