Defining Transcendental Functions

Date: 02/05/2001 at 21:10:33
From: Ruth
Subject: Defining transcendental functions

I am looking for a precise definition of a transcendental function 
and of an elementary function. I BELIEVE that the set of elementary 
functions is just the set of algebraic functions and the set of 
transcendental functions combined, but I cannot find it specifically 
defined anywhere in my calculus book or at any of the Web sites I 
have visited. (Most sites with mathematics definitions do not include 
elementary functions.)

My book (James Stewart's Calculus, 4th edition) defines transcendental 
functions as everything except algebraic functions (functions 
containing only addition, subtraction, division, multiplication, 
powers, and roots), then gives a few examples (trig and inverse trig 
functions, log and exponential functions, and hyperbolic and inverse 
hyperbolic functions), and then says, "it also includes a vast number 
of functions that have never been named...we will study transcendental 
functions that are defined as sums of infinite series." Are all 
infinite series transcendental functions?

Later, when defining elementary functions, it lists examples and then 
says that the integral of an elementary function is usually not an 
elementary function. Can you define a non-elementary function by some 
other means than as the integral of another function? Can the unnamed 
transcendental functions that the book mentions be expressed 
symbolically, or would that be considered naming them? How am I to 
distinguish between an unnamed transcendental function and a non-
elementary function? 

I am would LOVE a clear answer on this.  

Thank you, 
Ruth Costa

Date: 02/08/2001 at 23:30:46
From: Doctor Fenton
Subject: Re: Defining transcendental functions

Hi Ruth,

Thanks for writing to Dr. Math. You've asked some very good questions.

The definition of elementary functions seems to be a matter of 
consensus within the mathematical community, and it basically consists
of the "familiar" functions, and functions that can be generated from
the standard ones by addition, subtraction, multiplication, division, 
and composition. Like most definitions, this is somewhat arbitrary, 
but I think there is pretty general agreement on which functions are
"elementary."

The definition of an algebraic function is also fairly precise, as you
described (I might add that the powers and roots must be rational 
ones).

A problem with "transcendental" is that it is a catch-all definition, 
defined by exclusion rather than by direct characterization. Anything 
that is not an algebraic function is, by definition, transcendental. 
In that regard, it's somewhat like irrational numbers, which are 
defined by exclusion: they aren't rational. That makes it difficult 
to make general statements about them. For example, we can say that 
the sum and product of rational numbers is rational, but sqrt(2) and 
(2-sqrt(2)) show that the sum of irrationals can be rational, and 
sqrt(2)*sqrt(2) shows that the product of irrationals can be rational.

There is only a countably infinite number of algebraic functions, but 
it can be shown that there are uncountably many continuous functions 
on an interval, for example. Since there are only countably many 
finite strings of mathematical symbols, it isn't possible to 
describe "most" functions. These are some of the "unnamed 
transcendental functions" Professor Stewart is trying to describe. 
He can only give you a hint of the complexity of the set of 
transcendental functions. 

There are also "named" transcendental functions: Bessel functions and 
Hankel functions, to name a couple. These arise in certain specialized 
areas, and are well-known to experts in those fields, but the typical 
math student may never encounter them, so they are not put in the 
"elementary" list.

So basically, there is just no way to describe a "typical" 
transcendental function. Many of them are indeed generated by 
infinite series, but on the other hand, some infinite series describe 
algebraic and elementary functions, too. Elementary and algebraic 
functions are relatively small, precisely defined collections of 
functions, which can be characterized by common properties.  
Transcendental functions share only the property of being non-
algebraic.

I'm sorry I can't be of more help, but there just isn't any way to
characterize transcendental or non-elementary functions in general, 
other than that single exclusionary property.

If you have further questions, please write us again.

- Doctor Fenton, The Math Forum
  http://mathforum.org/dr.math/