Re: ways to define harmonic functions
- From: "G. A. Edgar" <edgar@xxxxxxxxxxxxxxxxxxxxxxxxxxx>
- Date: Sat, 29 Sep 2007 06:21:08 -0400
In article <1191031460.581720.164280@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Daniel J. Greenhoe <dgreenhoe@xxxxxxxxx> wrote:
There are at least three ways to define harmonic functions such as
sine, cosine, and the complex exponential:
1. as relations between sides and angles of triangles in plane
geometry
2. as solutions of second order homogeneous differential equations
with certain initial conditions as in
D^2 cos + cos = 0 with init.cond. cos(0)=1 [Dcos](0)=0
D^2 sin + sin = 0 with init.cond. sin(0)=0 [Dsin](0)=1
D^2 expi + expi = 0 with init.cond. expi(0)=1 [Dexpi](0)=i
3. as polynomials in a Taylor expansion
Does anyone know of any other definitions of harmonic functions on any
other mathematical structures?
Many thanks in advance,
Dan Greenhoe
The term "harmonic function" means something else in mathematics.
But as I understand you want to know various ways to define
sine, cosine, exp.
You have given three of them.
A fourth is in terms of integrals...this can be found in some calculus
textbooks...It works like this:
* define log z = integral(1 to z) dx/x, then exp is the inverse of it.
[also works, with care, for complex z]
* define arctan z = integral(0 to z) dx/(1+x^2), then tan is the
inverse of it. sin and cos then defined by identities from tan.
[Or I suppose you could directly use integral for arcsin.]
--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
.
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