Prove Injectivity of x^x Function?

[tehno]

mathwonk,this is strange to me ,but if that's really definition you're right (thank you!).
composition of standard elementary functions x,e^x,and ln(x) creates x^x.Indeed,I thought elementary functions were those obtainable from standard elementary functions only by means of finite number of arithmetical operations (+,-,*,:) among them.

 

[mathwonk]

mathwonk,this is strange to me ,but if that's really definition you're right (thank you!).
composition of standard elementary functions x,e^x,and ln(x) creates x^x.Indeed,I thought elementary functions were those obtainable from standard elementary functions only by means of finite number of arithmetical operations (+,-,*,:) among them.

throw in composition and you are there.

i.e. ln(ln(x)) is also elementary.

I guess the definition was made this way to try to encompass functions we usually try to antidifferentiate, like x^2 e^(x^3).


by the way in omitting trig functions, i was tacitly assuming the functions are complex valued, so that trig functions and their inverses are a special case of exponentials and logs.

e.g. arctan is the same as log except it twirls around i and -i instead of 0 and infinity. so if you compose with an automorphism like (1-i)/(1+i) they become almost the same.


to see this, just think of log as path integral of 1/z along paths that avoid 0 and infinity, and arctan as path integral of 1/(1+z^2) along paths that avoid i and -i.

 

[tehno]

You pro-mathematicians certainly know *all rules * of your game.And We computer scientists must obey it without objections .
Maybe,I'll be hang for telling you this but most of us consider only
polynomials over Q-field truly elementary functions