Partial Derivative of Okamoto's Function With Respect to the Parameter. Examples of Fractal Surfaces

[@strikeemblem2886]

At 12:17, where you obtained a power series expansion for K(x): Do you have anything more than pointwise convergence of the series? I do not think pointwise is enough to justify the geometric construction, though it is enough your results that come after based on the proof sketch.

[@CHALKND]

This is a bit of well-known (at this level of math) point-set topology and doesn't actually have to do with pointwise convergence of the sum. A way to see this is to first note that the ternary rationals in [0,1], that is the numbers that can be expressed as .a_1a_2a_3....a_N, where N is finite, and a_1,a_2,a_3...a_N are all in {0,1,2,} and represent a ternary expansion are dense in the unit interval [0,1], and K maps into the reals which is a Hausdorff space under the standard topology. The summation for K(x) when x is a ternary rational is a finite sum, since the terms are all zero after N, so the geometric construction limits to a function g where g(x) agrees with K(x) for all x which are ternary rationals. (At the nth step, g_n(x) is created by finding K(.a_1a_2....a_n) for all possible ternary rationals .a_1a_2...a_n in [0,1], and then connecting those points with straight lines in the natural way (i.e. from left to right as you move across the plane)). Now, since g(x) and K(x) are both continuous functions (g is continuous by construction, K is continuous by a standard ε-δ argument) and g and K agree on a dense subset of their shared domain [0,1] and both g and K map into the same Hausdorff space, the reals with the standard topology. Then a result from point-set topology says that g must equal K and thus the geometric construction is in fact an appropriate limiting construction of the graph of K.

[@strikeemblem2886]

@@CHALKND okay, the missing ingredient I was looking for is the agreement on a dense set of the domain. Now I'm convinced. Thanks!

[@NikolajKuntner]

You got the Chalkmann functional equation K(s) = -K(1-s), all the zero's in (0,1) are at s=1/2, I think it's time to collect your $

[@CHALKND]

But there’s so much more to investigate lol

[@Unemployed-Math-Major]

Is conference talk usually in such a format? Not by chalk board?

[@CHALKND]

For the most part yes, there are a few exceptions like with special seminars (the institute of advanced study has some seminar talks on their channel that are chalkboard talks) but most conference talks are done by beamer/slideshow because most of the time the talk is given at more than one conference/ its easier to prepare than a chalk talk etc.

[@Unemployed-Math-Major]

@@CHALKND thanks! do you think someday those KZbin beautiful math animations will be more popular among professional events?