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.
3,562
questions
0
votes
0answers
15 views
Performing Integration over Double Logarithmic function
I have been trying to perform the following Integration:
...
0
votes
0answers
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
...
6
votes
2answers
231 views
Prove $R^\top R = I_3$ and find skew-symmetric matrix $R^\top \dot R$
Consider the following operator defined over unit vectors of $\mathbb{R}^3$:
$$R(u,v) = (u\cdot v)I_3 + hat(u\times v) + \dfrac{(u\times v)\otimes (u\times v)}{1+u\cdot v}$$
...
0
votes
0answers
57 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
votes
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
$$
\begin{align}
\...
1
vote
2answers
68 views
Why does Integrate[] fail to give me an analytical expression
I am trying this integral to solve analytically, but Mathematica fails, and I get the integral back as it is defined. Although numerically it can be done, I need an analytical expression. Is there any ...
1
vote
1answer
46 views
Integral of complex function
How to compute a complicated complex integral? You can see the example in the figure. I set assumptions that all constants are positive and real, then I try to integrate the $pel(z)$ expression but ...
2
votes
0answers
54 views
Calculating double integral bounded by domain in Wolfram Alpha Notebook
If I want to compute Double Integral on a given region(See attached Picture)
How can I input this on Wolfram Alpha Notebook
1
vote
1answer
192 views
Program for efficient computation of given functional:
I need to plot the following functional with accuracy:
$$
I(x,s) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy,s) ā F(x ā\mathrm iy,s)}{\mathrm e^{2Ļy}-1},
$$
Where $ F(z,s) = \dfrac{1}{z^s\Gamma(\...
7
votes
4answers
242 views
Why the difference in the results of Integrate and NIntegrate?
Bug introduced in version 12.0.0, and persisting through 12.2.0.
Calculating the integral $$\int\limits_0^1 \frac{x^2\log(1-x^4)} {1+x^4}\,dx$$ symbolically
...
5
votes
0answers
118 views
Numerical verification of the estimate:
How to verify numerically with considerable accuracy in Mathematica the following :
$$\int_2^x\dfrac{1}{z\Gamma(\sin^2[Ļ\Gamma(z)/(2z)])}dz\sim\ln(\ln(x))$$
?
I need more suitable and better code ...
0
votes
0answers
25 views
Pure functions with derivatives and list elements [duplicate]
I have a list
list = {x, y, z}
and then the following line
D[#, list[[2]]] &
gives the output
...
4
votes
1answer
187 views
Plot of a function defined by an integral
(First of all, this is my first Mathematica question. I'm not used to Mathematica that much. So, apologies in advance.)
I need to plot the following functional with accuracy:
$$
I(x,s) =\int_0^\infty\...
1
vote
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:
...
4
votes
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 ...
12
votes
2answers
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 $y$ ...
0
votes
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}} + \...
0
votes
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 ...
13
votes
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 ...
12
votes
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 ...
2
votes
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:
...
1
vote
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 ...
0
votes
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).
0
votes
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
votes
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 ...
0
votes
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(...
0
votes
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 ...
0
votes
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
vote
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}], ...
2
votes
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 ...
0
votes
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?
...
1
vote
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?)
...
2
votes
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:
...
0
votes
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]
...
0
votes
0answers
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-...
0
votes
0answers
69 views
7
votes
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 ...
1
vote
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 ...
0
votes
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.
...
0
votes
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)=...
2
votes
3answers
41 views
ND gives expression instead of number
I have a function sol defined as a solution to an equation (computed numerically). Specifically
...
0
votes
1answer
29 views
3
votes
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 ...
6
votes
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
votes
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
...
0
votes
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^\...
1
vote
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 ...
4
votes
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
votes
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 ...
1
vote
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 ...
