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Jan
5
revised Is there a general algorithm to determine new contours for multivariable change of integration variables
deleted 10 characters in body
Jan
5
asked Is there a general algorithm to determine new contours for multivariable change of integration variables
Jan
2
accepted Small $\delta$ behavior of rational integral
Jan
2
comment Small $\delta$ behavior of rational integral
Ah, that's a shame. My IQ isn't high enough to figure out the right approach. Thanks, anyway.
Jan
2
comment Small $\delta$ behavior of rational integral
er.. my mistake, the sub-leading piece is actually a constant as $\delta\rightarrow 0$
Jan
2
comment Small $\delta$ behavior of rational integral
Wow, that is sneaky. Even more surprising to me is that the subleading piece is also singular as $\delta \rightarrow 0$. When I originally wrote the question, I thought the singular part is contained entirely in the leading term. Would you kindly show me how to obtain all singular terms up to the part that is regular as $\delta \rightarrow 0$. Also, where can I read about how to extract behaviors of such integrals?
Jan
1
asked Small $\delta$ behavior of rational integral
Dec
20
comment Finding a parametric form for the locus of points for a vanishing determinant
@Narasimham I just realized this determinant is proportional to the volume of a tetrahedron of sides $x$, $y$, $c_1$, $c_2$, $c_3$, $c_4$. (see mathworld.wolfram.com/Cayley-MengerDeterminant.html) Would that help?
Dec
13
comment Finding a parametric form for the locus of points for a vanishing determinant
I see, thanks! Would you kindly remove your answer and add just this opinion as a comment under my question?
Dec
13
revised Finding a parametric form for the locus of points for a vanishing determinant
added 19 characters in body
Dec
13
comment Finding a parametric form for the locus of points for a vanishing determinant
@AlexM. I see. So I am going to edit this and replace "three open sets" with "three disconnected sets". Would that be better? Also, would the whole "topology/preimage" stuff going to be helpful for getting my parametric equations?
Dec
13
comment Finding a parametric form for the locus of points for a vanishing determinant
Perhaps I didn't make myself clear: so here's a baby problem to which I have an adequate solution. Consider the locus of points satsfied by $x^2+y^2-1=0$. According to your No. 4, this gives explicit formulas for two curves: $y=\sqrt{1-x^2}$ and $y=-\sqrt{1-x^2}$. This is exactly what I want to avoid. The solution I am looking for is $(x(t), y(t)) = (\cos t, \sin t)$. This is one parametric equation describing the single set of points.
Dec
13
comment Finding a parametric form for the locus of points for a vanishing determinant
@AlexM. Well, I only assumed them to be open based on looking at sample graphs. The three V-shaped curves look like open sets to me, and the small O-shaped curve looks closed. I have no formal training in maths beyond multivariable calculus. Perhaps you can enlighten me as to how the three disconnected pieces are actually closed?
Dec
12
comment Finding a parametric form for the locus of points for a vanishing determinant
Thanks for your answer; but I don't want an implicit equation to describe three (or four) disconnected subsets. The reason is that the explicit solutions have branches that break up the separate sets in a yucky way (as you can see in the plot). I would like three parametric equations, each one of the form $(x(t), y(t))$ to describe the entirety of each disconnected piece.
Dec
11
comment Finding a parametric form for the locus of points for a vanishing determinant
@H.R. Yes, positive definite means non-negative. Maybe I should have used that.
Dec
10
awarded  Citizen Patrol
Dec
10
comment Explicit form for $\left(e^{-x^2}\left(\frac{d^n}{dx^n}e^{x^2}\right)\right)^2$
Is $P_n(x)^2 = \sum_{j=0}^{\lfloor n/2\rfloor}\sum_{k=0}^{\lfloor n/2\rfloor} \frac{(n!)^2}{j!k!(n-2j)!(n-2k)!}(2x)^{2n-2j-2k}$ good enough? :D
Dec
10
comment Antiderivative of $\frac{x^n \ln(a x^2+b x)}{(a x^2+b x)^m}$
@rubik Ok, I just realized that this integral is not suitable as a stackexchange question since it is just a laborious task with not too much insight. Can you initiate a vote to close?
Dec
9
comment Antiderivative of $\frac{x^n \ln(a x^2+b x)}{(a x^2+b x)^m}$
@rubik ooh.. I think it is best to make a substitution: $y=(ax+b)$.. still working on it...
Dec
9
asked Antiderivative of $\frac{x^n \ln(a x^2+b x)}{(a x^2+b x)^m}$