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  = f(x))
, but rather (if {p} represents an operator of p magnitude and }p{ reps its root inverse)  = f(x))
I ask because I've come across a curious set of "lowered operator" trigonometric function; if 0 <= q < 1, = \exp^{[-q]}(x))
;
 = x)
or )
is the identity function,:
 = q:ln(sin(-q:ln(x))))
 = q:ln(cos(-q:ln(x))))
they satisfy
) = sin_q(x))
) = cos_q(x))
they follow all the laws sin and cos follow only with lowered operators (using logarithmic semi operators); ie
}\, =\, cos_q(x)\, \{-q\}\, (q:ln(i)\, \{1-q\}\, sin_q(x)))
)\, =\, cos_q(x))
\,\{2-q\}\,2)\, \{-q\}\, (cos_q(x)\,\{2-q\}\,2)\,=\,S(1-q))
\, =\, (sin_q(x)\,\{1-q\}\,cos_q(y))\, \{-q\}\, (sin_q(y)\,\{1-q\}\,cos_q(x)))
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\, =\, f(0)\, \{p\}\, f(1)\, \{p\} ... f\(R\))
then
\, =\, \{-q\} \sum_{n=0}^{\infty} ([q:ln(-1)\, \{2-q\}\, n] \,\}1-q\{\, q:ln(2n+1!))\, \{1-q\}\, (x\,\{2-q\}\,2n+1))
\, =\, \{-q\} \sum_{n=0}^{\infty} ([q:ln(-1)\, \{2-q\}\, n] \,\}1-q\{\, q:ln(2n!))\, \{1-q\}\, (x\,\{2-q\}\,2n))
it can also be shown that if\, =\, \lim_{h\to\ S(-q)}\, [f(x\,\{-q\}\,h)\,\}-q\{\, f(x)]\,\}1-q\{\,h)
\, =\, cos_q(x))
I ask because I've come across a curious set of "lowered operator" trigonometric function; if 0 <= q < 1,
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