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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An alternative command to compute a logarithmic integral

I am trying to see if Mathematica can calculate: $$\int_0^1\frac{\ln(x)\ln(1-x)\ln(1+x)}{x}dx,$$ which has a well-known closed form. So I tried ...
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53 views

Algorithm for finding a function connecting points in a space

I have a set of points in a space and I need to find an algorithm which will find the function which connects these points. From current analysis I know that no three points lie in the same plane. So ...
2
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0answers
39 views

Finding region, between four hyperplanes, useful for integration

Consider the function $$ f(x,y,u,v) = \exp\bigl( -x^2 -y^2 -u^2 -v^2 \bigr), $$ and the region $\mathcal{R} = \cap_{i=1}^4 \mathcal{R}_i$ defined by four hyperplanes as follows $$ \begin{align} \...
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2answers
63 views

Why does Integrate[] fail to give me an analytical expression

I am trying this integral to solve analytically, but Mathematica fails, and I get the integral back as it is defined. Although numerically it can be done, I need an analytical expression. Is there any ...
1
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1answer
46 views

Integral of complex function

How to compute a complicated complex integral? You can see the example in the figure. I set assumptions that all constants are positive and real, then I try to integrate the $pel(z)$ expression but ...
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0answers
53 views

Calculating double integral bounded by domain in Wolfram Alpha Notebook

If I want to compute Double Integral on a given region(See attached Picture) How can I input this on Wolfram Alpha Notebook
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0answers
25 views

Pure functions with derivatives and list elements [duplicate]

I have a list list = {x, y, z} and then the following line D[#, list[[2]]] & gives the output ...
5
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0answers
118 views

Numerical verification of the estimate:

How to verify numerically with considerable accuracy in Mathematica the following : $$\int_2^x\dfrac{1}{z\Gamma(\sin^2[π\Gamma(z)/(2z)])}dz\sim\ln(\ln(x))$$ ? I need more suitable and better code ...
7
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4answers
241 views

Why the difference in the results of Integrate and NIntegrate?

Bug introduced in version 12.0.0, and persisting through 12.2.0. Calculating the integral $$\int\limits_0^1 \frac{x^2\log(1-x^4)} {1+x^4}\,dx$$ symbolically ...
4
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3answers
131 views

ApplySides Integrate to equations in differential one-form

Related to this Can we abuse notation and write equations in differential one-form? Is it possible to ApplySides a form of ...
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0answers
24 views

Integrating with indices

I am trying to integrate these functions in as inverse Fourier Transforms, where $q$ is the Fourier variable. $\int_{-\infty}^{\infty}{\frac{d^3q}{(2\pi)^3}e^{iq\cdot r}\frac{\delta_{ij}}{q^2}} + \...
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0answers
28 views

Derivative of an expectation function

For a general function $g$, define $f=\mathbb{E}_{(x,y)\sim \mathcal{N}(0, \Sigma)}[g(x)g(y)]$, where $\Sigma=\begin{pmatrix}a & c \\c & b \end{pmatrix}$, and $a>0,b>0,ab>c^2$. I am ...
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1answer
117 views

How can I make Mathematica calculate a volume given a set of inequalities on the the three dimensions?

How can I make Mathematica calculate a volume given a set of inequalities over three dimensional space? I know how to plot the 3D region of the volume that I need to calculate, like this: ...
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2answers
226 views
+250

Prove $R^\top R = I_3$ and find skew-symmetric matrix $R^\top \dot R$

Consider the following operator defined over unit vectors of $\mathbb{R}^3$: $$R(u,v) = (u\cdot v)I_3 + hat(u\times v) + \dfrac{(u\times v)\otimes (u\times v)}{1+u\cdot v}$$ ...
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1answer
67 views

How would I graph this function on Mathematica?

I could graph 1/n^2, but I don't understand how to graph the fractional part (x).
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0answers
44 views

How to solve this double integral analytically or numerically?

