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.
3,726
questions
0
votes
2answers
38 views
Triple integral: How do I include assumptions and conditions?
I would like to solve the following integral:
$$I=\int_{0}^{a} \int_{0}^{y} \int_{0}^{z} e^{(a-x)^{3}} d x d z d y$$
with the condition that:
$$a>0$$
So I tried to input the equation like below ...
0
votes
0answers
46 views
Is it possible to calculate the given area by an analytical exact expression?
I have the three two-variable curves $f,g,h$ with $0\leq x\leq \frac {\pi}{2} $ and $0 \leq y \leq \pi$
I use ContourPlot as
...
0
votes
0answers
53 views
Solving ODE with initial conditions uses a spurious and unnecessary initial value equal to unity
DSolve[ {f'[t] + \[Omega][t] f[t] == 0, f[t0] == 1}, f[t], t]
Gives,
...
0
votes
3answers
78 views
How to extract a factor from equation and rearrange the equation?
I want to extract a factor from the equations, it's not a common factor, just a factor defined by myself, like this:
(ac+b) --> c(a+b/c)
I have tried Factor,<...
3
votes
3answers
190 views
solution of $\frac{1}{x}=\infty$
Is there a way to obtain solution $x=0$ for the equation $\frac{1}{x}=\infty$ in Mathematica?
I tried Solve, Reduce, ...
0
votes
1answer
74 views
Different results in different versions of Mathematica (computing integrals)
I am working on computing the following integration:
aa = N[2, 20]
H = 4/160;
X = Range[-aa, aa, H];
Integrate[400/(1 + 40*x^2), {x, X[[k]], X[[k + 1]]
and ...
4
votes
1answer
75 views
Substitute known function's derivative
Suppose I know that the time derivative of $x(t)$ is $\dot{x}(t)=-x(t)$. I consider the function $V(x)=x^2$ and I want to compute the time derivative of $V$ along the trajectories of $x$, namely $\dot{...
1
vote
0answers
47 views
Closed form for integral with SinIntegral
MMA can't solve this integral:
$$\int_0^{\pi } \text{sinc}(x) \text{Si}(2 x) \, dx$$
The I use FourierSinTransform to find closed form,but I try a verify result is ...
5
votes
4answers
135 views
Are there any alternatives for NIntegrate to calculate the area for which $f(x,y)<0$?
I have this two-variable function
$$f(x,y)= (8 \cos (x+y)+7)\cos \left(\frac{x}{2}\right)+\cos \frac{x-2 y}{2}+2 \cos \left(\frac{3 x}{2}\right) $$
where $0<x,y<\pi$. I want to calculate ...
1
vote
0answers
45 views
Can you check analyticity of a complex function in Wolfram Mathematica? [closed]
Can you check analyticity of a complex function in Wolfram Mathematica?
0
votes
0answers
30 views
Integration with assumption [closed]
I am trying to evaluate the following integral on Mathematica 12.2 Student Version
$ \int_{0}^{\pi} xsin(nx) \,dx $.
I tried :
...
0
votes
1answer
49 views
Integral with exponential function
Could you please help me in solving the following integration. Thank you in advance.
Integrate[((Sqrt[g*(x + h)/y])^V) * Exp[-u * Sqrt[g*(x + h)/y]/n] * Exp[-L*x], {x, 0, Infinity}]
0
votes
2answers
220 views
Integrating an exponential with upper incomplete gamma functions
I would greatly appreciate calculating an integral consisting of an upper incomplete gamma function and an exponential function.
...
5
votes
4answers
168 views
0
votes
0answers
28 views
Nintegrate of a list of numbers
I have an issue. I have a function in two variables obteined from the table of a numerical integral (done with Nintegrate). Now I have to compute another integral (Nintegral) over it so I will have a ...
0
votes
2answers
86 views
Calculating the convergence value of summation
Could you please tell me what value the left side of the inequality converges to?
(H and P are constant values greater than 2 and n can go to infinity)
$$\sum_{n=1}^...
6
votes
3answers
305 views
Express MeijerG as integral
For definite integrals MMA gives identities in terms of Meijer G-functions, e.g.
$\begin{align}\sqrt{\pi}\int_0^\infty \textrm{e}^{-4x/t^2-t}\ \textrm{d}t &=
G_{0,\,3}^{3,\,0} \left( x\left.
\...
1
vote
1answer
44 views
How to generate assumptions automatically for definite integrals？
In Mathematica, a trick to speed up the process of definite integrals is by adding assumptions. For example, the following codes run faster if we add such assumptions:
...
