Internat. J. Math. & Math. Sci.
Vol. 8 No. 2 (1985) 247-256
247
FUNCTION THEORY FOR A BELTRAMI ALGEBRA
B. A. CASE
Department of Mathematics
The Florida State University
32306
Tallahassee, Florida
(Received July 16, 1984)
ABSTRACT.
Complex functions are investigated which are solutions of an elliptic
system of partial differential equations associated with a real parameter function.
associated with a particualr parameter function
The functions
form a Beltrami algebra denoted by the pair
developed in this algebra.
a
(D,g)
algebra.
g _--
For
(D,g)
g
on a domain
D
and a function theory is
A strong conformality property holds for all functions in
]z]
r
(D,r)
the algebra
is that of the analytic
func ions.
KEY WORDS and PHRASES: Elliptic system of partial differential equations,
Beltrami algebra, dilation.
1980 AMS SUBJECT CLASSIFICATION CODES: 30G30; 30G20; 30C60.
INTRODUCTION.
1.
The idea of defining classes of functions which are generalizations of the
analytic functions by means of systems of first order partial differential equations
goes back at least to a paper of Picard in 1891 [I].
The solutions of that system
were of the type later called "pseudo-analytic functions of the second kind" by Bets
[2] where for
u+iv
f
u
x
-Uy
where
a, b, c
and
d
a vx + b
V
a
b vx
y
(1.1)
Vy
are appropriate real-valued functions.
The relationship of
solutions of the elliptic system
u
x
-Uy
where
a, b, c
and
d
a v
x
+ b
c vx +
are real valued and
Vy
(1.2)
aVy
bc-a 2
to those of
system (l.l) is
well developed in papers by Caldwell, [3, 4]
Two current research areas related to the early investigations of elliptic
us[conforma| mappings nd function theoretic methods in
vste.s are the theory of
partial differential equations.
Works from the latter point of view which have
B.A. CASE
248
extensive references are Vekua, [5] and
Gilbert and Buchanan, [6].
Ahlfors
discusses, in his introduction to [7], analytic theory generalizations and rationale
concerning quasiconformal mappings in particular.
Examples from the algebras of
functions which we consider are found in very different areas of investigation,
indicating the possibility of an interesting and useful function
(e.g.,
theory.
Concerning quasiconformal mappings see [], p. 71; topological analysis, [9] and
[10], p.
lqg-
elliptic systems of partial differential equations,
[5],
The ftnctions considered are solutions of a Beltrami differential
complex dilar.tion at each point
f’
generalized derivative
existence of
f’
p.
131.
eot, ation
dependent on a real parameter function
z
with
A
g
Necessary and .ufficient conditions for
is defined
include the Beltrami efluation which reduces to the Cauchv-Riemann
Both re f and im f satisfy the same second order
euations for analytic
partial differentia equation with analytic atecedent the Laplace efl,ation.
Although the available enerlization of the Laplace eo,ation and its applications
are of intresv
In the system
i nv elliptic system,
o
the parallel
(n,g)
For functions in aebr
a
ime
z
and two fixed arcs the angle between
arcs is the same for all functions in the algebra.
gf
integral
line integral.
The eneralized line
shares in modified form much of the theory of the ordinary complex
Amon
applications in the last section is a function continuous
some
on the finite complex plane,
,
(D,g)
in a
bounded sets, and which maps
-{(0,0)}
algebra on
g-quas icon forma
onto a fixed open disk.
NOTATION AND DEFINITIONS.
2.
Complex valued f,nctions of one complex variable
f
im f.)
re
generalized conforma]itv holds which is
invarint in the algebra, that is, for given
the
(e.g.,
is not always so direct.
[1] a different generalized system holds for each of
and
re
[m
If[
v
denoted bv subscriuts.
O
q
ar f
and
are denoted
Standard definitions used are"
J
Ifzl 2
,
The complex plane is denoted
later in the paper,
+iy
Partial derivatives are
iO
fz e- fr (i/r)fo)
f eiO(fr + (i/r)fo)
(and,
re iO
z
IfEI 2
and a domain
D
.
