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Periodic functions that are periodic not by addition
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The title may sound a little bit odd, but I was wondering if anything has ever been documented about functions that aren't periodic in the sense , but rather (if {p} represents an operator of p magnitude and }p{ reps its root inverse)

I ask because I've come across a curious set of "lowered operator" trigonometric function; if 0 <= q < 1, ;
or is the identity function,:




they satisfy



they follow all the laws sin and cos follow only with lowered operators (using logarithmic semi operators); ie









Pretty much any trigonometric identity you can think of these lowered operator trigonometric functions obey.

They also have a logarithmic semi operator Taylor series very much the same as their sine and cosine counterparts.

if

then




it can also be shown that if

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