Definition of a Transcendental Function

[psholtz]

I had grown up thinking that the transcendental functions (to wit, e(x), sin(x), cos(x), etc) were (somewhat) arbitrary functions, "defined" simply by the fact that they could not be expressed as polynomials (or some such loose definition).

Indeed, from Wikipedia, we read that:

A transcendental function is a function that does not satisfy a polynomial equation whose coefficients are themselves polynomials, in contrast to an algebraic function, which does satisfy such an equation.

http://en.wikipedia.org/wiki/Transcendental_function

But now I'm reading from Ince's classic treatise on ODEs, and on page 23, he gets into a brief footnote about the nature of transcendental functions. Specifically, he writes:

The elementary transcendental functions are functions which can be derived from algebraic functions by integration, and the inverses of such functions.

-- Ince on ODEs

This is a rather different definition than that given by Wikipedia.

Indeed, Ince goes on to define the logarithm function as:

$$\ln(x) = \int_1^x t^{-1}dt$$
and then defines the exponential function as the inverse of the logarithm, and then tersely states that the remaining elementary transcendentals, like the sine, cosine, and their hyperbolic equivalents, can be defined in terms of the exponential.

Now, it's obvious to me how to define hyperbolic sine and cosine in terms of exponentials. That follows directly from their definition.

For the "regular" sine and cosine, he must have in mind Euler's formula, no?

$$e^{ix} = \cos x + i \sin x$$

I'm asking, just b/c I'm not sure how else to define sine and cosine in terms of an exponential..

At any rate, this was a little surprising to me, and I thought I would bring it up for discussion.

Is there a reference where I can read more about transcendental functions, and how they are defined? (i.e., defined in terms of integrating algebraic functions?)

Moreover, the imagination can easily conjure up a wide array of "transcendental" functions, to wit:

$$f(x) = x^\pi$$
$$g(x) = x^ {1/x}$$

and so on..

How might these functions be defined in terms of integration from algebraic functions?

 

[Office_Shredder]

I'm not familiar with the text you're citing, but I think the key here is "elementary transcendental functions". Not all transcendental functions can be defined by integration (any function defined by integration has to be continuous, and obviously there are transcendental functions that are not continuous).

As for defining sine and cosine, it does use Euler's formula

$$cos(x)=\frac{e^{ix}+e^{-ix}}{2}$$

$$sin(x)=\frac{e^{ix}-e^{-ix}}{2}$$

You can check by Euler's formula being plugged into the right hand side of each equation

 

[Hurkyl]

You know Euler's identity for e^ix. What about for e^-ix? Now solve for the two unknowns. (the unknowns being cos(x) and sin(x))

 

[nekronaut]

I'm not familiar with the text you're citing, but I think the key here is "elementary transcendental functions". Not all transcendental functions can be defined by integration (any function defined by integration has to be continuous, and obviously there are transcendental functions that are not continuous).

As for defining sine and cosine, it does use Euler's formula

$$cos(x)=\frac{e^{ix}+e^{-ix}}{2}$$

$$sin(x)=\frac{e^{ix}-e^{-ix}}{2}$$

You can check by Euler's formula being plugged into the right hand side of each equation

Almost right

$$\sin(x)=\frac{e^{ix}-e^{-ix}}{2i}$$