(9_.I)
is an open connected set
is required to be simply connected).
The classes
C
n [C
comprise the functions with continuous partial
derivatives of n Jail] orders.
Usage is enerally consistent with [I] and [lq] as concerns classical analysis;
[8], [7] and []4] as concerns ouasiconformal functions- and [15] as concerns
functional analysis.
The derivative
f’
defined here is the limit
quotient involving the parameter function
o
a generalized difference
which reduces to the difference
quotient for analytic functions.
For
C’
in a domain, existence of the limit
of this Reneral[zed d[fference ouotient is shown in [16] to be necessary and
sufficient for our defin[n condition (2.2 below).
DEFINITION I.
A parameter function
valued positive function of
is continuous on
D
zl
r
for
g
in the following discussion is a real
z e D
such that
g’
dg/dr
exists and
on
FUNCTION THEORY FOR A BELTRAMI ALGEBRA
DEFINTTrON ?.
(P,)
The ordered pair
C2(D)
h,meomorphisms
reDr,’sents
249
the set of complex valued
satisfying,
(2.2)
fz
e
(l+(g/r))
For such functions the eneralized derivative is defined
f e-
(2.)
r
A function is called g-analytic at a point z when there is a parameter function
(2.2) holds on some domain containinR z
f’ is called the -derivative.
so
(2.27
The Beltrami eauation
condition.
a, b, c
c
is equivalent to a
Reneralized Cauchy-Riemann
in rectangular coordinates in the form
2 2
)/mr 3
x r +y
xy( -r 2 )/mr 3 b
When stated
(y2 r2+x 2 2 )/r3
(1.27
the functions
22
2
a
are"
g
For computational convenience we state that condition as a
system in polar form
(ira
(re
-
f)0
f)0
or
f0
I,et
rEF1VlTON 3.
y
dfferenttable arc
i
(re
f)r
(ira
f)r
(7.4)
fr
be complex valued and continuous on (pieceise)
f
on which
# 0
r
A eneralized lne integral deoendent on
[s defined by"
oarameter function
(03/g)(ucosO-vsin0)dr- (vosO+usin0)dO
Y
Y
(ucosO-vsinO)dO
exJdr/R
,=
where
(2.5)
if(oo/t,.)(vcosO+usinO)dr +
The roperties of the lemma follow from the definitions
n
a
straiRht-
forward manner.
LEMHA 1.
in
When
h)
The constant
f
constant
f
f.
If
e)
If
f # O)
f
f’-=O
2
(D,)
if either
d)
fr 12
(/r)
J
and
f’
---0
c
or
then the sum,
f
If
h
(D,)
such that
is
Also,
is identicallv constant
(O,)
cf
uroduct and reciorocals are in
and the eneralized derivatives
(1/f)g
)
ar f
is a comolex number, then
(fh)’
f)
is constant.
then
el
im f
re
(D,.) and
h
(D,z)
and are the only functions
for a constant function
f+h)
function on
(D,tz)
functions are in each set
(D’I)n(D’2) l
c)
for
(D,)
a)
(D,)
and
(1/f
are"
f’ + h’
(?.6)
f’h + h’f
-f/(f)P
f’ E h’
on
D
then
f-h
is a constant
D
If nonconstant
(D,)
h e
(D,r)
then
hof (D,)
B.A. CASE
250
h)
(D,)
.
For
(2.5)
(The, multiplicative unit
is an algebra over
(D,)
E r
is the set of f,nct
ons
are, respectively, the usual derivative and line
Definition
that the function
sense oreservine.
,:]a
Oth,,r condirion
"nnlicntionq.
g.
or
The reouirement
larer ouasconformal
reversin functions.
n the definitions mav be weakened in some
e
[171, [1 for
See
and
be non-negative and hence that the functions be
Thi is c,nsistent wth practice in the
s rh,,r, mnV f these reultq hold for sense
nlthoh,
enref]
prop,,rtes of Belrnmi
tasic,nf,,rml
integral.
