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.
4,921
questions
2
votes
3
answers
104
views
How to modify the given codes (NIntegrate of Boole) to get the most accurate result possible?
I have five conditions con1 to con5 over the two-dimensional domain $t,u\in[o,2\pi]$. I want to compute this area $(con1 \cap con2 )\cup(con3 \cap con4 \cap con5)$ numerically. I try the two codes ...
- 349
0
votes
2
answers
81
views
Computing $\frac{1}{d}\int_1^d \int_1^d \mathrm{d}i\ \mathrm{d}j\ \left(f(i,s)-f(j,s)\right)^2$ for $d\to\infty$
I'm looking to compute the following integral
$$U(s)=\frac{1}{d}\int_1^d \int_1^d \mathrm{d}i\ \mathrm{d}j\ \left(f(i,s)-f(j,s)\right)^2$$
Where $f(i,s)=i^{-p}\exp(-2s i^{-p}), s>1, p>1, d\...
- 8,135
2
votes
2
answers
107
views
Integrating over Implicit region, trash output
I want to compute lebesgue measure of region {$(x,y,z) \in R^3 : (x^2+y^2+z^2+3)^2 \leq 16(x^2+y^2), y>0$}. So I type:
...
1
vote
1
answer
128
views
1
vote
2
answers
145
views
Finding closed form representations of floating point numbers?
Is there a trick to automatically recognize that ${-0.5771,1.2280}$ are approximately equal to {-EulerGamma, Log[2 + Sqrt[2]]}?
These were obtained by using ...
- 8,135
4
votes
2
answers
259
views
Displaying integrals with non-variable factors in front [duplicate]
In class I sometimes go through the steps in mathematical derivations using Mathematica. Some of the steps involve substitutions and assumptions that come conceptually outside a strict mathematical ...
- 39.9k
0
votes
0
answers
40
views
Poles of filter transfer function, fifth order equation [duplicate]
I am just trying to find the roots of the fifth-order equation. But I could get enough result from Wolfram.
Could anyone help to me about this issue?
...
2
votes
3
answers
101
views
Plotting $\frac{1}{d}\frac{\sum_{ij}^d(i^{-p}-j^{-p})^2}{\left(\sum_i^d i^{-p}\right)^2}$ for large $d$
I'm trying to understand large $d$ behavior for $F(p,d)$ with $1\le p\le 2$ and $d>10^6$
$$F(p, d)=\frac{1}{d}\frac{\sum_{ij}^d(i^{-p}-j^{-p})^2}{\left(\sum_i^d i^{-p}\right)^2}$$
The following ...
- 8,135
1
vote
0
answers
77
views
CaputoD as Caputo Fractional Derivative [closed]
I hope everyone is doing well. I have a question about mathematica that I have attached here. If you look into the attached picture, then you can see the Derivative of unknown function f[t] in ...
1
vote
1
answer
105
views
How to sort eigenvalues (from minimum to maximum) and their corresponding eigenvectors?
When using NDEigensystem, the first eigenvalue corresponds to the first eigenvector, the second eigenvalue to the second eigenvectors, and so on. I would like to sort the eigenvalues from min to max, ...
- 1,753
3
votes
2
answers
85
views
What is the meaning of adding/subtracting a fraction in a summation limit inside a limit?
In Wolfram Alpha entry 1 the following is claimed to be true:
$$\lim_{n\to \infty } \left(\sum _{k=1}^n \log (2 k)-\sum _{k=1}^{n+\frac{1}{2}} \log (2 k-1)\right)=\frac{1}{2} \log \left(\frac{\pi }{2}\...
- 1,149
6
votes
4
answers
406
views
Approximation of the Fabius function with a quotient of exponentials
Approximation of the Fabius function $f(x) = \text{FabiusF}[x+1]\cdot \text{HeavisideTheta}[1-x^2]$ - FabiusF[x] doesn't work in Wolfram-Alpha
I am looking to figure out how well the displaced version ...
- 163
4
votes
2
answers
192
views
Finding the sum of eigenvalues of a matrix depending on the parameters
There is a system that has the following Hamiltonian: $H=-\frac{1}{2}\Delta-\frac{1}{r}+\frac{B^2}{8}\rho^2-\frac{B}{2} (m + 2 m_s)$, where $r=(\rho,z,\phi), B=5, m=0, m_s=-1/2$. To find the ...
- 1,753
2
votes
4
answers
537
views
1
vote
2
answers
307
views
3
votes
3
answers
226
views
Approximating $Q^{-1}\left(\frac{1}{x+1},\frac{1}{2}\right)$ for large $x$
The following function appears linear on LogPlot, what is the an elegant way to get a large-x series expansion?
...