-2
votes
0answers
37 views
How can we write code to find the number of patterns except impossible code? [duplicate]
I'm writing a code to find the number of lock patterns on my Android phone.
In 3.가능한 패턴의 경우의 수(Number of possible pattern cases), How can we write code to find the number of patterns except impossible ...
5
votes
2answers
116 views
Efficient computation of Hopf index
I am trying to check using Mathematica the following result:
$$-\frac{1}{4\pi^2}\int_{0}^{2\pi}\int_{0}^{2\pi}\int_{0}^{2\pi} \mathbf{F}\cdot\mathbf{A}\;dk_x dk_y dk_z=1,\quad\text{where}$$
$$z_{\...
8
votes
4answers
271 views
Challenge: Computing the Fourier Series of a List Plot!
I have a bunch of data points (copied from Mathematica below), which I have 'ListPlotted'
I would now like to compute the Fourier series of this function (after doing a Fourier transform I assume? Or ...
3
votes
0answers
81 views
Integrate providing incorrect result for complex exponentials
Consider the integral
\[\int_{-\infty}^\infty\frac{(e^{iax}-1)(e^{ibx}-1)}{x^2}\quad a,b\in\mathbb R\]
which can be manually computed to be $\pi|\text{sgn}(a)-\text{sgn}(b)|\min(|a|,|b|)$.
However ...
1
vote
2answers
52 views
Series expansion for two limits of x [closed]
I have a function f($x$) given by the expression
$$f (x) = \frac{\left(1+x\left[1-\sqrt{1+x^2}\right]\right)^2-x+x^3\left[1-\sqrt{1+x^2}\right]^2}{1+x^2\left(1-\sqrt{1+x^2}\right)^2}$$
and would like ...
0
votes
1answer
79 views
Incorrect results of Integrate
The commands
Integrate[1/(z + Exp[I*t]), {t, 0, 2*Pi}, Assumptions -> Abs[z] == 1]
(2 \[Pi])/z
and
...
0
votes
1answer
29 views
How to write a function taking limit of multivariables in sequence?
I'm trying to write a function to calculate the limit of a function on a number of variables in sequence.
Here I take a function with four variables x, ...
0
votes
1answer
41 views
How to compute the max/min surface area of a donut-shape solid generated by a revolved 2D circle, as the volume of the solid doesn't change?
A donut shape solid is generated by revolving a circle $(x-a)^2+y^2=b^2$ around the y-axis.
$a$ is the distance from the center of the hole of the donut to the center of the circle revolved, and $b$ ...
3
votes
1answer
130 views
Analytical Integration
I have a question on analytical integration.
z = y/ps; n = 0.95;
A1 = Integrate[((a (a + z) - r^2)/r)^0.95 - (n z) ((a (a + z) - r^2)/r)^-0.05, r]
Answer:
...
6
votes
3answers
308 views
How to rotate a circle around the y-axis and calculate the volume of the 3D solid?
How do I rotate the equation ${(x-5)^2}+{y^2}={3^2}$ around the y-axis and calculate the volume of the plotted 3D solid?
I tried the code below but the result doesn't work.
...
-1
votes
1answer
56 views
How to plot the solutions of $f(z,a)=0$ for $z$ in the complex plane?
I have an equation $f(z,a)=0$ where $z$ is a complex variable $\frac {56}{10}<z<2\pi$ and $a$ is a real variable $0<a<\pi$.
I want to plot solutions of this equation on the complex plane $...
1
vote
0answers
51 views
Why Mathematica is not computing this integral? [closed]
Why is the following integral not computed
Integrate[(Exp[-t] - 1)/
Sqrt[2 + Exp[-2 t] - Exp[-t] -
Exp[t (-(1/2) - I a) - Exp[t (-(1/2) + I a)]]], {t, 0, 1}]?
...
0
votes
1answer
57 views
Complex values for a supposed real integral
I have an integral for which I have checked that it is supposed to be real but there some values of zm which gives a complex value.
First I checked the validity ...
-3
votes
1answer
93 views
5
votes
3answers
320 views
How do you find information regarding the derivative?
I have been struggling with an issue that I believe has a simple way to solve.
Given that I have extracted a term of a partial differential equation:
f(1,2,0) [x,y,z]. In this case, it is similar to ...
0
votes
2answers
63 views
How to illustrate schematically (animated) solutions of a system of $3$-variable equations by increasing one of the variables with step $0.01$?
Following my previous question
here, I am looking an alternative way to solve this problem (the method was not applicable to more complicated situations). I have a system of two 3-variable equations $...
