# Tetration Forum

Full Version: 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 $f(x + \tau) = f(x)$, but rather (if {p} represents an operator of p magnitude and }p{ reps its root inverse) $f(x\, \{p\}\, \tau) = f(x)$

I ask because I've come across a curious set of "lowered operator" trigonometric function; if 0 <= q < 1, $q:ln(x) = \exp^{[-q]}(x)$ ;
$x \{p\} S(p) = x$ or $S(p)$ is the identity function,:

$sin_q(x) = q:ln(sin(-q:ln(x)))$
$cos_q(x) = q:ln(cos(-q:ln(x)))$

they satisfy
$sin_q(x \{-q\} q:ln(2\pi)) = sin_q(x)$
$cos_q(x \{-q\} q:ln(2\pi)) = cos_q(x)$

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

$e^{x\, \{1-q\}\, q:ln(i)}\, =\, cos_q(x)\, \{-q\}\, (q:ln(i)\, \{1-q\}\, sin_q(x))$

$sin_q(x\, \{-q\}\, q:ln(\frac{\pi}{2}))\, =\, cos_q(x)$

$(sin_q(x)\,\{2-q\}\,2)\, \{-q\}\, (cos_q(x)\,\{2-q\}\,2)\,=\,S(1-q)$

$sin_q(x\, \{-q\}\, y)\, =\, (sin_q(x)\,\{1-q\}\,cos_q(y))\, \{-q\}\, (sin_q(y)\,\{1-q\}\,cos_q(x))$

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 $\{p\} \sum_{n=0}^{R}\, f(n)\, =\, f(0)\, \{p\}\, f(1)\, \{p\} ... f$$R$$$

then

$sin_q(x)\, =\, \{-q\} \sum_{n=0}^{\infty} ([q:ln(-1)\, \{2-q\}\, n] \,\}1-q\{\, q:ln(2n+1!))\, \{1-q\}\, (x\,\{2-q\}\,2n+1)$
$cos_q(x)\, =\, \{-q\} \sum_{n=0}^{\infty} ([q:ln(-1)\, \{2-q\}\, n] \,\}1-q\{\, q:ln(2n!))\, \{1-q\}\, (x\,\{2-q\}\,2n)$

it can also be shown that if $q:\frac{d}{dx}\, f(x)\, =\, \lim_{h\to\ S(-q)}\, [f(x\,\{-q\}\,h)\,\}-q\{\, f(x)]\,\}1-q\{\,h$

$q:\frac{d}{dx} sin_q(x)\, =\, cos_q(x)$