I know about hyperbolic trig functions, but I’ve never heard of a parabolic sine function. Is there some fundamental relationship between this and the circular and hyperbolic functions? The relationship between the circular and hyperbolic functions is quite clear to me. For example, [ul]i sin u = sinh iu
tanh u = -i tan iu
etc.[/ul]Is there a smililarly straightforward relationship that ties in the parabolic functions? Or are these parabolic functions something entirely different? Is it proper to say “sinp x” equals such-and-such?
First I’ve ever heard of them, and tehre sems to be essentially nothing else on the Web …
I think it’s just a term they made up for a function which graphs a little prettier than a parabola or an ordinary sine. The use of the double quotes around the term might indicate this.
Hrm… a parabola with a sine wave superimposed on it. Nothing really special here like JonF has already mentioned. Note that the article is supposed to orginate from a high school as teaching material. The contents suggest that this “parabolic sine” was just produced to help students understand the use of derivatives, and how the x[sup]2[/sup] term vanishes while the sine remians etc. etc.
It looks like there is a use, actually. I’ve been doing searches and have come across a few sites where it is used to define types of motion (Brownian, for example). The sites have been a bit too advanced for me, as well as using characters that don’t seem to translate very well(maybe need special fonts). Here’s one on minimal parabolic functions and time-homogeneous parabolic h-transforms and Molecular Mechanics/Molecular Dynamics.