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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15 views

Performing Integration over Double Logarithmic function

I have been trying to perform the following Integration: ...
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37 views

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 closed form found here. So I tried ...
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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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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 ...
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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
68 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 ...
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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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54 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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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(\...
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242 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 ...
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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 ...
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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 ...
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1answer
187 views

Plot of a function defined by an integral

(First of all, this is my first Mathematica question. I'm not used to Mathematica that much. So, apologies in advance.) I need to plot the following functional with accuracy: $$ I(x,s) =\int_0^\infty\...
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1answer
170 views

Simplify function with a derivative

This may be a very simple question: how can I substitute out a term involving a derivative in Mathematica? I have the following derivative: ...
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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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530 views

Can we abuse notation and write equations in differential one-form?

Consider a simple equation with a one-form on both sides (mathworld.wolfram is also aware of this): $$ y\mathrm{d}x = \mathrm{d}y $$ This is a perfectly valid abuse of notation. We can carry the $y$ ...
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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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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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3answers
2k views

Why does Integrate declare a convergent integral divergent?

When I try this command Integrate[1/Sqrt[(s^2 - u)^2 - 1], {s, m, Infinity}, Assumptions -> u > 2 && m > 10] Mathematica declares that the ...
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3answers
564 views

Assumptions allowing to calculate an elliptic integral

When I feed Mathematica the following integral: Integrate[Sqrt[(A - x) (B - x)/x], {x, 0, B}] it spits it back out without evaluating it. However, it can ...
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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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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 ...
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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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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^{-\...
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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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1answer
46 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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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
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 ...
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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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2answers
188 views

Alternative to Dump Save

When dealing with large matrices I like to store them in the memory once for all, such that when I restart Mathematica I do not have to wait for a very long computational time of order of minutes. To ...
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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
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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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: ...
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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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51 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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69 views

First derivative of curve and onset and endset points

If I have the following data: ...
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4answers
983 views

How to transform the result of this integral into powers of $\sin\,x$?

I want to calculate the integral $I = \displaystyle\int{\cos ^5 x\, \mathrm dx}$. Using the variable $t = \sin x$, I get $$I = \sin x - \frac{2}{3}\sin^3 x + \frac{1}{5}\sin^5 x + C.$$ With ...
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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 ...
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3answers
140 views

Problem with finding a potential function of a vector field

I have this code where I try to calculate the potential function of the vector field F. ...
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3answers
1k views

Revolution of two curves around the y-axis

I've looked through the other posts on rotating a function around the x or y axis and haven't had any luck. my question is this how do i rotate the area between the functions $f(x)=x^2+1$, and $f(x)=...
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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])): ...
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2answers
196 views

Reconstructing a function from its gradients

I have a list of the components of the gradients, $\partial f/\partial x_i$, of a function $f(x_1,x_2,\cdots)$. Is there some neat way to reconstruct the function $f$? One approach to doing this ...
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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 ...
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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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^\...
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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 ...
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1answer
277 views

Mathematica Can Find the Primitive Function But It Cannot Find the Closed Form for Corresponding Definite Integral

Consider the following function f = 1/2 (-3 + 4 u) Sqrt[-u (1 + u) (-1 - u + u^2)] where it is assumed that ...
4
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2answers
166 views

How to determine the frequency of oscillations in system of three ODEs?

I want to determine the frequency of oscillations in a system of three stiff ODEs (Oregonator model). That model describes a chemical oscillator. I have a slightly more advanced model of the default ...
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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 ...

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