- 8,135
2
votes
3
answers
171
views
How to write a function in the following form f[a_, b_, c_,...] so that it is flexible with respect to the number of arguments?
@lericr have written under several of my questions that the way I try to modernize the code could be made much simpler. Below is the original part of the code. The remaining parts have similar designs,...
- 1,753
2
votes
1
answer
86
views
How to correctly reassign variables {c11 -> p11, c12 -> p12} using a Table or some other similar structure?
Why is the value of the function Pxx[1,2] is a number but the function Pxxx[1,2] is an expression? Isn't this construction ...
- 1,753
0
votes
1
answer
76
views
Plugging in a value after taking partial derivative [closed]
I am trying to write a code where I have a function of two variables, take the partial derivative with respect to each variable and finally plugs in specific values for the variables. To be more ...
- 15
4
votes
2
answers
213
views
Integrate $x^{-d/2}(1-x)^{(d-3)/2}$
I am trying to find the result of $$\int x^{-d/2}(1-x)^{(d-3)/2}dx$$ where $0<x<1$ and $d\ge2$ is an integer.
Mathematica gives the following:
...
- 143
2
votes
2
answers
86
views
How to use sequence of variables (Sequence) correctly?
Why does the function EER[1,2] have a different value than the function EERR[1,2]? Isn't the entry ...
- 1,753
1
vote
2
answers
69
views
Why does Mathematica not see some of the function PxB1[a10_, b10_, qa0_, qb0_, c11_, c12_] variables?
Why does Mathematica not see the c11 and c12 variables in the function PxB1[a10_, b10_, qa0_, qb0_, c11_, c12_]? How to fix it?
...
- 1,753
2
votes
2
answers
75
views
The correct notation of a function that contains an expression with NIntegrate
I tried to set the function Px[a1_, b1_, qa_, qb_]. Why does Px[1.22, 0.44, 1.40, 1.81] not give out a specific number?
...
- 1,753
1
vote
0
answers
65
views
Asymptotics for implicit functions involving Zeta, Log, ExpIntegralEi
Edit Roman describes how to automate asymptotic inversion for problems of the form below under "without Solve" section of this answer
I'm interested in a streamlined way of obtaining ...
- 8,135
0
votes
1
answer
32
views
Is it possible to ask Mathematica to order this expression in terms of descending absolute values of the arguments of the exponential functions?
I have the given expression which is a sum of exponential functions with some parameters $a,t,u,y$ where $a>0$.
Question
Is it possible to ask Mathematica to order this expression in terms of ...
- 349
4
votes
2
answers
86
views
Why do I get a list from an integral over a region when my integrand is scalar?
In Mathematica 13.2.1.0 on Windows 10, this integral gives an unexpected result:
Integrate[EuclideanDistance[x, {2, 0}], x ∈ Circle[]]/ RegionMeasure[Circle[]]
<...
- 23.9k
3
votes
1
answer
84
views
Plot propagates MachinePrecision numbers despite `WorkingPrecision`
Edit solved, Plot calls candidate function using MachinePrecision for its argument, hence need to use SetPrecision to recover ...
- 8,135
4
votes
1
answer
230
views
Getting a series expansion for implicitly defined function
I have a function $f(r)$ defined as $x$ satisfying $g(r,x)=0$.
How do I get a series expansion of $f(r)$ around $r=\infty$?
Function $f(r)$ below is unexpectedly linear, and I'm trying to get a closed ...
- 8,135
1
vote
1
answer
76
views
a problem in numerical integration in Mathematica
I am trying to numerically compute an integral of the following type in Mathematica. I can show that this integral can be computed analytically. Still, I want to make sure Mathematica can also give me ...
- 109
0
votes
1
answer
53
views
Calculate double integral with boundaries [duplicate]
I'm working on calculating the following integral in Mathematica:
$$\iint_D (30 x^2 + 49 xy + 20 y^2)^3 \, \mathrm{d} x \, \mathrm{d} y,$$
where $D$ is a region enclosed by four lines: $6x+5y=3$, $6x+...
- 27
3
votes
0
answers
87
views
Calculate the integral of the Slater determinant
This is a Slater determinant:
$$
s=\left|\begin{array}{ll}
\psi_{1 s}\left(r_1\right) \alpha & \psi_{1 s}\left(r_1\right) \beta \\
\psi_{1 s}\left(r_2\right) \alpha & \psi_{1 s}\left(r_2\right)...
- 1,402
1
vote
1
answer
65
views
Double integral problem [closed]
I want to calculate this double integral using Mathematica:
$$\iint_D x^3 y^3 \ln(1+x^2+y^2) \, \mathrm{d}x \, \mathrm{d}y,$$
where
$$D=\left\{ (x,y): x^2 + y^2 \leqslant 4, \; x,y \geqslant 0 \right\}...
