[Sorry -- this needed some more edits.] Is anyone on this list aware of anyone who has studied the following functions: s2(x):=(2^x-2^(-x))/2 =sinh(x*log(2)) Generally, s2(x) acts like a sinh function, suitably fudged. as2(x):=sgn(x)*log(abs(x)+sqrt(1+x^2))/log(2) =asinh(x)/log(2) For x near zero, as2(x) is x/log(2) to a first approximation. For x very large, |as2(x)| approximates log(2|x|)/log(2) = log2(|x|)+1. s2(x)=-s2(-x) as2(x)=-as2(-x) Generally, as2(x) acts like the asinh function, suitably fudged. The cool thing about s2(n) is that its binary representation is a series of "1" bits exactly 2n bits long, with n-1 bits to the left of the binary point and n+1 bits to the right of the binary point. We can use linear combinations of these functions to create bit strings which are palidromes ! This bit representation also gives a bit more insight into the true nature of the sinh function. --- BTW, "asinh" now has a real paying job. I searched using Google and I found that astronomers are now using asinh instead of log to compute brightness of stars in a star catalog. I have advocated the use of asinh in audio & video processing to do "dynamic range compression" in a way which is somewhat smoother than current ad hoc methods and floating point methods. Another job is its use in genetic engineering & statistics to smooth data better than log when very small magnitudes are encountered, which would otherwise drive log nuts (i.e., very substantially negative). Asinh pops up in the calculation of Chebyshev filters in DSP's. Also asinh shows up in the design of certain magnetic coils.