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throwaway81523 • today at 7:16 AM • 2 replies • view on HN

It's news to me that "elementary functions" include roots of arbitrary polynomials, but the wiki article in fact says that they're included at least some of the time. I remember reading about the Risch algorithm (for finding closed form antiderivatives) a long time ago and elementary functions were just the ordinary ones found on calculators.

Interestingly, the abs (absolute value) function is non-elementary. I wonder if exp-minus-log can represent it.

Replies

reikonomusha • today at 7:26 AM

EML can represent the real absolute value, so long as we agree with the original author's proviso that we define log(0) and exp(-∞), by way of sqrt(x^2) as f(x) = exp((1/2)log x). Traditionally, log(0) isn't defined, but the original author stipulated it to be -∞, and that all arithmetic works over the "extended reals", which makes

    abs(0)
    = f(0)            ; by defn
    = exp(1/2 log 0)  ; by defn
    = exp(-∞/2)       ; log 0 rule
    = exp(-∞)         ; extended real arith
    = 0               ; exp(-∞) rule
If we don't agree with this, then abs() could be defined with a hole punched out of the real line. The logarithm function isn't exactly elegant in this regard with its domain restrictions. :)
traes • today at 7:25 AM

abs(x) = sqrt(x*x), no?

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