Can someone help me to solve this integral or explain me how to set it on mathematica? \begin{equation} \int_{-\infty}^{+\infty} dk_\alpha dk_\phi \frac{1}{2\pi} e^{-\frac{k_\alpha^2}{2}} e^{-\...
2
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1answer
70 views

Overflow during computation

The following code display the Predictor-Corrector approach to solve the four dimensional Fractional-Order Delay Differential system, The below code works fine for some values of $\tau$ but it gives ...
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0answers
42 views

Evaluation of a double summation invovlving hypergeometric and exponential functions

I am trying to compute the following double summation over the indices, $m$ and $n$, which involves the hypergeometric function, ${}_2 F_1$, an exponential function and, factorials as a part of a ...
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1answer
45 views

Integration shows RootSum and really long expressions

I am doing this integration and when I use $C_\alpha(T)$ as just constant numbers(...
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1answer
54 views

Plotting function and its approximation function

I have a problem which I have not been able to solve. I want to plot a function and and operator which approximates it when you let w to infinity. I will give all ...
1
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1answer
67 views

limit of an expression including BesselK function

i want to calculate the limit of the following expression when 'w' tend to zero. I have used the Limit function, it takes a lot of time for running without any result. could you please help me how to ...
1
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0answers
65 views

Compute the Hessian matrix of a function [duplicate]

To compute the gradient of a function, it can be made with matematica using Grad[f[x,y],{x,y}] Suprisingly, it's not possible to compute the Hessian matrix (something as HessianMatrix[f[x,y],{x,y}], ...
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0answers
36 views

NSolve as integration limit

I need the (numerical) solution of equation to be integration limit as below. How to make it work? ...
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1answer
61 views

Understanding higher order derivative result

I have problems in understating the result of the computation of higher order derivates. D[(w - 1) f[w], {w, n}] returns: ...
1
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1answer
129 views

How does Mathematica evaluate these sum and integral?

How does Mathematica internally evaluate the following (interrelated) sum and integral, and how does it do the subsequent simplification? (I mean, based on what mathematical facts?) ...
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0answers
49 views

Why that difference in outputs of FunctionInjective?

Trying a new in 12.2 command FunctionInjective, I execute FunctionInjective[{Sin[x]^2, Element[x, Integers]}, x] ...
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0answers
50 views

Numerical integration of oscillatory integrand

Having two points $\vec{r_1}$ and $\vec{r_2}$ I interested in calculating the following integral $\int_{0}^{\infty} exp(-ar)r_1r_2dr_1dr_2d\phi_1d\phi_2$, where $r = |r_1-r_2| = \sqrt{r_1^2+r_2^2-...
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0answers
69 views

First derivative of curve and onset and endset points

If I have the following data: ...
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0answers
37 views

Is there a function to quickly find the Cauchy main value of a function at a pole?

Suppose, f[x] has a pole at a. How do I quickly find Limit[(f[a+h]+f[a-h])/2,h->0] ? I ...
2
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3answers
41 views

ND gives expression instead of number

I have a function sol defined as a solution to an equation (computed numerically). Specifically ...
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1answer
29 views

Plot of derivative does not show

I have following function 4/((1 + v1[h])*(1 + v2[h])): ...
3
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0answers
59 views

About what does FunctionSurjective think so long time?

Having tried a new in 12.2 command FunctionSerjective, I executed FunctionSurjective[z^(5/2), z, Complexes] // AbsoluteTiming ...
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0answers
67 views

How to get a result for $\int{\sqrt{a^2 \left(\sin{a x}-2 \sin{2 a x}\right)^2+1} \, \mathrm{d}x}$?

I want to obtain the result of this integral where $\mathbb{R}\ni a,x>0$ $$\int{\sqrt{a^2 \left(\sin{a x}-2 \sin{2 a x}\right)^2+1} \, \mathrm{d}x}$$ but Mathematica does not give an answer. Of ...
0
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1answer
61 views

Finding an approximate solution to this integral: $\alpha(\phi,r,p,d)=\int_0^\infty w(z,r,p,d)Q(z,r,\phi)dz$

I'm working on a physics problem and encountered a rather complex integral for which I'm trying to find an approximate solution. The integral is of the following form: $\alpha(\phi,r,p,d)=\int_0^\...
6
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1answer
175 views

Could anyone please suggest me MATHEMATICA code to solve Fractional-Order Time Delay Differential System?

