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From: gwyn@brl-tgr.ARPA (Doug Gwyn <gwyn>)
Newsgroups: net.puzzle
Subject: Re: More on Interview Questions...derivative of x!
Message-ID: <9551@brl-tgr.ARPA>
Date: Thu, 28-Mar-85 15:27:21 EST
Article-I.D.: brl-tgr.9551
Posted: Thu Mar 28 15:27:21 1985
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> > ... A question one
> > could ask is can you really find ALL functions, defined in any way,
> > that equal their derivatives and  which are they?
> 
> I think the exponential is the only one, but I don't know.

Suppose f(.) and g(.) are any two such functions.

Df = f		Dg = g

D(f - g) = Df - Dg	(by linearity of D)
	= f - g

So (f - g) is another such function.  So what?  Well, it's cute.
I think a uniqueness proof could be based on it.  However, let's
assume the availability of a text on ODEs:

Let f(.) be such a function of its single (real) argument.

Df = f

Df - f = 0

(D - 1.)f = 0		(linear (not just affine) operators)

Two C-infinity solutions:

(A)	f == 0		(constant-zero function)

(B)	f == c exp(.)	where c is any constant
			((B) includes (A) as a special case, actually)

The theory of linear ODEs tells us that (B) is the most general
solution under very general continuity assumptions.