(2.3)
and
be posit[w, is equivalent to reeu[rin that the
(P,)
Jacobians for functions
is f
D
analyt ic on
discussion of topological and
svqtem.
kAPP ( pRooRrq
Functi,)ns ,,hich
TI, dilatation quotient
,n
bounde’l on
A
point
is bounded on every
is
.
preservation
(2.P)
of the ge]trami eQi,tion
regular
compact
D
subset of
g-nuasiconforma!
are ]ocallv
and, if it is
mappin for
> un ax{Ir, r/}
zcD
eneraliation ,)f c,,nF,;rmaIitv holds for ,-analvtic f,nctions.
nd the parameter f,nction
is the
arcs
then
stronq
z
qolttions
are
sam,,
the
(D,
whichever
in the conformal case.
available for the lar.er
claqs
maps
Oiven a
anle between the im,.es of two fixed
the arcs.
This reduces to anle
It is more soecific than the inequality
of o,asiconform.l mal)pin.s in eneral.
Theorem 4, which is an enuivalence to nuasi-conforma] itv.)
THF, ORFM
1.
Arcs
Y1
and
make anles respectively of
the radial direction
z
(
throuRh
2’
y?
intersect at
and
[
Tf
z
O
(7
and
is -analvtic in a domain
T
with
containinl
f(y1) and f(Y2) is determined by"
?, s in O c o scc o sB
/cos
=.
((e/rsin(B-O)[fcos20+(/r
+ (sinPO+(/r)cos ? O)sinccos6
cosOsinO(1-(,/r)2)(sin(0t+3)] -I
then the nle
’
between
((,lr(sin(T_o)l(cosccosx+(/r)P
(lonverselv,
where
.
if
The
"
C
qetisfies
PROOF.
at
0
and
Definition 1,
q,.cond
f(z)
the point
(3.2)
(7=
c+II-O
sin(Tsinx)
holds on some domain containinlz
is ,-anlvtic at
then
(’.9)
of the forms of
first then Follows, since
f(yl
where they hnue taneonts which
z
with the rel axis and of
T=
Let
sino’
rcSOr
p(dqb/dp)
+
sin(TqbO
rcosp r + sin(p 0
nOVO )u- (rcos(Tu r
sin(TVo)V- (rcos(TUr
(rcSJVr
+ si
+
(rcosclv
+
+
r
(7’
with the radial direction throu.b
aleebraicallv ith allowable anRles, we hve
coso’
z
is most convenient to derive.
f+II-O.
sinOuo)V
sinOuO)u
z,
The
be the anzle made
f(z).
ManiD,latin.
251
FUNCTION THEORY FOR A BELTRAMI ALGEBRA
and, using similar expressions for
T’
T
write
sin(T’-O’)
cds(’r ’-’)
s int’
cos’
(rc oSOVr+Snovo)(rcos Ur/S inxu
rcoSOUr+Si nou@ rcosrVr+S inky 0
(rcoSOVr+SinOVo)(rcosl;Vr+Sina;v 0)
im(]r f0) (sin(x
r
0
(rcosOur+SinOu 0)(rcosa;ur+sinxu 0)
+
(3.a)
cY)
(r? fr
.qince
(D,)
f e
by
hvlothesis,
if
Conversely,
(?.4)
substitution usinR
(’.2)
Rives
is a function for which the expressions
(3.2)
(3.3
are
eouivalent for all arcs, then
r 2 (e
2
fr
m
(r f0) )+r (.(r e ( r f0 )) (t a nO+
a n))+((R
Ire]2- ? im(fr-
0
and hence we h.ve the system
1%12
i( o)
re(frf @)
which is e0uivalent to the comple form of
DEFINITION 4.
is
(2.4)
since
J
>
0
is sense
is -analytic.