12
votes
2answers
299 views
Total flux of the gradient of the numerical solution of a PDE through a surface
Trying to solve the following PDE with BC T==1 on a spherical cap of a unit sphere and T==0 at infinity (approximated as ...
5
votes
1answer
129 views
How to plot solutions of a system of equations by changing colours when one of the variables increases by step?
I have a system of two 3-variable equations $SysEqs=\{f(x,y,z)=0\;,\;g(x,y,z)=0\}$ where the domain of variables are
$$6 < x < 2 \pi \\
- \frac{15}{100} < y < 0\\
0 < z < \frac 32$$
...
0
votes
5answers
131 views
How to calculate the total area of $\frac{\cos (2 y) \cos (2 x-y)+\cos (3 y)}{\cos (2 x-y)+\cos (y)}$ enclosed in the range $[0,1]$?
I have this two-variable function
$$f(x,y)=\frac{\cos (2 y) \cos (2 x-y)+\cos (3 y)}{\cos (2 x-y)+\cos (y)}$$
where $0<x,y<2\pi$. I would like to calculate numerically the total area of the ...
1
vote
1answer
37 views
Computing Partial Derivatives of a Function wrt Another Function
I have two functions $P_1,Q_1$ which are both depending on the variables $r_{T_m}, r_{i_f}$, which are theirselves functions of $T_m, mi_f$. I am interested in computing the Jacobian
$$\frac{\partial (...
4
votes
1answer
180 views
Gradient of the numerical solution of a PDE
Solving the following PDE, defined in the same domain of a previous question:
...
1
vote
1answer
60 views
How to obtain numerical solutions of a system of two equations with two unknowns?
I have two equations with two unknowns $-10<x<0$ and $0<y<15$
$$3 y \cosh (4 x) \cos (y)+(2 x+5) \cosh (x) \sin (y)=0,\\3 x \cosh (x) \cos (y)-(2 x+5) \sinh (x) \cos (y)=0$$
I would like ...
1
vote
0answers
38 views
Derivative of unknown functions and constants [closed]
I'm trying to do symbolic differentiation with Mathematica for expressions with unkown functions and constans.
For example the following expression:
$$\text{expr} = \omega_1 Q_1^2 Q_2 + \frac{Q_1^3}{\...
0
votes
0answers
39 views
Calculating correlational dimension using Mathematica
I am modelling such a system:
$T:(x,y)\rightarrow (d+C_2(x\cos\tau - y\sin\tau),C_2(x\sin\tau+y\cos\tau))$,
$\tau = C_1-C_3/(1+x^2+y^2), C_1=0.4, C_2=0.9,C_3=0.6,d=0.6$.
I am applying $T$ to point $(...
0
votes
0answers
41 views
Findroot and arc length,similar situation but does not work
In another problem I was presented with this situation
It is to find the arc length of the curve
...
1
vote
0answers
62 views
Step-by-step solution for a Limit
Is it possible in Mathematica to get a step-by-step evaluation of some functions? Such that Mathematica is outputting not only the result but all the stages that have led to it? If so, how does one do ...
2
votes
0answers
53 views
How to find the regions for which the derivative of my function changes sign?
I have an energy function of the angles of the electrons' spins. th1 is vector with (2l+2) elements and each element represents the angle of an individual electron spin. I need to eventually find the ...
2
votes
2answers
122 views
Efficient way of calculating the arc length of a parameterized curve?
give me the curve :I am asked for the arc length between 0 and 2 (although I think it is between 0 and 2Pi) and it gives as a result 100
...
1
vote
0answers
26 views
How to output divergence or gradient of a symbol in the form of partial derivatives? [closed]
As being new to divergence and gradient operators, I am trying to make them more readable, so could anyone tell me how can I output the divergence of u={{ux},{uy},{uz}} in the form of partial ...
2
votes
2answers
130 views
Two-Points Padé Approximant
My Question is about two points Pade approximant
From Mathematica References. I just find the Pade approximant for a real function in one point
Example :
...
0
votes
2answers
29 views
Supremum of a complex periodic signal in symbolic form
For complex periodic signal:
$-\frac{8 T \left(2 T \omega \sin (2 t \omega )-e^{-\frac{t}{T}}+\cos (2 t \omega )\right)}{\alpha ^2 \left(4 T^2 \omega
^2+1\right)}$
where $T,\alpha,\omega$ - ...
0
votes
1answer
38 views
Numerical value of Derivative of Dirichlet L function at some particular points [closed]
I want to find the numerical value for the derivative of Dirichlet L function to compare with an another formula. However I am not able to obtain any numerical values. I tried to use
...
