When we do Cauchy's Integrals what are we supposed to picture in our heads?
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.
This integral is clearly a prime target for contour integration. (We could factorise into its real quadratic factors, use partial fractions, and integrate the simpler summands, but contour integration seems faster).
Note that since the quartic denominator is both even and real, the poles of the integrand occur at where is an arbitrarily chosen root, say the one in the first quadrant of the the complex plane.
Now if we take a semicircular contour (radius R) with diameter on the real axis centred at the origin, and semicircular arc in the upper half-plane, then the integral around this contour (with positive orientation) is just equal to I, because the integrand decays as 1/R^2, and so the contribution from the curved segment of length O(R) tends to zero.
On the other hand, for sufficiently large R this integral is just equal to
It remains to compute , which is just the principal square root of .
Writing , we get
The resulting biquadratic yields , where is the golden ratio.
Hence
Last edited by seanieg89; 1 Feb 2016 at 11:57 AM.
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.
Yeah, it can be proved that way: https://en.wikipedia.org/wiki/Sophomore%27s_dream#Proof .
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.
Didn't look like a joke to me. Looked like taking credit where credit was not due.
But I'll humour you. Explain the your first "wild guess" then.
I'm sure we're all fascinated to see how you guessed such an answer, whilst refusing to provide at least any sort of outline of a method.
From above: As for Ahmed's Integral, I didn't know the answer was exactly that until I put the numerical answer into ISC+.
Also, putting the integral into Wolfram Alpha doesn't give you the closed form. Google... well you can't exactly search tex code easily. So I resorted to the above, taking the numerical value of the integral to 50 decimal places.
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.
Last edited by Carrotsticks; 21 Feb 2016 at 1:49 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.
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