preservinR, hence
then
(.4)
0
If
(D,g)
f
R-conformal
satisfies
f
and
(3.2)
locally for all
D
z
D
on
(D,R)
For all functions in a iven
alzebra, the -conformalitv Rives the
between the images of intersectin arcs de0endinz on the tanRents to the arcs
themselves and the point at which they intersect. In the algebra of analytic
(D,r)
functions
but, reducin
the relation is not only also independent of the intersection point
to anle
with
The similarity to the analytic case is less obvious in
the case of the effect of the g-analytic function on arc length.
are elements of arc in the
z
dS 2
The modulus of the
(y1,y2)
preservation, is the same for all airs of arcs
the same anle between them.
and
f(z)
Where
and
ds
dS
planes, respectively,
Ifgl2(dr2+
g2d021
(3.a)
eneralized derivative retains position as
a mgnif[cation
factor,
although not in the Euclidean metric.
4.
PROPERTIES OF THE FUNCTION THEORY
In adaition to some theory of the zeneralized derivative
(.)
and
we
Rive
im f
for
a
(2.3)
and intezral
meneralized harmonicity relation which is satisfied bv both
re
(D,R)
in a
examples which are the most
alRebra. It is helpful to first construct two
Reneral functions in a (D,R) alRebra under certain
cond it ions.
From now on, we assume a domain
EAMOLE 1.
If at
z
f(z)
where
C2(D)
c
then
c
2
is a simply connected.
)(r) exp(
are real constants,
is
D
D
R-analvt ic
at z
c]
and
i(clO *c?)
0
(4.])
and nonconstant real valued
R
Cl)/),
%(r)
On the other hand, when
B.A. CASE
252
(4.1)
is a -analytic function of form
exn(ClJdr/z
fl(r)
c!
when
and a branch of
(D,g)
h
st-tement
0
(irn
qince
O
indenendent of
and
(re
0
i(Cl0+ c2))
+
f?)0
then there exist real constants
13
+
i(cl0+c 2)
b _= if
where
2
is independent of r and
f:?O
o that there exists real
fl
f’learly
f?
ho
conositio, of functions
such that
c
form ,f
(.1)
require the
’
the
I-th
2
and
2.
Let
positive function
is
nd
t,l-nalvtic
(4.17
c1%/%’
el
for
(n, 1)
(D,>,)
=6
and
D
z
f’e
induction that
(See Theorem 3b.)
f’
is .-analvt ic at
z
for real
only if at least one of the followinR conditions holds"
k such that X(r)
r+k
for this case (
r+k
There exists real ,,alued differentiable
h(0)
(ira
such that
f)r
f)r
(re
Fither
f)o
Let
PROOF.
then
f)r
re
is zero.
f
cosOu r +sin0v r
hold for some real
nO
0
sin0
(re
or
u+ iv
(2.)
Rec!uir[nR that
’r
-cos
X(r}
r
’r
c)
osOv r -sin0u
f’
r
(a.4)
o
rri
(See xample 2, (t.a).)
implies the relation of
im
gives the matrix eouation
If the third matrix is the zero matrix then we have
0
Now, in eneral, if we
we note that
h[her order derivatives.
then
h)
we have
By I,emma 1 on
It follows bv mathematical
There is a real number
c)
,/g’
-1
in
to be
a)
h(O)(:-e
and
f2
di_scusq the existence of
e
(4.9)
is the exponent ial function.
h
derivative exists under atpropriare conditions.
Theory,ms
f?O
The
is
(f2).
function of form
TqEORE.M
f?)r
(ira
functionq of
fl
nd hence,
exp(i((cl-1)O+c2)
(4.a)
aq
(re
is .-analvtic if and only if
is e-analytic. Also,
(c /.)e -i0
is given by
which is
function of the
f2
the -derivtive of
where
is
f?
then
,e consider two relationshits between the example
(4.’)
(a.)
such that
clJdr/
f2(z)
(re h)r
and
,)llows
(D,)
ar. z
f2
If
2.
