Symmetry principle
Schwarz' symmetry principle, Riemann–Schwarz symmetry principle, for analytic functions
Let a domain in the extended complex plane
be bounded by a closed Jordan curve
, part of which is an arc
of a circle
in
. Further, let
be a function defined and continuous on
, analytic in
, and on
take values belonging to some circle
in
. Then
can be extended across the arc
into the domain
that is symmetric with
relative to
, to a function analytic in
. Such an extension (across
) is unique and is defined by the following property of the original function
: If
and
are symmetric (inverse) relative to
, then
and
are symmetric relative to
. In particular, if
and
coincide with the real axis in
, then
for
. By circles in the extended complex plane one understands both proper circles and lines. Continuity also can be taken as usual and in a generalized sense, that is,
is called continuous at
if
as
, independently of the finiteness or infiniteness of
. The curve
, as well as
, may pass through the point at
. From the conditions,
, but it is not necessary that
. In addition, if
and
have a common interior point, then the continued function need not be single-valued at these points.
The symmetry principle for harmonic functions for the same ,
,
,
is formulated as follows: If a function
is harmonic in
, continuous on
and equal to zero on
, then
can be extended across
into
to a function that is harmonic in
. Here, if
and
are symmetric relative to
, then
.
The generalization of the symmetry principle to the case of an analytic arc (and
) is the Schwarz principle of analytic continuation of analytic and harmonic functions (see [1], [2]). The generalization of the symmetry principle for harmonic functions to the case of a function of any number of variables is called the reflection principle. The symmetry principle is widely used in applications of the theory of analytic and harmonic functions (under conformal mappings of domains with one or more axes of symmetry, in the theories of elasticity, hydromechanics, electrostatics, etc.).
References
[1] | H.A. Schwarz, "Gesamm. math. Abhandl." , 2 , Springer (1890) |
[2] | I.I. [I.I. Privalov] Priwalow, "Einführung in die Funktionentheorie" , 1–3 , Teubner (1958–1959) (Translated from Russian) |
[3] | M.A. Lavrent'ev, B.V. Shabat, "Methoden der komplexen Funktionentheorie" , Deutsch. Verlag Wissenschaft. (1967) (Translated from Russian) |
Comments
The symmetry principle has interesting generalizations in the theory of holomorphic functions and mappings of several complex variables.
Examples are the edge-of-the-wedge theorem (see Bogolyubov theorem) and reflection principles for holomorphic mappings, which lead in many cases to smooth extendibility of such mappings to the boundary of the domains involved. See also Biholomorphic mapping.
References
[a1] | R.M. Range, "Holomorphic functions and integral representation in several complex variables" , Springer (1986) |
[a2] | C. Carathéodory, "Theory of functions" , 2 , Chelsea, reprint (1981) |
[a3] | E. Hille, "Analytic function theory" , 2 , Ginn (1959) |
Symmetry principle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Symmetry_principle&oldid=14999