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.
3,821
questions
0
votes
1answer
36 views
Distributive property of Integration
As it is known that
Integrate[A+B]= Integrate[A] + Integrate[B]
I am facing problem with the following integral, when I integrate
...
0
votes
0answers
24 views
Replace time fractional derivative approximations algorithm to another algorithm, but why the values of the approximations have not been calculated? [closed]
I am trying to replace fractional derivative approximations algorithm in another algorithm, but the values of the approximations have not been calculated.
Please help to replace approximate correctly ...
0
votes
1answer
47 views
Derivative with respect to a variable is leaving unevaluated derivatives of functions
I have the following
...
1
vote
1answer
63 views
Incorrect result of integration with GenerateConditions
The GenerateConditions->True option of the Integrate command is useful
and works properly in most cases. However, there are ...
3
votes
1answer
155 views
Why does this lead to zero when integrating a non-negative function?
When I try to integrate this $\int_0^{1} \frac{\log ^2(x+1) \log ^2(1-x)}{x} \, dx$, it leads to 0.
And when I use NIntegrate, it turns into a positive number, ...
0
votes
1answer
56 views
How do I code up nested derivatives?
I want expressions for the spherical Bessel functions in terms of sinusoids using Rayleigh's formulae:
$$
j_{n}(x)=(-x)^{n}\left(\frac{1}{x}\frac{d}{dx}\right)^{n}\frac{sin(x)}{x}\\
y_{n}(x)=-(-x)^{n}\...
0
votes
0answers
100 views
How can I integrate a function containing divergence? [closed]
I want to calculate the embedding surface in the wormhole scenario. However, the integrand diverges at the throat ($r = r_0$) of this object. I will summarize the process. With the shape function $b(r)...
0
votes
1answer
28 views
Suppress "integral does not converge" error message
Is there a way to suppress this error message and beep?
0
votes
0answers
36 views
Derivative where the variable is a function, e.g. D[x^4,x^2] [duplicate]
How can I express in MMA a derivative if the variable is a function?
Example:
The first derivative of $x^4$ with respect to $x^2$, would be
$\dfrac{\mathrm{d}{x^4}}{\mathrm{d}{x^2}}=2x^2$ ?
...
1
vote
0answers
53 views
solving ordinary coupled differential equation
I have a set of 8 ordinary differential equations containing matrices which I need to solve using initial conditions. I have written the equations in this form:
...
1
vote
2answers
60 views
Integrate just spits out the input
Hi new to Mathematica here. I am trying to integrate this function. I am not so sure if it can be integrated or I have maybe messed up the syntax somewhere. I would be very happy for any help. The ...
1
vote
1answer
79 views
3
votes
2answers
215 views
0
votes
2answers
96 views
How plot the domain of multivariable function? [closed]
I want to get and plot the domain of the following function with mathematica :
$$Arccos(xy)$$
Any idea ?
0
votes
1answer
86 views
Solving an ODE with paramters and taking the limit of the solution
I am very new to Mathematica and already spent a lot of time trying to do this but failed.
I am trying to solve an ODE:
...
5
votes
2answers
134 views
Defining a new wavelet (Fibonacci wavelet)
I want to define a new wavelet (Fibonacci wavelet) in the reference.
So, I read the tutorial on Wolfram Site, @Jason B. ' s answer, and also @Sektor 's answers.
But I still have some problems in my ...
0
votes
0answers
28 views
How to calculate the derivatives of absolute value? [duplicate]
I need to calculate derivatives of absolute value as follows
$$
\frac{d}{dr_1}\lvert r_1-r_2\rvert
$$
somewhere in my calculations. When I simply use ...
1
vote
0answers
42 views
What do do if Mathematica stops without error messages?
I am trying to run the relatively simple code
...
0
votes
1answer
84 views
Unable to solve this differential equation with NDSolve. NIntegrate::inumr error
My code goes as follows:
...
-1
votes
2answers
154 views
How to calculate this limit?
For didactic purposes I study the function
$$f(x):=\lim\limits_{k\to \infty} \frac{x^{2k}+\left(\frac \pi 3\right)^{2k}}{x^{2k-1}+\left(\frac \pi 3\right)^{2k-1}}$$
with help of Mathematica. ...
