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Topic: Re: Definition of the exponential and natural logarithm
functions [was....doozi]

Replies: 3   Last Post: Oct 19, 2001 1:28 AM

 Messages: [ Previous | Next ]
 SANTU DESILVA Posts: 160 Registered: 12/4/04
Re: Definition of the exponential and natural logarithm
functions [was....doozi]

Posted: Oct 17, 2001 4:05 PM

On Mon, 15 Oct Lou Talman wrote in part:

:There are three principal methods for giving rigorous definitions of the
:exponential/logarithm pair of functions:
:
: 1. Definite integral
: 2. Series
: 3. Extension by continuity.

Listowner Flashman replies:

: A fourth approach to the definition of the natural exponential function
: can be based on the differential equation y'=y with y(0)=1 - motivated by
: population growth models, amd made credible using a mixture of
: differential equation heuristics and properties of solutions to DE's that
: are accessible to first year calculus students.

The operative word here is *rigorous*.
There certainly is a rigorous theory of differential equations,
based on existence and uniqueness theorems. It would be
somewhat misguided to introduce freshmen to these things.

An axiomatic approach, in contrast, based on, say, a
functional equation and continuity, obviates the need for
developing machinery, and encapsulates the fundamental
facts based from which the properties of the function are derived.

However, if rigor is not an important consideration (and of course
there are definitions of rigor that are very forgiving indeed) then
I would say the differential equation approach is at least on a
par with the other three.

It is almost impossible to predict which of these approaches will
be most attractive to a given set of students!

P.S. How about the definition exp(x) = lim( {1 + x/n}^n, n -> infinity}?

Arch

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Date Subject Author
10/17/01 SANTU DESILVA
10/17/01 Victor Steinbok
10/17/01 Kazimierz Wiesak
10/19/01 Jerry Uhl