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.

Filter by
Sorted by
Tagged with
12 votes
2 answers
2k views

Symbolic scalar-by-matrix derivative

Let's say I want to calculate the following scalar-by-matrix derivative $$\frac{\partial}{\partial A} \text{tr} \left[(\vec X^T A)^T (\vec X^T A)\right],$$ with $\vec X$ and $A$ being a $n \times 1$ ...
0 votes
3 answers
235 views

Computing the elliptic integral $\int \frac{1}{x \sqrt{f+\frac{a}{x^3}-\frac{k}{x^2}}} \, dx$

I don't know how to solve this with or without Rubi?!! ...
0 votes
0 answers
32 views

Getting a wrong plot after NIntegration

I am trying to plot this function numerically. R2T = ...
1 vote
1 answer
27 views

Getting the same result from Integrate as Probability

I'm trying to solve a Baysian inference problem (although the details are not all that important for this question) I define two probability distributions P and <...
0 votes
1 answer
59 views

Numerical Plot for an example

I would like to compare the numerical and analytical solution with plot for this example but something is wrong I don't know how to fix it?!! ...
1 vote
0 answers
62 views

What's the meaning of Global?

I'm calculating an integral, Integrate[(t e^(-z t) e^(-q t))/(1-e^(-t)), t], and the result is ...
0 votes
0 answers
54 views

Settings that integrate gives also distributions as solutions

How can I tell Mathematica that I also want distributions as solutions for Integrate? Example: $$\int_{\mathbb{R}} \mathrm{e}^{\mathrm{i}kx}\mathrm{d}x=2\pi\delta(x)$$
0 votes
2 answers
99 views

Integrate: slow and wrong results

Very often I struggle to understand what is happening when I try to integrate stuff in Mathematica. Generally, it deals very well with ugly symbolic integrals. But I have an example in which it deals ...
0 votes
0 answers
39 views

Replace Derivative[ i ][ f ][ t ] with f[ i, t]

Assume the PadeApproximant is carried over f[t0] * g[t0] with the order {1, 1}, that is <...
-3 votes
1 answer
67 views

Taking sentences with a spatial factor [closed]

To ask my question precisely, I should say that I have got many terms, among them some have coefficients $M^n$. Some things like : $$f = f_1(M) + f_2(M) + M f_3(M)+ M^2 f_4(M)+f_5(M)+...$$. I need to ...
0 votes
2 answers
66 views

Incorrect result of AsymptoticSum

Trying in version 13 on Windows 10 ...
10 votes
3 answers
2k views

Can Mathematica calculate this elliptic, triple integral?

Integrate[(4 a b/Pi) (a^2 + b^2 - 2 a b Cos[c])^(1/2), {a, 0, 1}, {b, 0, 1}, {c, 0, Pi}] I'm using basic plan, it gives me the result like that. The ...
0 votes
0 answers
57 views

DSolve doesn't solve a second order differential equation

I am trying to solve an ordinary differential equation: ...
0 votes
1 answer
74 views

How come I can't get $\int f(t)$ from performing derivative/anti-derivative operation on $\int f(t)$?

I know the the area under the curve of some function between the coordinates $t_0$ and $t_1$, is: $$\int_{t_0}^{t_1} f(t) \space dt=\frac{\left(t_0^2+t_1^2\right)^2}{4t_1^2}$$ How do I find $f(t)$? I ...
2 votes
1 answer
118 views

How to integrate this product of Exp[] and Cos[] using Mathematica

I would like to integrate the following product of Exp and Cos: ...
2 votes
1 answer
69 views

Selecting a solution range that is independent of a variable value

This symbolic computational question derives from this mathematical one on math.SE: Find all values of $a \in \mathbb{R}$ for which $f(x;a) = x^3 - 3 a x^2 + 3 a x - 2$ is non-decreasing. My ...
5 votes
1 answer
108 views

Evaluating PlusMinus to 2 decimal answers

How do I evaluate and expand an expression containing PlusMinus such that: N[PlusMinus[1, 1]] returns 2, 0 I'm stuck at ...
1 vote
1 answer
284 views

Integration not working

Can't seem to get this integration to work. s, n, b, b, l are all constants. ...
0 votes
0 answers
94 views

Composite function derivatives

I would like to calculate the numerical partial derivatives (numerical gradient) of this function: ...
0 votes
0 answers
72 views

Sort a List a Delete case [closed]

I try to sort a list in this way I have a polynomial DeleteCases[Table[Cos[(2 π)/3 k] z^k, {k, 0, 8}], 0, {-1}] gives ...
2 votes
0 answers
42 views

Why does Mathematica not know some integrals diverge?

