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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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I would like to solve the following system of differential equations numerically for two one-dimensional functions $R(x)$ and $\phi(x)$: \begin{eqnarray} c_1 \left(R''(x) - (\phi'(x))^2 R(x) \right) - ...
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How does Mathematica define the indefinite integral $\int f(x) dx$? For example, if you input into Mathematica Integrate[Sin[x], x] it will return $-\cos(x)$ and ...
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When I ask Mathematica (version 14.1) to do the following symbolic integration: ...
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In V 14.3 Quit[] ode=2*y[x]*D[y[x],{x,2}]==1+D[y[x],x]^2; DSolve[ode,y[x],x,IncludeSingularSolutions->True] Gives Is it valid for DSolve to return ...
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This is problem 150, page 54, Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983. ...
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I am trying to write a Mathematica program to compute the following: For a given Hermitian matrix $\rho$, the operator $L_\theta$ with respect to a parameter $\theta$ is defined as: \begin{equation} ...
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I study the behavior of spatial curves and it is very convenient to write curvature and torsion as pure functions (PF). It is often necessary to obtain their combinations, integrals and differentiates ...
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Although there is a ready-made code for the inverse Laplace transform in Mathematica, I want to manually write the code to define the inverse Laplace transform so I can modify it. This is my attempt: <...
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In my question on MathOverflow, I was looking for a closed form result of the following sum: $$\sum _{k=0}^n \frac{(-1)^{n-k} x^{2 k} (2 (n-k)-1)\text{!!}}{(2 k)\text{!!}}.$$ Someone suggested me to ...
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I don't use Mathematica as much and only use it for some specific tasks from time to time (mostly simplifying expressions and calculating integrals and derivatives). Lets say I have an large ...
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I the following ODE with parameters \begin{align} B_e\: \theta''(s)+2(s-1)\cos\theta(s)=S_e\: f\left(\theta(s)\right), \end{align} with $0\leq s\leq 1$ and \begin{align} \theta(0)=0\:\:\:\text{and}\:\:...
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I was trying to see if I can trick DSolve for the ode $y'=0$ which has solution $y=c_1$, so all solutions are constant lines (horizontal lines). But then I asked it ...
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There is an integral whose leading order behaviour in terms of $p$ is what I want. $$I(p) = \int_0^{D(p-1)} \log(1-Q^2e^{-x}) \, \mathrm dx,$$ where $D$ is really large and $p$ tends to 1. For the ...
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Mathematica does not have builtin function to determine if ode is linear or not. Currently I use the code below, but it can give false negative. For example, the ode $\frac{1}{y'(x)} = x$ is linear ...
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I'm working on a big integral which I want to define in terms of a wedge of differential forms. I had been using D[x] as a substitute for dx, but I can see based on ...
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Consider the following integral: $$\int_{-27}^{27}\left(9-x^{\frac{2}{3}}\right)dx$$ In WolframAlpha if I assume the real-valued root, the answer is $\frac{972}{5}$. Now rewrite the above integral via ...
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WRI CASE:5291279 The textbook says that the differential equation \begin{align*} t y' &= 3 y\\ y(0) &= 0 \end{align*} has a solution $y= c_1 t^3$ for any $c_1$....
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I'm encountering numerical accuracy issues when computing the infinite series: NSum[Sin[Sqrt[k]]/Sqrt[k], {k, 1, ∞}, WorkingPrecision -> 20] Despite setting <...
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I wanted to integrate $\phi$ with respect to $\eta$. But Mathematica cannot perform the integration. The problem is with the $\sinh$ function but I could not replace it with any simple function except ...
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I was trying to find the root of a function which is in-turn given only by the solution of FindRoot. The following does work for some values. For others it produces the same error as without the ...
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Below are two equivalent definite integrals. $$\begin{align*}S&=2\pi\int_0^\pi b\sin t\sqrt{a^2\sin^2t+b^2\cos^2t}\text dt\\ &=4\pi b\int_0^\frac{\pi}{2}\sin t\sqrt{a^2-(a^2-b^2)\cos^2t}\text ...
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(Edited to emphasize the point about assumptions). Consider a simple integral: Integrate[Exp[I x/2], {x, 0, 2 Pi}] Mathematica produces the following answer: ...
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I've been experimenting for a while with Claude, DeepThink, Copilot and Chat, and all are great for helping quickly beginner programmers with bad memory like me, but they also waste a lot of your time ...