0
votes
0answers
45 views
Leibniz Rule with Inverse of CDF
Suppose $X$ is a random variable with PDF and CDF, $f$ and $F$, respectively. Now, we define $Z(w)=F^{-1}\left( \frac{p+g-w}{p+g} \right)$, and $\Lambda(Z(w))=\int_{0}^{Z(w)}(Z(w)-x)f(x)dx$ and $\...
0
votes
1answer
42 views
Making equal left and right hand side of the some equality
I think that there is a similar posts in here. But I don't find right keyword.
My question:
After some calculations, I get equality. For a simple example:
$$list=\{a,b,c\}==\{1,2,3\}$$
Can we extract $...
0
votes
1answer
82 views
Integral of Sqrt[1/x-1/a] Cosh[Sqrt[x]]
MMA delivers:
$$\int_0^a\sqrt{\frac{1}{x}-\frac{1}{a}}\cosh\left(\sqrt{x}\right)\textrm{d}x=\pi I_1(\sqrt{a})$$
where $a>0$ and $I_1$ is the modified Bessel function of $1^\textrm{st}$ kind and ...
1
vote
2answers
83 views
I want to integrate the following expression:
I want to integrate the following expression:
$$\int_0^t v_{xx\tau}(x,\tau) d\tau$$
Why I can not get the following ?
$$v_{xx}(x,t)-v_{xx}(x,0)$$
or how to suppose that $v_{xx\tau}$ is continuous in ...
1
vote
0answers
55 views
New symbollicaly integral operator [closed]
I have an u=u(x,t) that is an unknown function.
Suppose that the function u can be written as follows:
**
...(Eq. 1)
**
In here, $\Psi(x)$ is an Nx1 known vector, and C is an NxN matrix. (N is an ...
0
votes
0answers
31 views
Solving constraint optimization problem for different intervals using VariationalD
I have been trying to solve for a constraint optimization problem (which eventually yields to a Laplace distribution).
Consider the following equation (1)
\begin{equation*}
J=\int^{\infty}_{-\infty} p(...
4
votes
1answer
141 views
Proof associated Legendre polynomials are orthogonal: integral doesn't solve
Wondering why Mathematica can't solve this integral:
Integrate[
LegendreP[l1, 1, x] *
LegendreP[l2, 1, x], {x, -1, 1}]
Mathematica outputs: $\int_{-1}^1 P_{\...
0
votes
0answers
23 views
Derive composite function w.r.t. inner function and variables of inner function
I need to do some calculation using a general function F[ξ] where ξ is defined as
...
1
vote
1answer
62 views
Evaluate $c_k = \frac{1}{k!}\Big\{ \Big(\frac{d}{dz}\Big)^{k-1} f(z)^k \Big\} \Big|_{z=0} $ for $k=1,2,\dots$
I would like to compute
$$c_k = \frac{1}{k!}\Big\{ \Big(\frac{d}{dz}\Big)^{k-1} f(z)^k \Big\} \Big|_{z=0} $$
for $k=0,1,2,\dots$,
where
$$ f(z) = \frac{z[\cos( \alpha z ) +\sin(\alpha z)]}{\sin(\alpha ...
0
votes
1answer
65 views
Leibniz Integral Rule for probability distributions
Suppose $Y$ is a probability distribution with PDF and CDF, $f$ and $F$, respectively, and it is defined over nonnegative real numbers. I want to take the derivative of this function $F(\frac{x}{a}) \...
0
votes
1answer
60 views
Composite function derivatives
I would like to calculate the numerical partial derivatives (numerical gradient) of this function:
...
0
votes
1answer
93 views
Analytic result for the given integral
I want to calculate analytically this integral
$$ \int_{0}^{\pi} (F-G) \,dy , $$
where the functions $F$ and $G$ are given in the code below. The parameter $c$ is a constant which varies from $-1$ ...