- 27
0
votes
1
answer
59
views
Calculate definite integral
I have the following integral that I want to calculate in Mathematica:
$$\int_x^y t^z \ln t \, \mathrm{d}t,$$
where $z$ is an integer and $0<x<y$.
What code can calculate this integral?
- 27
0
votes
0
answers
33
views
Computing Asymptotics of a Fresnel Integral At Infinity
I am very new to Mathematica, and am trying to write code that will give me the series asymptotics for the function
$$ F(y) = \int_0^y f(y)\;dy, $$
where $f(y) = y^2 \cos(y^2) C( (2/\pi)^{1/2} y)^2$, ...
- 101
1
vote
1
answer
142
views
Never ending Integrate computation
I am studying the result of a Feynman Diagram, trying to apply the Renormalization Group to a Field theory for Active Matter. I have arrived at a point where I want to integrate with Mathematica a ...
- 13
1
vote
2
answers
58
views
Integrate function [closed]
I want to find the constant in front of the term of the integral that only contains $x$ in the following function:
$$f(x) = \frac{(56-15 \sin x + \sin^2 x)^2}{6-5 \sin x + \sin^2 x}$$
This is my input ...
- 27
3
votes
2
answers
163
views
Optimizing execution time and exploring closed form solutions in Mathematica's DSolve
I am encountering two issues while running the following code in Mathematica:
Slow execution: I have been running the code for approximately 30 minutes, but it is still executing. I am using Windows ...
- 507
0
votes
0
answers
53
views
how to integrate the function {Exp[2/r + 2 r] - 1}^(1/2) with respect to r?
I have tried to integrate the function {Exp[2/r + 2 r] - 1}^(1/2) using Integrate command but no result came, can anybody please ...
3
votes
1
answer
55
views
Help in solving high-dimensional definite integral
I am trying to solve a 6D integral over this integrated
...
- 165
0
votes
2
answers
167
views
An integral with residues [closed]
I am trying to calculate the following integral using the residual method
$$ \int_0^{2\pi}dx\frac{n \cos(x)(-a + 2r\cos(x))\sin(n(x+t))-(a-2)\cos(n(x+t))\sin(x)}{(a-2r\cos(x))^2} $$
where $n$ is a ...
- 9
0
votes
0
answers
151
views
Inconsistent results in symbolic integration with Mathematica
I'm encountering a problem with symbolic integration in Mathematica that is giving me inconsistent results even when using assumptions and clearing the variables.
...
1
vote
1
answer
52
views
How to modify NIntegrate code below so that it gives the most accurate/reliable numerical result?
Imagine that I have two conditions con1 and con2 over the two-dimensional domain $x,y\in[-2,2]$. I want to compute the area of the union of these two conditions i.e. the region $(con1 )\cup(con2)$. I ...
- 349
2
votes
1
answer
133
views
Mathematica doesn't solve the integral [closed]
I need to find primitive function to:
$$f(x)=\frac{(168-26\sin{x}+\sin^2{x})^2}{6-5\sin{x}+\sin^2{x}}$$
I'm very new to Mathematica and its syntax, but I think I wrote the code correctly.
...
- 187
0
votes
1
answer
41
views
Derivative with respect to a sequence
I want D[Subscript[a, k],Subscript[a,j]] to return something like the Kronecker delta. Is this possible?
0
votes
0
answers
41
views
Finding the first and last intersection points from a lists of plots
I am trying to reproduce Figure 4 from this paper (page 7), for which I need to use the line $\tilde{T_p}$ from this figure:
in the output of this figure:
Now by varying the value of $\tilde{Q}$, I ...
- 899
0
votes
1
answer
47
views
Resubstituting variables back into the calculated
I have lengthy second order derivative of a function that is defined with multiple lines of variables. When I compute the derivative how can I put back into the computed result the auxilary variables ...
- 1,580
8
votes
3
answers
456
views
Inverse Laplace Transform of Hypergeometric function
Any tips how to massage the following to get computed by Mathematica for $p>1$? I suspect the result should be expressible in terms of exponential integral
...
- 8,135
2
votes
1
answer
64
views
Problem with integration with indexed variable
I wonder why I can not have a correct integration form on Mathematica with some indexed variables.
Indeed, when I write :
...
- 23
0
votes
1
answer
71
views
InverseLaplaceTransform producing unexpected imaginary terms
The following code computes inverse Laplace Transform (trueLaplace) manually and using Mathematica (observedLaplace) of ...
- 8,135
3
votes
1
answer
64
views
Make a new list from another list
I have a list like this:
{{1, 2}, {3, 4}, {6, 8}, {9, 0}, {10, 11}, {12, 15}}
I use this code and make a new list.
...
- 115