producte integral sinus-y-cosinus elíptic

sin[n](x) = ...
... ∑ ( e^{(2/(n+1))·pi·i} )^{k_{1}...k_{n}}·( 1/((2·k_{1}...k_{n})+1)! )·x^{( (2k_{1}...k_{n})+1 )}
cos[n](x) = ...
... ∑ ( e^{(2/(n+1))·pi·i} )^{k_{1}...k_{n}}·( 1/(2·k_{1}...k_{n})! )·x^{(2k_{1}...k_{n})}


d_{x}[sin[n](x)] = cos[n](x)
d_{x}[cos[n](x)] = e^{(2/(n+1))·pi·i}·sin[n](x)
d_{xx}^{2}[sin[n](x)] = e^{(2/(n+1))·pi·i}·sin[n](x)
d_{xx}^{2}[cos[n](x)] = e^{(2/(n+1))·pi·i}·cos[n](x)


( cos[n](x) )^{(n+1)} + ( sin[n](x) )^{(n+1)} = n^{(n+1)}
d_{x}[cos[n](x)]^{(n+1)} + d_{x}[sin[n](x)]^{(n+1)} = n^{(n+1)}
d_{xx}^{2}[cos[n](x)]^{(n+1)} + d_{xx}^{2}[sin[n](x)]^{(n+1)} = n^{(n+1)}


f(x) [o( sin[n](x) )o] sin[n](x) = f(x)


S[n]_{x}[f(x)] = ( d_{x}[f(x)]/( n^{(n+1)}+(-1)·( f(x) )^{(n+1)} )^{(1/(n+1))} )


f(x) [o( cos[n](x) )o] cos[n](x) = f(x)


C[n]_{x}[f(x)] = e^{( (2n)/(n+1) )·pi·i}·( d_{x}[f(x)]/( n^{(n+1)}+(-1)·( f(x) )^{(n+1)})^{(1/(n+1))} )


teorema:
sin[n]( f(x) [o(x)o] g(x) ) = sin[n](f(x)) [o( sin[n](x) )o] sin[n](g(x))
cos[n]( f(x) [o(x)o] g(x) ) = cos[n](f(x)) [o( cos[n](x) )o] cos[n](g(x))