On Thu, 15 Jan 1998, Dave Huang wrote:

> On Wed, 14 Jan 1998, Colin Wood wrote:
> > I've often wondered where the name came from, but it's just stuff like
> > sine, cosine, and tangent (and all the variations that are supported).
> 
> Transcendental functions are functions that aren't algebraic :) I guess
> maybe they transcend algebraic operations or something...
> 
> Algebraic functions are functions that can be made by performing a finite
> number of algebraic operations (i.e. +, -, *, /, reciprocal, power, root)
> on the argument of the function and rational constants. So like sqrt(x^5 -
> 1/(sqrt(2/3 + x^9))) is freaky, but still an algebraic function. 

I agree with the jist of what you're saying, but I wouldn't put root and
power to a non-integral value in there. Power to an integer exponent is
just shorthand for multiplying and maybe reciprication.

I'd say the big difference is in the rules about combining things. Like is
the operation associative and commutative (sp?). +, -, and * are.
Reciprication doesn't fit nicely. Nor does x**y.

> Some (maybe all?) transcendental functions can be expressed as an infinite
> sum of algebraic functions though. Taylor series and stuff like that...
> Like sin x = x/1! - x^3/3! + x^5/5! - x^7/7! + x^9/9! - ... forever

I think they all do, if you look in a good-enough reference book. :-)

x**y is often written as exp(y*ln(x)), which is why I'd say it's a
transcendental function too. And there's a Taylor series for sqrt too.
sqrt(1+x) = 1 - 1/2 x + 3/8 x^2 ... (I'm pretty sure it's 3/8).

Take care,

Bill