So, I have several functions I've been working on, and was hoping to find a more general form. I guess they're maybe a series, of sorts, or could be.
I've been a bit out of the Mathematics game since High School and a little bit in College. So, I'm a bit rusty on my terminology, etc.
Here are a couple examples:
PolarPlot[(Abs[Cos[0]] + Abs[Cos[2 Pi/4]]) / ((Abs[Cos[x]] + Abs[Cos[x + 2 Pi/4]])), {x, 0, 2 Pi}]
PolarPlot[(Abs[Cos[Pi/6]] + Abs[Cos[3 Pi/6]] + Abs[Cos[5 Pi/6]]) / (Abs[Cos[x + Pi/6]] + Abs[Cos[x + 3 Pi/6]] + Abs[Cos[x + 5 Pi/6]]), {x, 0, 2 Pi}]
PolarPlot[(Abs[Cos[0]] + Abs[Cos[2 Pi/8]] + Abs[Cos[4 Pi/8]] + Abs[Cos[6 Pi/8]]) / (Abs[Cos[x]] + Abs[Cos[x + 2 Pi/8]] + Abs[Cos[x + 4 Pi/8]] + Abs[Cos[x + 6 Pi/8]]), {x, 0, 2 Pi}]
PolarPlot[(Abs[Cos[Pi/10]] + Abs[Cos[3 Pi/10]] + Abs[Cos[5 Pi/10]] + Abs[Cos[7 Pi/10]] + Abs[Cos[9 Pi/10]]) / (Abs[Cos[x + Pi/10]] + Abs[Cos[x + 3 Pi/10]] + Abs[Cos[x + 5 Pi/10]] + Abs[Cos[x + 7 Pi/10]] + Abs[Cos[x + 9 Pi/10]]), {x, 0, 2 Pi}]
PolarPlot[(Abs[Cos[0 Pi/12]] + Abs[Cos[2 Pi/12]] + Abs[Cos[4 Pi/12]] + Abs[Cos[6 Pi/12]] + Abs[Cos[8 Pi/12]] + Abs[Cos[10 Pi/12]])/(Abs[Cos[x + 0 Pi/12]] + Abs[Cos[x + 2 Pi/12]] + Abs[Cos[x + 4 Pi/12]] + Abs[Cos[x + 6 Pi/12]] + Abs[Cos[x + 8 Pi/12]] + Abs[Cos[x + 10 Pi/12]]), {x, 0, 2 Pi}]
What I'm hoping to do is to generalize these.
That is, my hope is that each of these corresponds to a specific iteration of some super-function. Such that maybe by specifying a certain variable, it would reproduce the associated function.
It would be cool if Mathematica could take these individual functions and compare them and generalize them for me. It's save a considerable amount of work and maybe let me get at some other related problems more quickly.
So, what Mathematica function(s), if any, are capable of this?
It seems to me like maybe we should be able to rewrite JUST the functions being plotted above as something like:
f(4) = (Abs[Cos[0 Pi/4]] + Abs[Cos[2 Pi/4]]) / ((Abs[Cos] + Abs[Cos[x + 2 Pi/4]]))
f(6) = (Abs[Cos[Pi/6]] + Abs[Cos[3 Pi/6]] + Abs[Cos[5 Pi/6]]) / (Abs[Cos[x + Pi/6]] + Abs[Cos[x + 3 Pi/6]] + Abs[Cos[x + 5 Pi/6]])
f(8) = (Abs[Cos[0 Pi/8]] + Abs[Cos[2 Pi/8]] + Abs[Cos[4 Pi/8]] + Abs[Cos[6 Pi/8]]) / (Abs[Cos] + Abs[Cos[x + 2 Pi/8]] + Abs[Cos[x + 4 Pi/8]] + Abs[Cos[x + 6 Pi/8]])
f(10) = (Abs[Cos[Pi/10]] + Abs[Cos[3 Pi/10]] + Abs[Cos[5 Pi/10]] + Abs[Cos[7 Pi/10]] + Abs[Cos[9 Pi/10]]) / (Abs[Cos[x + Pi/10]] + Abs[Cos[x + 3 Pi/10]] + Abs[Cos[x + 5 Pi/10]] + Abs[Cos[x + 7 Pi/10]] + Abs[Cos[x + 9 Pi/10]])
f(12) = (Abs[Cos[0 Pi/12]] + Abs[Cos[2 Pi/12]] + Abs[Cos[4 Pi/12]] + Abs[Cos[6 Pi/12]] + Abs[Cos[8 Pi/12]] + Abs[Cos[10 Pi/12]])/(Abs[Cos[x + 0 Pi/12]] + Abs[Cos[x + 2 Pi/12]] + Abs[Cos[x + 4 Pi/12]] + Abs[Cos[x + 6 Pi/12]] + Abs[Cos[x + 8 Pi/12]] + Abs[Cos[x + 10 Pi/12]])
The question is can Mathematica then use those 5 examples of the "series" (I have more examples, up through at least about f(20) ...) and be able to predict the equivalent form of f(14), f(16), f(18), ..., f(n)? Actually, I'd be really interested if it managed to come up with any of the intermediate f's of "odd" values >= 3, as I've had no luck with figuring those out as of yet (or whether they're even possible, though I suspect they should be).
How would I go about this? It's been so long since I've had to try and do this kind of thing, I've all but forgotten how... Which sucks, 'cause way back when, I used to be pretty good at this kind of thing, when it was all fresh in my head.
Thoughts? Seems like this would/could/should be something Mathematica can handle. I just don't know what functions and parameters to use to even get it to
try.
Thanks,
~MG
