Transcendental and implicit functions

Question

Def. An implicit function is a function that is defined implicitly by an implicit equation, by associating one of the variables (the value) with the others (the arguments). For example, $$F(x,y)=0$$$$ e^x+x+y-\sin(y)=0$$ Def. A transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function. For example, $$e^x-y=0$$

Can we state that

1. all transcendental functions of more than one variable are implicit functions

and vice versa

2. all implicit functions are transcendental functions of more than one variable

Note. I understand that the 1. point is incorrect, but if throw out "all", under what conditions will the 1. point be true?

Answer 1

Implicit and transcendental functions are rather different, as

1. Implicit function: is a relation of the form $R(x_1,...,x_n)=0$, where $R$ is a function of several variables (often a polynomial, but sometimes transcendental)

$\ \ \ \ x^2+y^2-1=0 \ \ \ \ $ implicit (polynomial)

$\ \ \ \ x^2+e^y-1=0 \ \ \ \ $ implicit (transcendental)

2. Transcendental function: is analytic function that has not polynomial structure, as an example:

$\ \ \ \ x^2+e^x-1=0 \ \ \ \ $ transcendental

$\ \ \ \ x^2+e^y-1=0 \ \ \ \ $ transcendental (implicit)