Half Window Functions and Formulas
A use of such window functions would be to window data that "goes straight to business" starting at the first sample and then gradually dies off, like an impulse response of some system. See What is the meaning of half window functions? A smooth rise in the window function would be detrimental as it would attenuate the important peak, unless the rise in the window is quick enough and the peak delayed enough. However, I think it would be a stretch to call something like that a half-window, if it is not a half of some window. So start directly at 1 at the peak of a symmetrical window I think.

Then again, beware that a sine window a.k.a. cosine window may also be called half-sine window or half-cosine window, as it represents half a cycle of a sinusoid (sine or cosine).