If I am a conic section, then my e = ∞
Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.
I think you guys should enrol in Extreme Integration (MATH1052), or if you're feeling ambitious try Insane Integration (MATH3025)!
PM me for details.
(Try ignoring the lengthy 4U way in this thread :P)
Last edited by leehuan; 15 Nov 2017 at 8:38 PM.
Prove the integral diverges if the argument of the exponential does not describe an ellipse.
Challenge: Formulate a generalisation to the above integral in higher dimensions. (Experience with Linear Algebra will be very handy)
Last edited by Paradoxica; 15 Nov 2017 at 11:30 PM.
If I am a conic section, then my e = ∞
Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.
If I am a conic section, then my e = ∞
Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.
Oh didn't see you edited it, was the divergence thing always there and I just blind before?
In any case, we can still represent the quadratic form by a symmetric, hence diagonalisable matrix Q. If the quadratic form does not represent an ellipse, then at least one eigenvalue is non-positive, which implies that at least one of the integrals in the above product of single-variable integrals is infinite and we are done.
Easy difficulty
If I am a conic section, then my e = ∞
Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.
Last edited by seanieg89; 27 Nov 2017 at 10:20 AM.
(Not sure if already asked)
Really hope I'm right coz this took me forever. Also sorry for bad LaTeX.
Let
where
This is easily evaluated depending on the different cases:
If or , we have , if then and , and if then and .
Case 1:
hence for some constant .
Evaluating the integral for , we obtain that:
when .
Case or .
(which follows from Case 1).
when or .
Case 3:
Let
By symmetry,
Let
By symmetry,
Therefore , so .
Subbing the values back in, we obtain .
Case 4:
This can be done by the same method as case 3, and so .
Hence, when , , and otherwise.
Last edited by calamebe; 4 Dec 2017 at 11:43 PM.
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