I want to analyse the convergence of some complicated 1d integrals that may have divergences at $x=0$, similarly to the harmonic integral $\int_0^a 1/x \ \mathbb{d} x$, which I know is divergent. When ...
0 votes
1 answer
68 views

The inverse Laplace transform alters parameter constraints

I have this Laplace transform: $$\left( w \frac{L}{L+s}+(1-w) \frac{Q}{Q+s}\right)^n \ for \ L>0, Q>0,0<w<1.\ (1)$$ ...
0 votes
2 answers
36 views

`Assuming` combined with mathematical notation of `Integrate`

I am trying to write down the following expression Integrate[Exp[-a/2/x]/x/(x + 1), {x, 0, b}, Assumptions -> a > 0 && b > 0] using mathematical ...
0 votes
0 answers
49 views

splitting integrand based on variable dependence

For multiple integration, I would like to automatically check whether the integrand can be split up to a product of terms, each dependent on as small a subset of the variables as possible. E.g., given ...
0 votes
0 answers
45 views

Algorithm: Permutations & Signature

Theoretical side Simple example: If I have two sets $A_1=\{1,3\} ,A_2=\{2,3\}$ and Permutations[{1, 2, 3}]= $\left( \begin{array}{ccc} 1 & 2 & 3 \\ 1 &...
0 votes
0 answers
33 views

Integer parameter assumption gives incorrect result when integrating

There are some dangerous pitfalls that can happen when evaluating some integrals, assuming that some parameters are integers. For example, consider the following: ...
1 vote
2 answers
82 views

Error complex function ERFI(X): looking for alternative function representations?

I have some analytical results from a physics problem, where the Mathematica gives the results in terms of complex error function. I would like to explore another function representation using ...
2 votes
1 answer
239 views

(Numerical) calculation of nested integrals

The problem I'd like to (numerically) compute nested integrals of the type $$I(a;k) \equiv \left( \prod_{i=1}^k \int_0^\infty \mathrm{d}x_i \right) \exp \left( - \frac{1}{4a} \sum_{i=1}^k \left( x_i - ...
1 vote
1 answer
61 views

How to perform the first and second derivatives of Nth polynomials?

Suppose  $P(x,y)=\displaystyle\sum^{N}_{i+j=0}\alpha_{ij}x^{i}y^{i} \equiv \alpha_{00}+\alpha_{10}x+\alpha_{01}y+a_{11}xy+\ldots+a_{0N}y^{N}$ is a multivariable polynomial of $N$th degree. I want to ...
0 votes
1 answer
120 views

Accurate Numerical values of a given functional

Consider the following functional : $$ I(x,s) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy, s) − F(x −\mathrm iy, s)}{\mathrm e^{2πy}-1}, $$ where $ F(z, s) = \dfrac{\sin^2[π\Gamma(z)/(2z)]}{z^s} $....
1 vote
1 answer
340 views

How do I visualize $\exp(z)$ as a complex mapping? [closed]

How do I visualize $\exp(z)$ as a complex mapping? How can I ensure that it does not miss any value on the complex plane as it's value (is in the best condition of Picard's theorem). Can anyone help ...
3 votes
1 answer
101 views

Inconsistent results between NIntegrate and Integrate

I get these inconsistent results What is going wrong? Fast oscillating terms? Small exponential factors? But then, why the result seems to toggle between values 1 and 2, rather than being a real ...
5 votes
1 answer
243 views

Poincare section for a perturbed string

I am trying to find the Poincare section of a perturbed string in a gravitational background with a magnetic field. The plots that I am trying to reproduce are from this paper (page 11). The code that ...
0 votes
2 answers
44 views

Limit is not actually evaluated, but wolframalpha does [closed]

I'm trying to evaluate the limit $\lim_{x \to a} f(x)$ of the function $f(x)=\dfrac{1}{1+\tfrac{a^2}{4x(a-x)}\sin^2 \left(b\sqrt{a-x} \right)}$ using mathematica, and it should yield $\dfrac{1}{1+ab^...
1 vote
1 answer
73 views

I need some help debugging a calculation command

I am trying to make a calculation but I have a few bugs and I do not understand them/how to fix them. Below is my syntax, the output, and the error messages. I am trying to calculate the point in my ...
1 vote
1 answer
68 views

How do we just have $\log$ solution?

