On the transcendency of function
iThe main goal of the paper is to prove the transcendency of functions by means o{ functional equations. AMS ClMsirication 1980<39B20.Secondar3r 26A30. 1. It is well known that a function / iD »R with D C R is called an algebraic function if it satisfies the equaüon (1 ) P ( x , / ( x ) ) = 0 , x C D . where is a polynomial in two variables and where all ^ i i ( x ) ( k = 0 , l , " , n ) are polynomials in x. All the other functions are called transcendental functions. In other words.if the function /|D-»R is not the Solution of any algebraic equation ( l ) , i n its dotnain of definition or partial domain of definition.then /iD-^R is a transcendental function. REMARK. If /|D»R is an algebraic function. then the Solution set of the equation f(.x) = c (where c is any constant) is finite. In this paper.we present a rule which can be used to teil whether a function is transcendental. This criterion is built on the base of certain functional equations. 2. We put R ' =R'^\A.where R'*' is the set of all positive real numbers and A is an enumerable set contained in R"*". We shall prove the following THEOREM 1. Suppose that / is a nonconstant function with its domain of definition DZ)R ' .and / iD »R satisfies the Poincari functional equation (2 ) / ( < u : ) G ( / ( x ) ) . x 6 D where a > l , G « E — R and E 3 / ( D ) . I f /and G fulfil the conditions. ( i ) there exists a fixed point f e E f l / C D ) of G , i . e . G ( ( ) = ^ . ( i i ) there exists a xo€R* such that / ( x o ) = l t ( iü) o * x , 6 R ' f o r n 1 , 2 , . then / is a transcendental function.