Re: Special Functions

On Feb 27, 3:42 pm, mike3 <mike4...@xxxxxxxxx> wrote:
On Feb 27, 9:27 am, A N Niel <ann...@xxxxxxxxxxxxxxxxxxxxx> wrote:



In article
<71536251-53d6-47d7-aa2f-fce8795c5...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,

<umum...@xxxxxxxxx> wrote:
In the present article, Special Functions are defined as functions
with
_one_ of the following properties.

For all real x and y (where c is a real, sometimes positive,
constant):

1. Linear     : F(x+y) = F(x) + F(y)  for example:  F(x) = c.x
2. Logarithmic: F(x.y) = F(x) + F(y)  for example:  F(x) = ln(c.x)
3. Exponential: F(x+y) = F(x) . F(y)  for example:  F(x) = e^(c.x)
4. Power      : F(x.y) = F(x) . F(y)  for example:  F(x) = x^c

So it is rather obvious where the labels come from. But the QUESTION
is: are the functions mentioned indeed the ONLY ones with the special
properties.

The only *continuous* (or *measurable*) ones.  With some little
objections, like the constant zero in 3. and only for positive x,y in
some.

<snip>

Does this also mean they're the only *constructable* ones?

That depends on what you mean by "constructable". For example, a
recursive function on the reals must be continuous wherever it
converges, but a piecewise linear function may not be.
.