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Jan
18
comment Using Solve to solve the equation $x^{1/3}=-1$
I don't think there is a solution. I believe that the image of the complex plane under $z\to z^{1/3}$ is $\{r e^{i\theta}: r\geq 0, -\pi/3 < \theta \leq \pi/3\}$, given the location of the branch cut.
Jan
17
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11
awarded  Enlightened
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Dec
24
comment Revolution of Koch Snowflake
@ubpdqn Thanks - and same to you!
Dec
24
revised Revolution of Koch Snowflake
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Dec
24
answered Revolution of Koch Snowflake
Dec
11
revised Numerically evaluating an integral related to Cantor's staircase
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Dec
11
revised Numerically evaluating an integral related to Cantor's staircase
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Dec
11
comment Numerically evaluating an integral related to Cantor's staircase
@J.M. Glad you like it!
Dec
10
answered Numerically evaluating an integral related to Cantor's staircase
Dec
7
comment Basins of attraction of equilibrium points
For the system $x'=V_x,y'=V_y$, an equilibrium will be attractive iff it's a local minimum of the potential function. Before proceeding further, though, you should make sure that's the system you're interested in. Perhaps, you're really interested in a second order system, which is likely in celestial mechanics. That's a more complicated question, as you've got to deal with initial conditions on $x'$ and $y'$ as well.
Dec
7
comment Basins of attraction of equilibrium points
As you have shown, the algebraic system $V_x=0,V_y=0$ has four numeric solutions and, if we apply Newton's method to that system, each of those solutions has an interesting basin of attraction. But that is a different question from whether those equilibrium points are attractive under a system of differential equations like $x'=V_x,y'=V_y$.
Dec
7
comment Basins of attraction using Newton's method Part II
I agree with Quantum_Oli - it would be nice to know where this came from. It appears there might be an interesting question here but the image looks like the Julia set of a polynomial - I can't imagine how your non-differentiable function generated it.
Dec
7
comment Basins of attraction of equilibrium points
I'm quite sure that the basins of attraction of Newton's method in two real variables have been well studied; there's an elementary discussion on pages 486-487 of Gil Strang's calculus text. If you're interested in studying the basins of attraction of the smooth dynamical system with a given potential, however, that's quite a different thing. Only one of your equilibrium points is attractive for that system.
Dec
5
comment Basins of attraction using Newton's method
The first thing my answer to that question does is generalize the code to arbitrary functions. I guess that's what @Saurav is referring to.