# 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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### How to Handle First Variation with Symbolic Computation in Mathematica?

Background I am currently working on a problem related to surface tension and found a research paper that appears to have the information I need. I am attempting to replicate the calculations in this ...
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### Why does Replace function not work the first time I try it?

Please explain why in the following mathematica code, the cos term was not replaced in the first try, but got replaced in the second try? ...
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### How to handle undefined Integral results for excluded conditions in Mathematica

I have a mathematical function where both input parameters x and y are real numbers within the interval [−π,π]: ...
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### LineIntegrate seemingly does not work

The scalar line integral function seemingly does not work, the both expressions remain unevaluated: ...
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### How to run Integrate[Sqrt[(x - a)^2 + (y - b)^2 + (z - c)^2], Element[{x, y, z}, Ball[{0, 0, 0}, 1]], Element[{a, b, c}, Ball[{0, 0, 0}, 1]]]

I find that MMA cannot run the integral below ...
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### Issue solving recurrence relation

I´m trying to solve the following recurrence relation: ...
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1 vote
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### Allan Variance integral for a first order Markov process

I am using Mathematica to solve this definite integral: 4*(q*T)^2*Integrate[(Sin[Pi*f*t])^4/(Pi*f*t)^2/(1 + (2*Pi*f*T)^2), {f, 0, Infinity}] $T$ and $t$ are real ...
283 views

### Problem computing a limit

I have the following function in Mathematica: $\frac{x! \sum _{i=1}^j (-i+j+1) S_{j+1}^{(-i+j+2)} x^{-i+j+1}}{(x-j)! \left((1-x)_j\right){}^2}$ Defined as: ...
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### Troublesome output of summation's derivative

I am trying to compute a derivative of a nested sum, but I can't avoid Mathematica to output Kronecker deltas and unexpected cases on complex numbers. Consider the following setting: ...
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1 vote
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### Convergence of $\left\{n!\sum_{k=1}^{n!}\frac{1}{k^{\frac{3}{2}}}\right\}$ as $n\to \infty$ using Mathematica or otherwise

Using Mathematica or otherwise, I need to find the convergence or divergence of$$\lim_{n\to\infty}\left\{n!\sum_{k=1}^{n!}\frac{1}{k^{\frac{3}{2}}}\right\}$$ where $\{x\}$ denotes the fractional part ...
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### NDsolve does not solve equation

I have a question for system of ordinary differential equations, because Mathematica does not give the solution. To be more specific, it does not give any sign that there is some problem. There is the ...
1 vote
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### Integration of Gaussian functions takes ages

I am trying essentially to compute a fourier transform with Integrate. 1]Is integrate faster or slower than FourierTransform? 2]Does anyone know how to efficiently calculate these integrals? My code ...
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1 vote
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### What methods can quickly determine the coefficient of determination of the data?

The given data is: x = {0.04, 0.06, 0.04, 0.08, 0.08, 0.05, 0.05, 0.07, 0.07, 0.06}; y = {0.25, 0.4, 0.22, 0.54, 0.51, 0.34, 0.36, 0.46, 0.42, 0.4}; Is there a ...
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### Trivial Mathematica integral on a cube not computing

I was trying to evaluate some intergal over some cube with variable side length, but for some reason Mathematica spit out the input. The interand was sufficiently nice that I couldn't believe it didn'...
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### Mathematica integral wrong

I need to evaluate the following integral: ...
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### Numerical Integral and Partial Differentiation Of a Complicated Expression

I need help plotting Figures 1 and 2 solving EQ 36 and 37 from https://arxiv.org/pdf/2310.12445. here I am trying to find \Gamma,\phi and their Q values for ...
53 views

### Finding an increasing or decreasing function

I have a problem where I am given the derivative of a function, asked to graph it, then state where it is increasing or decreasing using Mathematica. f'(x)=(x^2-3^x)/2x 0<x<4 I've tried the line ...
1 vote
146 views

### Confused about the output of CosIntegral

I am interested in the properties of the cosine integral function, in this case the anti-derivative of Cos[Pi*x]/x, which Mathematica evaluates to ...
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### Checking the sign value of the derivatives of a complicated function

Consider the following function: ...
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### Integration with respect to functions

I observe some strange behaviour of Mathematica when pulling functions into the integrator. I know that Mathematica's capabilities are limited when computing Stieltje or Lebesgue Integrals. But even ...
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### How is the result of this integral obtained by the function Integrate?