Mathematica solves this but I want the solution just as Ln[..]?!! DSolve[y'[t] == 3/2 a*y[t] ((y[t] + b)^2 - f), y[t], t] thanks
0 votes
0 answers
107 views

Dictate Mathematica to use the contour integration method

I want to evaluate an integral, which can be in principle computed exactly. For $a>0$ and $k_1, k_2 >0$, define $G(x) = (x+ia)^3$. The integral that I have is $$I = \int_{\mathbb R^4} dy_1dy_2 ...
0 votes
1 answer
74 views

How to get the derivative from this implicit equation with differentials? [duplicate]

The idea is here to have a implicit equation and take from both sides of the equation the differential and solve out of this the derivative The derivative is the ratio of two differentials and can be ...
4 votes
2 answers
146 views

The jumping problem when rolling reuleaux triangle

I'm making an animation of a Reuleaux triangle rolling on a straight line like this The animation generated by my code is not continuous, is there a simple way to eliminate jumping? ...
0 votes
0 answers
39 views

How to analytically solve for an intersection of two functions on an interval?

I would like to analytically solve for the intersection of two functions on a given interval and store the solution as a function (of parameters) for later use. How can I tell Mathematica to look for ...
1 vote
0 answers
23 views

Commutation of the partial derivatives on scalar in xAct

I do not know how to automatically commute partial derivatives on a scalar and simplify it. For example, if I have: $$\partial_a \partial_b \partial_c f- \partial_c \partial_a \partial_b f $$ where $f$...
0 votes
1 answer
107 views

How can I calculate the asymptotic value of this function correctly?

I'm trying to reproduce the results of a paper which in one part of it, I have to calculate the asymptotic value of a function but I can't reproduce that result exactly. I will be so grateful if ...
2 votes
1 answer
125 views

Hypergeometric2F1 in Mathematica

I would like to see some limits and plots of the Hypergeometric2F1 function in Mathematica 13.0. I don't have experience with hypergeometric functions, but need to compute the following limit and plot ...
0 votes
2 answers
77 views

How to set integral starting from 0 in the symbolic solution to ODE

I have the following ODE Bfull'[t] == - (p0/( p0 + (1 - p0) E^Integrate[α[s], {s, 0, t}])) * α[ t] (S[t] - Bfull[t]) + α[t]*c + δ * Bfull[t] where α and S ...
2 votes
2 answers
108 views

Plot the general and a particular solution for this differential equation

In a previous question, I had obtained a differential equation from: Starts with this implicit equation: (x - a)^2 + y^2 == 1(1) ( circles on x -axis ) $(x-a)^2+y^2=...
1 vote
1 answer
103 views

Integrals on symbolic regions

Let's define a region as r=Polygon[{{-a,0},{a,0},{a,h}}]. I want to integrate some function over that area. Say x^2. So, ...
2 votes
1 answer
93 views

How Implicit differentiate in order to get a differential equation [closed]

Starts with this implicit equation: (x - a)^2 + y^2 == 1(1) ( circles on x -axis ) $(x-a)^2+y^2=1$ Is the set of equations of the given circles. This set contains ...
3 votes
3 answers
98 views

How to test a recursive formula using Mathematica?

Through taking the derivative of $\binom{n}{k}$ w.r.t $n$ repeatedly, I found the recursive formula: $$\frac{\partial^a}{\partial n^a}\binom{n}{k}=\sum_{j=0}^{a-1}\binom{a-1}{i}\frac{\partial^j}{\...
0 votes
1 answer
109 views

Second order differential operator in Mathematica

I am trying to write the code for the following expression in Mathematica: $$\partial_{x} e^{-g x}=-g e^{-g x}+e^{-g x} \partial_{x}\,.$$ I was able to do so with the help of the following links with ...
1 vote
1 answer
61 views

Solve functional integral equations

I encountered one problem which is to find $F(\lambda)$ which satisfies $1 = \int_{-\pi}^\pi e^{i k \lambda} dF(\lambda), \forall k \in \mathbb{Z}$ and the answer is $F(\lambda) = 1$ if $\lambda \ge 0$...

1
2 3 4 5
84