|Re: (Cos x)^x plot|
Message #4 Posted by Karl Schneider on 1 Dec 2007, 2:19 p.m.,
in response to message #3 by Chuck
Hi, Chuck --
Thank you for the plot, which would be laborious to produce using an HP-15C and graph paper...
It's actually defined when negative, as long as the values aren't complex.
The function is defined continuously in the real domain for positive or negative x as long as "cos x" is positive. If "cos x" is negative, then only integer values of x can produce a real-valued result.
Also, it resembles secant for x<0.
Since cosine is "even", [cos x = cos(-x)], for negative x we can write
(cos x)x = 1 / [(cos |x|)|x|
Of course, sec x = 1 / cos x.
The spikes and "U"'s get narrower for large x's.
As expected, since larger exponents "reduce" smaller arguments. That is, xa < x for x < 1 and a > 1.
I understand the solid lines of the plot, where the function has a real-valued result, but the dotted parts (e.g., between pi/2 < x < 3*pi/2) don't seem to represent Re[f(x)].
Edited: 1 Dec 2007, 3:35 p.m.