I have stumbled across something beautiful! I haven’t the time to write of it now, but I can allude to it without fear. Eventually, I will reproduce a very fascinating formula for $latex \pi$.
But first:
Consider the following expression:
$latex \cos \dfrac{\xi}{2} \cos \dfrac{\xi}{4} \cos \dfrac{\xi}{8} … \cos \dfrac{\xi}{2^n}$
It can be simplified into a very simple quotient of $latex sin$ in terms of $latex \xi$.
August 7, 2020 at 4:15 pm on AboutThat's great! If there are particular plots that you're interested in, let me know. But I (very recently) posted a library for making the plots https://davidlowryduda.com/phase_mag_plot-a-sage-package-for-plotting-complex-functions/ Alternately, I make sage/python notebooks available for many of my visualizations. These are usually linked to within the post. If you see something that you want to replicate and find the code, leave a ---
April 11, 2020 at 4:53 pm on AboutHi Could you please share the python code to generating beautiful plots? I plan to use it for my son to get interested in python programming. Son October 17, 2019 at 3:17 pm on Notes from a talk at the Maine-Quebec Number Theory ConferenceThanks Peter! You're right --- and it's a nice reference. 
October 7, 2019 at 4:03 am on Notes from a talk at the Maine-Quebec Number Theory ConferenceOmega results of this form using this method are presented quite nicely in Chapter 15 of Montgomery and Vaughan; they give applications towards sign changes of the Chebyshev psi function and of partial sums of the Mobius function. 
June 16, 2019 at 1:55 am on Choosing functions and generating figures for “When are there continuous choices for the mean value abscissa?”At first, we chose 2 because it was the other "obvious" point that we hadn't yet specified. We already controlled the values at 0, 1, and 3. But I later realized it doesn't matter what point we choose. What we're really doing is adding one extra degree of freedom and using it to minimize the L2 norm of the derivative ---
June 16, 2019 at 1:14 am on Choosing functions and generating figures for “When are there continuous choices for the mean value abscissa?”Why did you choose 2 for the pt in the L2 part? What if you chose 1.5 or something? 
September 27, 2018 at 3:55 pm on AboutI appreciate your taking the time to reply. The version of FreeCell I play online does count taking a card from the tableau (and the Freecells) as a move when adding them to the foundation. I never realized others did not. September 27, 2018 at 8:20 am on AboutI believe you are referring to: this post on MathSE. Yes, that's right. In the typical language, a "move" consists of one mouse-click-and-drag action. When there are open freecells, it is possible to move multiple cards using one mouse operation (as a shortcut to doing all the card moves individually). As an example, if all 4 freecells are open, you --- September 27, 2018 at 1:38 am on AboutI see nowhere to submit an email so I am asking my question here. On another site you answered a question about the average number of moves to solve Freecell. You gave an in-depth answer about the studies that had been done (thank you) which was 45. What I cannot comprehend is how anyone can move 52 cards, one at --- September 24, 2018 at 6:33 pm on An intuitive introduction to calculusThank you. You could also write $P(t) = ae^{kt + C_1}$. But this may give the impression that $a$ and $C_1$ carry different data --- but they don't. To see the equivalence, you can write $ae^{kt + C_1} = a e^{kt} e^{C_1} = (a e^{C_1}) e^{kt} = A e^{kt}$, where $A = a e^{C_1}$. There are two fundamental degrees of ---
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