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comment What is the notation for taking negative imaginary values for roots of negative numbers?
@gebruiker Your answer is helpful (so +1). The answer that I found most useful was not provided by a math.se member, but by a colleague: to use $\sqrt(x-i\varepsilon^+)$. I answered my own question with the intent of accepting it, but it was deleted by math.se members and was merged into my original question. So I am unable to accept.
Apr
1
comment Symmetrizing an integral representation of a symmetric function
This is a nice answer. Yes, $a$ and $b$ must be positive if they have no imaginary part. But, if $a$ and $b$ are complex, the substitution means that the new contour is actually a ray in the complex $x$-plane $0\rightarrow\infty/a$. This is again not symmetric under $a\leftrightarrow b$, but if the contour can be deformed to a ray that is symmetric (without crossing any poles/branch points), then the answer is perfect. I'll think about it some more and maybe you may have some ideas...
Apr
1
asked Symmetrizing an integral representation of a symmetric function
Mar
31
comment Derivation of the general forms of partial fractions
This is answer is very helpful, and the proof provided in the link is clear. Quick question: suppose I want to decompose $1/q(x)$ where its factorization is known to be $q(x) = \prod_{i=1}^m (x-r_i)^{l_i}$ with $r_i$ and $l_i$ are known. Then do you know if the coefficients $a(i,h)$ are known in closed form (as sums/products) [I'll probably ask this as a separate question]?
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
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