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Here is a difficult definite integral that Mathematica cannot seem to solve symbolically: $$\int\limits_{-1}^1 \frac{1}{x} \sqrt{\frac{1+x}{1-x}} \ln \left( \frac{x^4 + \sqrt{2} x^3 + x^2 + \sqrt{2} x ...
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Bug introduced in 14.3 or earlier and persisting through 14.3.0 or later Consider the simple example $Version Series[\[Chi],{\[Chi],0,0}] Before version 14.3, the ...
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Consider a function like this ...
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It would be nice to find Hopf bifurcations in Mathematica by minimizing distance of eigenvalues to the imaginary axis. Since I always start from a stable fixed point, it suffices to NMaximize the ...
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Let me use as an example a cubic function (the original problem is much more complicated) f[x_] := a x^3 + b x^2 + c x + d; I know that the function has: zeros ...
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I am reading an interesting paper One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational by Zudilin. We fix odd numbers $q$ and $r$, $q\geq r+4$ and a tuple $\eta_0,\eta_1,...,\eta_q$ of positive-...
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So I found this definition in this paper, but I wanted to double check that this is actually correct because I heard that the Partial derivative of a mxn matrix with respect to a p vector is supposed ...
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I would like to do some abstract power series manipulation. I am defining a power series like this func = Sum[Subscript[a,i] * x^i, {i, 0, M}]; When I try to take ...
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I would like to reproduce the Poincaré sections presented in this and this papers, which look like The figures are for energy values of E=0.2 (a) and E=0.25 (b) in the Hamiltonian, which reads $$H=\...
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Related MSE post I'm trying to make Mathematica demonstration of the paper The expected volume of a random polytope in a ball. In the $d$-dimensional Euclidean space $E^d$ ($d \geq 2$), consider the ...
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I was wondering if there is a way to make this function faster ...
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I am having trouble overriding when I compose functions. Here is a simple example. I would like the first output below to be 10 g[x] as opposed to ...
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I am trying to recursively define the following function: ...
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I am reading an interesting paper One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational by Zudilin. Define $$\varphi_0(x,y):=\sum_{j=1}^{3}([y]+[\eta_0x-y]-[y-\eta_j x]-[(\eta_0-\eta_j)x-y]-2[\...
Max's user avatar
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It's easy to prove that Limit[Cos[π Sqrt[k^2 + k + 1]], k -> ∞] equals 0, however the latest Mathematica version claim it's ...
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I am perplexed by the output of the following code: ...
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Bug introduced in 14.2 or earlier. I am trying to determine if my solution converges, by calculating the residual of my PDE. ...
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I have expressions $\log\frac{z}{z-1}$ and $\frac{1}{z \left( 1 + W_0\left(-\frac{1}{e z}\right) \right)} $ visualized below. There's a branch cut along the Im[z]=0 ...
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I'm having unexpected difference applying inverse Mellin transform after a simple algebraic rearrangement (1/(-1 + 2 s) vs ...
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6 answers
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I'm trying to solve the equation \begin{align} \epsilon_b\:\theta''(s)-(l-s)\cos\theta(s)=\epsilon_\gamma\sin\theta(s)\cos\theta(s), \end{align} with $0\leq s\leq l$ and \begin{align} \theta(0)=0\:\:\:...
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I want to analytically integrate this integration $$\int_0^{1 - 3/rs}\left(\frac{9 \sqrt{3} \sqrt{\frac{1}{\left(1-\frac{2}{\eta }\right) \eta ^2 \left(\frac{\eta ^2}{1-\frac{2}{\eta }}-27\right)}}}...
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As a result of repeated updates to Mathematica (12.3.1), the usage of integrate has become practically useless to an extent I consider ridiculous. To illustrate the problem take the following example ...
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Mathematica 13.2 (on windows 10) is giving strange results for a simple integral, after making contradictory assumptions on a variable which is not in that integral. In: ...
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I am trying to construct a linear differential operator programmatically. My approach is based on the worked-out example in the NonCommutativeMultiply documentation....
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I have the following ordinary generating function OGF $$\frac{1}{(1-s)\left(1 + W_0\left(\frac{s}{e(1-s)}\right)\right)}$$ ...
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On Mathematica ver.14.2.1.0 on a Raspberry Pi 5 with 8GB of RAM for the following simple integral: Integrate[Sqrt[5 x - x^2], x] I get the following result: ...
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I found a bug when computing the following integral $\int_0^{2\pi} \log ((e^{ix}-r)(e^{-ix}-r)) dx = 0$ for $0 \leq r < 1$ using Mathematica 14.1: ...
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