Okay, so I'm still working on ye olde investigation and have decided upon a new tack to the problem of solving for the "odd" functions.

I've decided to shift the frequency of all of my graphs down to the same frequency as the first graph, but clearly the amplitude does not match when that happens.

GraphStart = 4  (* First graph to plot. Must be even integer. *) 
GraphEnd = 50  (* Last graph to plot. Must be even integer. *) 
xStart= 0 
xEnd = 2 Pi 
xTicksTable := Table[xt, {xt, 0, xEnd, Pi/8}]; 
zTicksTable := Table[zt, {zt, 0, GraphEnd, 2}]; 
f[z_?EvenQ] := With[{n = z/2}, (Sum[Abs[Cos[((n + 2 + 2 (k - 1)) Pi)/(2 n)]], {k, 1, n}])/(Sum[Abs[Cos[(2/n) x + (((n + 2 + 2 (k - 1)) Pi)/(2 n))]], {k, 1, n}])] funcs = Table[{x, z, f[z]}, {z, GraphStart, GraphEnd, 2}]; 
ParametricPlot3D[funcs, {x, 0, 2 Pi}, PlotRange -> {Automatic, {0, GraphEnd}, {0.5, 1}}, BoxRatios -> {1, 1, 1}, ViewPoint -> {8, -20, 10},  AxesEdge -> {{-1, -1}, {-1, -1}, {-1, -1}} , Ticks -> {xTicksTable, zTicksTable, Automatic}, PlotPoints -> 200, AxesLabel -> {"x axis", "z axis", "y axis"}]

So I want to plot some reference points at the same places on each graph slice, and perhaps output their precise values (fractions, with Pi or whatnot) to a table and display the table?

But, it's been a while since I looked at this and my brain's a little bit fried. How would I go about something like that?

I'd probably want to say something like for z from GraphStart to GraphEnd, incrementing by 2, at each value of Z, I want to see what the precise value of the function is at the points, say from 0 to 2Pi, incrementing by Pi/8.

So, it would maybe output something like:

for z=4, @Pi/8=datum1, @2Pi/8=datum2, @3Pi/8=datum3, @4Pi/8=datum4, @5Pi/8=datum5, @6Pi/8=datum6, @7Pi/8=datum7, @8Pi/8=datum8, ... , @16Pi/8=datum16. for z=6, @Pi/8=datum17, ... , @16Pi/8=datum32 ... for z=GraphEnd, ..., etc.

How would I go about adding this extra output table below the 3D function slices graph?

Thanks in advance!

~Michael

...the results it spits out seem to in some cases be a bit too simplified (has hoped for everything to be simplified as far as possible in the Cosine format, without necessarily evaluating to radicals, etc.), and other parts feel like they're maybe not quite simplified enough...

For example:

GraphStart = 4;
GraphEnd = 14;
SamplingRate = Pi/32;
f[z_?EvenQ] := With[{n = z/2}, (Sum[Abs[Cos[((n + 2 + 2 (k - 1)) Pi)/(2 n)]], {k, 1, n}])/(Sum[Abs[Cos[(2/n) x + (((n + 2 + 2 (k - 1)) Pi)/(2 n))]], {k, 1, n}])];
FunctionValuesTable = Table[{f[z]}, {x, 0, Pi, SamplingRate}, {z, GraphStart, GraphEnd, 2}];
Grid[FunctionValuesTable, Frame -> All]
Simplify[Grid[FunctionValuesTable, Frame -> All]]

Even using simplify here, when the simplified grid table comes up, once can see that things are "simplified" inconsistently...

In particular, the 4th column of the 2nd grid / table, where the numerator seems inconsistently simplfied.

Sometimes the left term is simplified to Sqrt[10-2Sqrt[5]], sometimes it's simplified to Sqrt[2(5-Sqrt[5])], whereas the right term of the numerator appears to ALWAYS simplify to Sqrt[2(5+Sqrt[5])] and NEVER to Sqrt[10+2Sqrt[5]], despite the terms being nearly identical in form.

Likewise, it SOMETIMES leave the Sine / Cosine functions in the denominator, but sometimes switches to a Sqrt[] equivalent value. The Numerator only rarely seems to simplify to the Sine / Cosine functions, and almost always simplifies to some Sqrt[] equivalent...

And it seems to ignore my equation's original formattingsolely in terms of Cosine, and "simplifies" to a combination of both Sine AND Cosine, which is much harder to parse, since I'd have to figure out which term the Sine's came from and possibly convert them back to see what's actually going on.

It's a bit annoying. Seems like a deficiency in the program to render such inconsistent results? Dunno... Especially things like "factoring," I mean, one would think it would factor consistently, so as to either ALWAYS render a result like Sqrt[2(5-Sqrt[5])] with the 2 factored out, or ALWAYS render a result of Sqrt[10-2Sqrt[5]] with the 2 multiplied out across the inner terms. Weird that it doesn't handle these things consistently...

Yeah?