0
votes
0answers
63 views
Fourier transform of $ \exp{\left(\text irx/\sqrt{1-\text ix}\right)}/(1-\text ix) $
I want to calculate the following Fourier transform:
$$u(r,t) = \int_{-\infty}^{\infty}\text d\omega\ \frac{1}{1-\text i\omega}\, \exp\left\{\frac{\text ir \omega}{\sqrt{1-\text i \omega}} +\text i\...
0
votes
1answer
31 views
Coloring in ComplexListPlot
I want to plot the points of a complex sequence with colors changing from point to point. I tried
...
0
votes
0answers
94 views
Is mathematica integrating $\frac{1}{x}$ correctly?
When $x<0, \quad \int \frac{1}{x}\,dx = \ln (-x) + \text{const.}$
Mathematica calculates this
Assuming[x < 0, Integrate[1/x, x]]
as
...
0
votes
1answer
127 views
Find where there is no solution involving three variables
I have two equations SvA[zm, zs, zh]==0 and tzmzs[zm, zs, zh, tb, b]==0.
I will plot these two equations and find their ...
0
votes
2answers
123 views
Deriving an integral reduction formula programmatically
There are a number of integral reduction formulas from basic calculus, including several involving trigonometric or exponential functions. These are of the form where some term in an integrand is ...
2
votes
3answers
287 views
How to ask Mathematica to subtract each adjacent pair in a list, and then, sum them?
If I have a list of numbers as (the number of elements in this list is even)
list={1,23,32,54,65,76,87,98,109,110,...}
How can I ask Mathematica to subtract each ...
0
votes
1answer
57 views
Laurent Series with Assumptions
I want to obtain the Laurent of the function f(z) = 1/(z(z-1)) valid first in 0 < Abs[z] < 1 and second in ...
0
votes
0answers
121 views
0
votes
0answers
28 views
0
votes
4answers
95 views
Integration challenge
For positive real $a \leq \pi$ and $b \leq \pi$
$\int_{-a/2}^{a/2} \frac{\sin{(b/2)}}{\sqrt{1+\cos^2 (b/2)\tan^2 (\phi )}} \; d \phi = 2 \arcsin \left[ \sin (a/2) \sin (b/2) \right] $
as can be ...
0
votes
0answers
51 views
2
votes
0answers
55 views
Using Curl and Grad in cylindrical coordinates gives different result than in Cartesian
I am doing a quite elementary calculation to verify
$$
\vec \nabla\cdot V = \vec\nabla \times \vec A
$$
Where $V$ is just some scalar function in $\rho$, $\phi$ and $z$ and $A$ is defined as
$$
...
3
votes
5answers
269 views
How to solve numerically a double integral that converges slowly?
I'm having trouble trying to numerically solve the following double integral in Mathematica, which I know, a priori, has a solution for values of $mh \in[0.4,2.0]$ and $tmer = 0.017$, $tmel = 0.044$.
<...
0
votes
1answer
71 views
Increase convergence near singular point
I have this integrand given by tinteg[z,zm,zh] where I want to integrate it in the range [0,zh]. However, the integrand is ...
2
votes
1answer
142 views
Does this series converge or diverge?
Let us consider the series $\sum_{n=1}^\infty \frac {\{\sqrt n\}-\frac 1 2} n$, where $\{\cdot\}$ denotes the fractional part
of a real number. On the one hand,
...
2
votes
2answers
108 views
Unpacking a Mathematica expression returned by DSolve
I was trying to solve a system of differential equations in Mathematica and had troubles understanding what the solution looked like. So I wanted help to unpack it.
I had a system of two coupled ...
0
votes
1answer
93 views
Integrations done by residues and numericals don't agree
I want to integrate
$$
\int_{-\infty}^{\infty} dw \hspace{0.51em}\frac{w^{n} e^{i wt}}{(w^{2}-\Omega^{2})^2 +(\gamma w)^2}
$$
I did so using contour integration because it has the poles:
$$w=\mp \frac{...
1
vote
1answer
78 views
Some numerically divergent sum
I try to get a numerically evaluation of the following sum:
$$\sum _{j=0}^{\infty } \frac{j^2}{\sqrt{j \left(j^2+\frac{1}{2}\right)}}$$
using different methods. With Euler-Maclaurin we get
...