We are familiar that the Predictor-Corrector Algorithm is applied to solve the Fractional-Order Differential systems. I need to solve the following Fractional Order Delay Differential System. Please ...
1
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1answer
192 views

Program for efficient computation of given functional:

I need to plot the following functional with accuracy: $$ I(x,s) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy,s) − F(x −\mathrm iy,s)}{\mathrm e^{2πy}-1}, $$ Where $ F(z,s) = \dfrac{1}{z^s\Gamma(\...
1
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1answer
43 views

How to continually simplify derivatives of function defined in a system? [closed]

For example, if I input something like b'[t_] = a[t]; a'[t_] = -Sin[b[t]]; a''[t] it outputs just a''[t]. Is there a way to ...
0
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0answers
28 views

How to return an inverse function based on parameters (for formal differentiation)

I want to return a different function for any given parameter in order to later calculate a value of the inverse of that function given the parameter as an indeterminate. That is, I want, for each $d$,...
2
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2answers
124 views

Asymptotic expansion of an Euler-type integral

I am looking for an asymptotic expansion for large $x$ for the following integral: $$\int_{0}^{1} \frac{t^{ia}(1-t/2)^{-ia}}{\sqrt{1-t}} e^{ixt} dt$$ $a$ and $x$ are real and positive. I tried using ...
1
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0answers
66 views

Strange behaviour of Integrate [closed]

Mathemtaica correctly computes the following definite integral: Integrate[(q Log[q]^3)/(1 - q), {q, 0, 1}] (* Out: 6 - \[Pi]^4/15*) But, when I change the ...
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0answers
34 views

Differential operator in Mathematica

I am fairly new to Mathematica and I wanted to know if I can define the following operator $E:=(x^2\sqrt{1-K(x^2+y^2+z^2}+y^2+z^2)\gamma^{1}\partial_{1}+(y^2\sqrt{1-K(x^2+y^2+z^2}+x^2+z^2)\gamma^{2}\...
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1answer
40 views

Best way to integrate over a list of data

I have a list given by: ...
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2answers
95 views

How to extend this integration to all the reals?

Doing an exercise on calculus with 12.2 on Windows 10 Pro, I obtain a = Integrate[Sqrt[1 + Sin[t]], {t, 0, x}, Assumptions -> x \[Element] Reals] ...
2
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1answer
78 views

Can Limit take limits of unknown functions?

For example, when I input this f[0] = 0; Limit[f[x], x -> 0] , it does not output 0. Is there a way to get it to output <...
0
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1answer
49 views

Derivatives of trigonometric functions

Let's use the following sample case Clear["Global`*"]; t1 = ArcTan[y/(x - x1)]; f = (3*(Cos[t1])^2 - 1); der = D[f, x] which gives ...
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3answers
96 views

How to calculate an iterated derivative in Mathematica?

I try to calculate an inverse Mellin transform for $s^n \Gamma(s)$: $$x\frac{\mathrm{d}}{\mathrm{d}x}\left(x\frac{\mathrm{d}}{\mathrm{d}x}\left(e^{-x}\right)\right)$$ for $n=2$ for $n$ as $$\left(x\...
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0answers
27 views
-1
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1answer
41 views

Why does the function Integrate give a different result?

It seems that Integrate gives a different result when a integrand is in an different form. ...
4
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2answers
90 views

Inefficient Manipulate code with plots and integrals

I did this manipulation piece based on Boltzmann distribution but I noticed that is rather slow, any ideas how to improve this code? ...
2
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2answers
98 views

How to compute numerical derivative after numerical integral?

Here's a simple example code to demonstrate my issue: f[x_, a_] := a x^2; g[a_] := NIntegrate[f[x, a], {x, 0, 2}]; ND[g[a], {a, 0}] Running this tells me that the ...

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