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Can't be given in terms of elementary functions, with a finite number of terms. Seems a bit arbitrary though -- neither can exp(x) or ln(x) or sin(x), so introd
by oxymoron 10y ago
Can't be given in terms of elementary functions, with a finite number of terms. Seems a bit arbitrary though -- neither can exp(x) or ln(x) or sin(x), so introducing erf(x) and defining it using an integral seems more par the course than cheating.
- kmill 10y agoYeah, I meant finitely many terms. To me, things like "infinite sums" are special operations since they don't always converge. In case and exp, ln, sin seem special: they are solutions to simple linear differential equations x'-x=0, tx'=1, and x''-x=0. Though, erf comes from x''+2tx'=0, so I can't say I really understand what makes a function "elementary." Without knowing more, I'd exclude logarithms, and then say that elementary functions are solutions to homogeneous linear differential equations with constant coefficients. Edit: looking at MathWorld, it seems the definition has the set of elementary functions closed under inverses, which is why logarithms are included.
- OscarCunningham 10y agoI don't know a precise definition of "elementary", but one thing about the elementary functions (exp,log,sin,cos,arcsin,arccos,+,x,-,/) is that they interact with each other in nice ways: log(ab)=log(a)+log(b) cos(a+b)=cos(a)cos(b)-sin(a)sin(b) sin(log(x))=(x^-i - x^i)i/2 etc. This means that if your integral yields and answer in terms of log, exp or trigonometric functions then there is a good chance that any further algebra you have to do will work out nicely. But erf satisfies no such relations and so an answer in terms of it is often an algebraic dead end.
- qntty 10y agoexp, ln and sin are elementary functions, so they can be given in terms of elementary functions.