       pair sums applied to trignometry sums

• To: mathgroup at smc.vnet.net
• Subject: [mg52487] pair sums applied to trignometry sums
• From: Roger Bagula <tftn at earthlink.net>
• Date: Mon, 29 Nov 2004 01:22:34 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```I had used the mechanism with Bailey type of sequences
and their sums in the work on b normalness in iteratives functions.

It occurred to me that by adding the variable x , I could get
functiond that used the nonlinear Cantor pair {1/(n+1),n/(n+1)}
to split the sine and the cosine down the middle.
The result is entirely new trignometric sum functions that converge very
well.

(* pair sums applied to trignometry sums: {1/(n+1),n/(n+1)} modulo 2
switched sums*)
(* these sums break the trignometry of a circle into four functions
(* these are subharmonic functions of a nonlinear Rational Cantor type*)
fs[x_,n_]:=
If[Mod[n,2]==1,(-1)^(n)*n*x^(2*n+1)/((n+1)*(2*n+1)!),(-1)^(n)*
x^(2*n+1)/((n+1)*(2*n+1)!)]

gs[x_,n_]:=
If[Mod[n,2]==1,(-1)^n*x^(2*n+1)/((n+1)*(2*n+1)!),(-1)^n*n*
x^(2*n+1)/((n+1)*(2*n+1)!)]

fc[x_,n_]:=
If[Mod[n,2]==1,(-1)^(n)*n*x^(2*n)/((n+1)*(2*n)!),(-1)^(n)*
x^(2*n)/((n+1)*(2*n)!)]

gc[x_,n_]:=
If[Mod[n,2]==1,(-1)^(n)*x^(2*n)/((n+1)*(2*n)!),(-1)^(n)*n*
x^(2*n)/((n+1)*(2*n)!)]

digits=100;

fsin[x_]:=N[Sum[fs[x,n],{n,0,digits}]];

gsin[x_]:=N[Sum[gs[x,n],{n,0,digits}]]

fcos[x_]:=N[Sum[fc[x,n],{n,0,digits}]]

gcos[x_]:=N[Sum[gc[x,n],{n,0,digits}]]

Plot[fsin[x],{x,-Pi,Pi}]

Plot[fsin[x],{x,-Pi,Pi}]

Plot[gsin[x],{x,-Pi,Pi}]

Plot[fcos[x],{x,-Pi,Pi}]

Plot[gcos[x],{x,-Pi,Pi}]

Plot[(fsin[x]+gsin[x]),{x,-Pi,Pi},PlotRange->All]

Plot[(fcos[x]+gcos[x]),{x,-Pi,Pi}]

ParametricPlot[{fsin[x],gsin[x]},{x,-Pi,Pi}]

ParametricPlot[{fcos[x],gcos[x]},{x,-Pi,Pi}]

ParametricPlot[{fsin[x],fcos[x]},{x,-Pi,Pi}]

ParametricPlot[{gsin[x],gcos[x]},{x,-Pi,Pi}]

ParametricPlot[{fsin[x]+gsin[x],fcos[x]+gcos[x]},{x,-Pi,Pi},PlotRange->All]
Respectfully, Roger L. Bagula