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Journals ACM SIGAPL APL Quote Quad Volume 28 Issue 2

On Continuing Powers, Superexponents and an Exponential Iterator --by gTan Ludwig Leeuwarden, The ~(etherlands UCH HAS BEENSAIDon the unsurpassed power of APL as a tool for solving numerical problems. T h e interactive nature of the workspace together with the straight machinc-executability of the notation offer unequalled possibilities for experimental mathematics. Recently, in a contribution on Power Reduction [1], Graham Parkhouse noticed that for large N the expression */NpA converges to a certain value, provided that A is within 0.0659... < A < 1.44467 .... W h e n A exceeds the upper limit then even for small N the result becomes immensely large, while for A < 0.0659... bifurcation takes place. For consecutive values for N the result of the expression oscillates between two boundary values. Because these results looked rather puzzling, I decided to investigate more closely the behavior of these continuing powers as an exercise in experimental mathematics. For the sake of an unequivocal description the following supplementary notation and terminology is introduced. The superexponent "A is the expression of n-1 continuing powers of a number A, represented posifionally by: A 1.1 1.3 1.444 1.4446678 Table I I nmi~ 16 S~ 1.1117820 1.4709890 2.5874703 2.71697986

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On continuing powers, superexponents and an exponential iterator

Ludwig, Fan
ACM SIGAPL APL Quote Quad , Volume 28 (2)
Association for Computing Machinery – Dec 1, 1997

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