When I use Mathematica to calculate $$\int_0^{+\infty } \frac{\log (1-\mathrm i x)}{1+x^2} \, \mathrm dx ,$$ I get different results using functions NIntegrate ...
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### Partial Derivative after Numerical Integration of a Complicated Expression with Singularity at Zero

I want to generate Fig no. 2,3,4 from https://arxiv.org/pdf/2212.08237 with the help of the following equations. In Fig 3 t_(opt) is the time Q_T reaches its max value Q_T(opt) for each T. Here is ...
1 vote
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### StreamPlot3d for the magnetic field of a loop

I am trying to go beyond StreamPlot in the case of a current loop that produces a magnetic field. My code works for StreamPlot, but does not end when I use the same field expressions with StreamPlot3D....
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### Integration of the product of two exponential functions

How can I find the value of this integral? Integrate[E^((-b)*t^a - k*t), {t, 0, Infinity}]
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### Integrate diverges for convergent integral, and returns unnecessary condition

I'm trying to solve an infinite integral using Mathematica. For some reason, Mathematica claims the integral diverges although when solving to a finite limit L and ...
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### Describing the region of integration $\mathbb{R}_+^n$ for arbitrary $n$

Let $f_n:\mathbb{R}^n \to \mathbb{R}$. I would like to know an elegant way of writing the integral $$I_n = \int_{\mathbb{R}^n_+} f_n(x)~\mathrm{d}x$$ in Mathematica. For fixed $n$, the best I could ...
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### Possible bug in partial derivative [closed]

In 14.0 on Windows 10 let us consider D[Min[x, y], x] Piecewise[{{1, x - y <= 0}}, 0] As I understand it, the result is ...
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### Numerical approximation of the integral by using data

I want to use the numerical approximation of the integral of a function given a list of data: $$\int_a^bf(x)dx\approx\sum_{k=1}^N\frac{f(x_{k})+f(x_{k-1})}{2}(x_{k}-x_{k-1}),$$ where $f(x_0)=f(a)$ and ...
250 views

### Integration involving Piecewise function and DiracDelta function

I want to calculate an integration, which reads where $\delta\left(q_{23}^{01}\right)=\delta\left(1+q_1-q_2-q_3\right)$ and $\mathrm{min}(1,q_1,q_2,q_3)$ means the minimum of $(1,q_1,q_2,q_3)$. What ...
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### Checking the formula for solving the Poisson equation with an integral over all of space

EDIT: According to 2.2.3 Rigorous derivation, if we find the integral $u(x,y) = \int \int G(x,y,m,n)f(m,n) dm dn$ then we obtain a function that satisfies the Poisson equation $\nabla^2u(x,y)=f(x,y)$. ...
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1 vote
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### How can I check integration $\int_0^c \exp\left(-ax+\frac bx\right)~dx$?

I am trying to integrate $$\int_0^c \exp\left(-ax+\frac bx\right)~dx$$. I proceed manually in this way (according to https://math.stackexchange.com/questions/2978887/primitive-of-exp-a-times-x-frac-bx?...
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### How to add a custom analytical integration to integration library

How can I add an unknown analytical integral in Mathematica to its integration library so that the Integrate routine, e.g. ...
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### Issue in HypergeometricPFQ function:

I have a solution from integral: A = Integrate[x^n*(1 + x)^n*Exp[-n*x^2] /. n -> 1, {x, 0, Infinity}] //Expand (*1/2 + Sqrt[π]/4*) %//N (*0.943113*) Then I ...
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1 vote
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### Why is DSolve unable to solve this first order homogeneous ode with IC?

This is first order homogeneous ode from undergraduate textbook, which can be solved using standard $y=u x$ change of variables. But DSolve can't. Could someone figure out why? I show below that ...
• 147k
1 vote
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### Integration by parts for Green's first identity

How should one implement $$\int _{\Omega }\nabla u\cdot \nabla v\,d\Omega \ =\ \int _{\Gamma }v\,\nabla u\cdot {\hat {\mathbf {n} }}\,d\Gamma -\int _{\Omega }v\,\nabla ^{2}u\,d\Omega\,,$$ Symbolically ...
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### Problem Derivative (D) of function defined with Piecewise or UnitStep functions - SIMPLE BEAM PROBLEM, 3 SUPPORTS

I am solving a simple structural beam problem using slope-deflection approach - 3 support A(left- fixed), B(middle- pin), C(right - pin). Beam AB - uniform load, Beam BC - concentrated load in ...
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### How to calculate this improper integral?

How to calculate the Integral involving MarcumQ function whose Integral interval is (0, +Infinity)? Any help (code or reference) would be greatly appreciated. ...
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