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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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 ...
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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 ...
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1answer
47 views
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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 ...
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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, ...
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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}\...
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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)...
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1answer
28 views

Suppress "integral does not converge" error message

Is there a way to suppress this error message and beep?
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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 $\dfrac{\mathrm{d}{x^4}}{\mathrm{d}{x^2}}=2x^2$ ? ...
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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: ...
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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 ...
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1answer
79 views

Non-numerical values for integral

I am trying to solve the following integral numerically ...
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2answers
215 views

Meaningless output of Limit

The output of the command ...
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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 ?
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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: ...
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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 ...
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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 ...
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0answers
42 views

What do do if Mathematica stops without error messages?

I am trying to run the relatively simple code ...
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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. ...
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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 $\...
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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 $...
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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 ...
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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 ...
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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 ...
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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
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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_{\...
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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 ...
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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 ...
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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}) \...
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1answer
60 views

Composite function derivatives

I would like to calculate the numerical partial derivatives (numerical gradient) of this function: ...
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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 $-1$ ...
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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\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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0answers
121 views

Is this function concave?

The result of the command of Mathematica 12.3.1.0 ...
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0answers
28 views
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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 ...
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0answers
51 views

4 Dimensional Numerical Integration

I have a pretty complicated function: ...
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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 $$ ...
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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 $tmer = 0.017$, $tmel = 0.044$. <...
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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
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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
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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 ...
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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{...
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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 ...

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