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 sqrt(!) and sqrt(exp) Kouznetsov Fellow Posts: 151 Threads: 9 Joined: Apr 2008 07/29/2009, 03:41 PM (This post was last modified: 07/29/2009, 03:52 PM by Kouznetsov.) I did not found where to put these pics, to, I open this tread for plots of square root of factorial (left) and that of exponential (right).     $f=\sqrt{!}(z)$ and $f=\sqrt{\exp}(z)$ are shown with lines $p=\Re(f)=const$ and $q=\Im(f)=const$. Levels $p=-4, -3, -2, -1, 0, 1, 2 3,4$ are shown with thick black lines. Some intermediate Levels $p=$const are shown with red lines of the nefative values and blue for the positive values. Levels $q=-4, -3, -2, -1$ are shown with thick red lines. Level $q=0$ is shown with dark pink. Levels $q=1,2,3,4$ are shown with thick blue lines. Some intermediate levels $q=$const are shown with green. Cuts are shown with black dashed lines. Such functions are solutions of the equations $\sqrt{!~}\Big( \sqrt{!~} \big(z\big) \Big)=z!$ and $\sqrt{\exp}\Big( \sqrt{\exp} \big(z\big) \Big)=\exp(z)$ Base-Acid Tetration Fellow Posts: 94 Threads: 15 Joined: Apr 2009 07/29/2009, 05:03 PM (This post was last modified: 07/29/2009, 05:08 PM by Base-Acid Tetration.) wow. the similarity between $!^{1 \over 2}$ and $\exp^{1 \over 2 }$ strikes me. just a question... why do you show the $!^{1 \over 2}$, not the $\mathrm{\Gamma}^{1 \over 2}$, the square root of gamma function? no, it doesn't really matter which choice, because !(z) and Gamma(z) are bassically the same: !(z-1) = gamma(z). but is it easier to extract the root of $!$? Kouznetsov Fellow Posts: 151 Threads: 9 Joined: Apr 2008 07/30/2009, 01:44 AM (This post was last modified: 07/30/2009, 02:39 AM by Kouznetsov.) (07/29/2009, 05:03 PM)Tetratophile Wrote: ... the similarity between $!^{1 \over 2}$ and $\exp^{1 \over 2 }$ strikes me.I had similar feeling about these two pictures. Various holomorphic fastly growing functions shouild give similar pictures. Quote:just a question... why do you show the $!^{1 \over 2}$, not the $\mathrm{\Gamma}^{1 \over 2}$, the square root of gamma function? Beacuse $\sqrt{!}$ is LOGO of the Physics Department of the Moscow State University, and there was an opinion that such a function cannot have any sense. I do things which are declared to be "not doable". If one publishes an article with statement, that the "half-iteration of Gamma cannot be evaluated", then I hope to evaluate it as well. Quote:... !(z) and Gamma(z) are bassically the same: !(z-1) = gamma(z). but is it easier to extract the root of $!$? The procedure for sqrt of Gamma should be similar to that for the factorial. So, I do not think that $\sqrt{!}$ is much easier; basically, the same algorithm should work for sqrt(Gamma), and I expect it to give the similar picture. We can evaluate the half-iteration of various holomorphic functions, not only exponential or factorial; and not only the half-iteration, but any iteration (even the complex one). I suggest the examples: In nonlinear acoustics, in may have sense to characterize the nonlinearities in the attenuation of shock waves in a homogeneous tube. (This could find an application in some advanced muffler that uses nonlinear acoustic effects to withdraw the energy of the sound waves without to disturb the flux of the gas). The analysis of the nonlinear response, id est, the Transfer Function H, may be boosted with the superfunction; in particular, the TransferFunction of a tube, which is half-shorter, is sqrt(H). If the snowball rolls down from the hill, getting heavier, and the mass at the bottom can be characterzed with the TransferFunction H of the initial mass, then the TransferFunction of the half-hill is sqrt(H). If a small drop of liquid diffuses through the oversaturated wapor in a vertical tube of length h, and its mass at the output is TransferFunction H of the imput mass, then the TransferFunction of the similar tube of length h/2 is sqrt(H). If the logarithm of power of light at the output of the homogeneous nonlinear optical fiber is TransferFunction H of the logarithm of its input power, then the TransferFunction of a half of this fiber is sqrt(H). Can you suggest more such examples? Kouznetsov Fellow Posts: 151 Threads: 9 Joined: Apr 2008 08/05/2009, 08:30 AM (07/29/2009, 05:03 PM)Tetratophile Wrote: .. why do you show the $!^{1 \over 2}$, not the $\mathrm{\Gamma}^{1 \over 2}$, the square root of gamma function?One more reason, why Factorial is "better", than Gamma: Factorial has integer fixed points; we have no need to introduce the irrational constant. By the way, the description of the figure of $\sqrt{!}$ and $\sqrt{\exp}$ is at http://www.ils.uec.ac.jp/~dima/PAPERS/2009supefae.pdf Catullus Fellow Posts: 205 Threads: 46 Joined: Jun 2022 06/08/2022, 05:32 AM (This post was last modified: 06/08/2022, 05:35 AM by Catullus.) (07/29/2009, 05:03 PM)Base-Acid Tetration Wrote: no, it doesn't really matter which choice, because !(z) and Gamma(z) are bassically the same: !(z-1) = gamma(z).!(z-1) = subfactorial(z-1). Not Γ(z). https://mathworld.wolfram.com/Subfactorial.html ฅ(ﾐ⚈ ﻌ ⚈ﾐ)ฅ Please remember to stay hydrated. Sincerely: Catullus « Next Oldest | Next Newest »

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