Harmonic entropy: Difference between revisions

Line 615:

The problem is that we're really stretching the boundaries of complex analysis with this. With unnormalized HE, we were able to analytically continue the Fourier transform of exp-UHE to obtain a concrete expression in terms of the Riemann zeta function. While complex analysis makes no guarantees on the behavior of the Fourier transform of the analytic continuation of a holomorphic function, we did see the result seemed to converge on exp-UHE in the limit of large <math>N</math> when transforming back from the Fourier domain, confirming empirically that our analytically continued expression seemed to make sense.

But in the case of "normalized HE," we analytically continued the Fourier transforms of the numerator and denominator, separately, transformed both out of the Fourier domain, and then took the quotient. Complex analysis ''really' makes no guarantee on the behavior of the quotient of two Fourier transforms of the analytic continuations of holomorphic functions, and in this case the behavior is very strange. A different approach to analytically continuing the expression would be required.

But in the case of "normalized HE," we analytically continued the Fourier transforms of the numerator and denominator, separately, transformed both out of the Fourier domain, and then took the quotient. Complex analysis ''really'' makes no guarantee on the behavior of the quotient of two Fourier transforms of the analytic continuations of holomorphic functions, and in this case the behavior is very strange. A different approach to analytically continuing the expression would be required.

This same principle explains why we plotted the exp of UHE, rather than UHE itself. Were we to take the log of finite UHE, we would be taking the log of a strictly positive function. However, the analytically continued exp-UHE snaps back to the x-axis, so that there are points where the function is zero or even negative. Taking the log of the analytically continued exp-UHE would yield a complex-valued function where it is negative, due to this snapping effect. However, looking at exp-UHE directly has no such problem.

@@ Line 615: / Line 615: @@
 The problem is that we're really stretching the boundaries of complex analysis with this. With unnormalized HE, we were able to analytically continue the Fourier transform of exp-UHE to obtain a concrete expression in terms of the Riemann zeta function. While complex analysis makes no guarantees on the behavior of the Fourier transform of the analytic continuation of a holomorphic function, we did see the result seemed to converge on exp-UHE in the limit of large <math>N</math> when transforming back from the Fourier domain, confirming empirically that our analytically continued expression seemed to make sense.
-But in the case of "normalized HE," we analytically continued the Fourier transforms of the numerator and denominator, separately, transformed both out of the Fourier domain, and then took the quotient. Complex analysis ''really' makes no guarantee on the behavior of the quotient of two Fourier transforms of the analytic continuations of holomorphic functions, and in this case the behavior is very strange. A different approach to analytically continuing the expression would be required.
+But in the case of "normalized HE," we analytically continued the Fourier transforms of the numerator and denominator, separately, transformed both out of the Fourier domain, and then took the quotient. Complex analysis ''really'' makes no guarantee on the behavior of the quotient of two Fourier transforms of the analytic continuations of holomorphic functions, and in this case the behavior is very strange. A different approach to analytically continuing the expression would be required.
 This same principle explains why we plotted the exp of UHE, rather than UHE itself. Were we to take the log of finite UHE, we would be taking the log of a strictly positive function. However, the analytically continued exp-UHE snaps back to the x-axis, so that there are points where the function is zero or even negative. Taking the log of the analytically continued exp-UHE would yield a complex-valued function where it is negative, due to this snapping effect. However, looking at exp-UHE directly has no such problem.