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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33 views

How can I do partial derivation for F(y,y') with respect to y'? [closed]

This is a part of a question on the master/doctor degree graduate school entrance test for Tokyo Univ. in 2016. The conditions are below ( ) Here is the question. I have to make a differential ...
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1answer
97 views

Why are the results of integrations of a DiracDelta related integrand and its approximations strikingly different?

Let us consider a double integral, treating it as a functional which corresponds a number to a function, ...
2
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2answers
305 views

center of mass of a 2D square

Using Mathematica, find the center of mass of a 2D square not centered at the origin. center: (x0, y0) ...
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1answer
54 views

Principal value of a double integral numerically

I have several integrals of the form : where and . where a,b,wc and T are constants and PV denotes the Cauchy principal value I tried to integrate one of this form using NIntegrate however when ...
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0answers
30 views

Assume some expression is false

I need to Reduce a complicated function and I want to exclude some conditions A and B. How do I assume that the expression A is ...
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1answer
64 views

Simplify not working

I am trying to simplify the following expression code: ...
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3answers
114 views

How to get out coefficient of term in series?

Suppose I have a function $f(s,t) = [(1-t^2)(1-s^2t^2)]^{-1/2}$. Is there a way to get the general coefficient in this power series of the form $s^{2k} t^{2n}$?
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0answers
34 views

Four dimensional symbolic integration using HalfSpace

Consider the regions $r1 and $r2 defined as follows, ...
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2answers
57 views

Error in Integration of special functions using mathematica 12.0

When I try to integrate the following, Integrate[-GegenbauerC[22,-1/2,x]/(1+k*x),{x,-1,1}] where -1<k<1 and k!=0, Mathematica gives different results if I ...
3
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0answers
97 views

How to perform integration with Dirac Delta in Mathematica?

I am trying to calculate the integral $$ y(x)=\int^b_a dz\delta(x-w_z) $$ were $\delta$ is the dirac delta function, $a=-3$, $b=3$ and $w$ is a one-dimensional matrix such that for some values of $z$ ...
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0answers
31 views

Integration of rational function with small coefficients

I have to integrate a rational function $\frac{f(T)}{g(T)}$ which has really small numbers as coefficients. like this: $\frac{3.4*10^{-23} p T^5 \sin ^2(2 \theta ) m_s^4}{\left(1.\, +e^p\right) \left(...
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2answers
51 views

Assumptions for Definite Integral of Log[]

I have a given function: sol = t D[ri t/(v - b) - (1/v - 1/(v + b)) a/(b Sqrt[t]), t] // FullSimplify I now need to integrate this function across $v$ within a ...
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1answer
69 views

Code does not evaluate in version 12.1 in a reasonable amount of time

I just installed Mathematica version 12.1 on my desktop, and I have Mathematica version 11.3 on my laptop. There's a line of code that runs fine (although it takes a while to evaluate) on my laptop, ...
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0answers
26 views

PDE solution positivity constraint [duplicate]

I have a 2d PDE and NDSolve returns a solution that takes negative values but I want the function to be positive f[x,y]>=0 everywhere. How do I include ...
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1answer
48 views

Why doesn't Mathematica evaluate this simple integral involving the logarithm function? [closed]

I am trying to perform the following integral in Mathematica: ...
7
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0answers
130 views

Series with ArcTan gives wrong symbolic answer in Wolfram Language

Bug introduced after 9, persisting through 12.2. Recently, I have found a very bad problem with Wolfram Language. It gives the wrong answer for a quite simple expression! When calculating ...
2
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1answer
29 views

How to use one-way ANOVA for three groups?

I want to reproduce the Example illustrated here using Mathematica 12.0. In this example one-way analysis is applied to the following three groups of values: ...
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1answer
52 views

Asymmetric multivariable Taylor expansion

I want to expand a two-variable function up to asymmetric orders in two expansion variables, i.e. $$f(x,y) = T[f(x,y)] + \mathcal{O}(x^2,y^3,xy,xy^2).$$ Note that, while quadratic terms in $y$ are ...
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0answers
30 views

Performing Integration over Double Logarithmic function

I have been trying to perform the following Integration: ...
2
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1answer
144 views

An alternative command to compute a logarithmic integral [closed]

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 ...
2
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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} \...
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2answers
83 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
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1answer
48 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 ...
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0answers
71 views

Calculating double integral bounded by domain in Wolfram Alpha Notebook [closed]

If I want to compute Double Integral on a given region(See attached Picture) How can I input this on Wolfram Alpha Notebook
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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 ...
5
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0answers
121 views

Numerical verification of the estimate:

How to verify numerically with considerable accuracy in Mathematica the following : $$\int_2^x\dfrac{1}{z\Gamma(\sin^2[π\Gamma(z)/(2z)])}dz\sim\ln(\ln(x))$$ ? I need more suitable and better code ...
7
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4answers
258 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 ...
4
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3answers
133 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 ...
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0answers
27 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}} + \...
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0answers
33 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 ...
2
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1answer
120 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: ...
8
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2answers
255 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 $\mathbb{R}^3$: $$R(u,v) = (u\cdot v)I_3 + hat(u\times v) + \dfrac{(u\times v)\otimes (u\times v)}{1+u\cdot v}$$ ...
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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).
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0answers
48 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
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1answer
75 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 ...
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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 ...
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1answer
48 views

Integration shows RootSum and really long expressions

I am doing this integration and when I use $C_\alpha(T)$ as just constant numbers(...
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1answer
55 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
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1answer
68 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 ...
1
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0answers
79 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}], ...
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0answers
37 views

NSolve as integration limit

I need the (numerical) solution of equation to be integration limit as below. How to make it work? ...
2
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1answer
63 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: ...
1
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1answer
131 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?) ...
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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] ...
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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-...
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0answers
73 views

First derivative of curve and onset and endset points

If I have the following data: ...
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0answers
39 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 ...
2
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3answers
43 views

ND gives expression instead of number

I have a function sol defined as a solution to an equation (computed numerically). Specifically ...
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1answer
29 views

Plot of derivative does not show

I have following function 4/((1 + v1[h])*(1 + v2[h])): ...
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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 ...

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