EXbgPLE
,c 2
then
a)
When
u
r
or
Otherwise, the renuirement that
is
zero
UrVrr-VrUrr_
r
b)
THEOREN q.
a)
If
f
(O,R)
for
(r)
orders of higher derivatives exist
r+k
h)
cl/),’
If
f
theo the
(1,1
nth
r+k
in
then
(D,r+k)
fr+k
e
(D,R)
3rn
)exp(i(clO+C 2
where
derivative exists for
max{d
max{d
inte.ers
f
e-in0 3nf
is of the form
rain{c]
If
and are defined by
integers
(d)
exists)
exists}}
c
<
0
c
> n
gl
C(D)
all
FUNCTION THEORY FOR A BELTRAMI ALGEBRA
In particular, where
(Cl-n+l)X (n-!)/x(n)
Un
((f,), ...),
(When
<" 0
c
c)
%
nd
C
(D,X)
E
exist.)
hiher derivatives
all orders of
f nonconstnt
then
exp(i((cl-n)O+c2))
n
el 2
253
(re
th
first derivative exists nH reduces to case
f)o
0
(re
or
f)r
0
(See the dcusson
b)
then the
follon
aple 2.)
a)
PROOF.
If
s,cod order partia]
n
f,)
h)
and
can be proved
(D,p)
is in an algebra
usin mathematical induct ion.
then both
re f
im f
and
satisfy the
am,,
differential eoation which is the Laplace eouation for analytic
a simply connected domain construction of
coniuate -ana]vtic functions
ows.
THnRM
(D
h
g
rr
p-h,rmonic.
Real
col led
tes.
-c)n
FORE 5.
)ld
r@
’
+
+ C11)
r
00
v(r,O)
and
n
iS -harmonic in
Tf
ech sat sfv
n
(.5)
saeisfvin
u(r,0)
functions
im
which satisfy
th,,n
is called
(2.&)
are
has at least one
conuate.
PROOF.
Without loss of enera]ity, let
v(r,O)
wh,r(
z
is a fixed point
O
r(r,@
!r -ue(
zo
satisfy
r,O o )4rl +
(A.5)
-
and conider
u (r O)dO
0
roeXP(iOo)
It is immediate that
vo(r,O
and we also compute
Vr (r
e)
-ue(r 0o1/
which ater ubstitutio of
(XUr r (r ,8)
+
(4.5)
reduces to
v
r
X’Ur(r
O))dO
-u0/
(2.4)
satisfying
for
u+iv
of the g=itegral as an Riemann path integral
Examination of Definition
the function
ivotvin
g
shows that the usual operational properties regarding
multiples and path components continue to hold.
the generalized line integral;
and path independence,
Weierstrass type.
THEOREM 6.
D then
g[f
PROOF.
We derive a few useful properties of
in particular these are analogues of Cauchy’s Theorem
Morera’s Theorem,
and some convergence theorems of the
Domains are simply connected.
If f
(D,) and T is a simple closed curve homologous to
0
0
sing the polar for of
reens
Theorem, re
f
and
i
f
ech
enerlized Cauchy Rienn cndition (2.)
The stronger for of Cuchys Theorem, lloing exceptional points in the doin,
reduce to ero after substitution of the
[16], p. 62-66.
f
hen
(,g)
is deeloped in
gb
eOROLLR7 1.
a
and
b
PROOF.
in
then
f
is independent of the pth jonin
D
If to paths fov
a
to
b
[n D
do not cross or cros a finite number
of t[es, the result follows by Theore 6 and the definitions and operational
properties of the path intevals. If the paths cross infinitely often a tiit[nR
process [s available. (See [13], p. 315).
B.A. CASE
254
Let
THEOREM 7.
Zf
W
(D,g)
F
then
(D,g)
f
zo
and
D
be fixed in
zo
and define
and further
((exPrlrdrlg)Ig)
Fg’
(4.6)
f
I
Since the generalized integral is path independent we apply properties
PROOF.
of line integrals in the plane to compute the partial derivatives of the real and the
(2.4)
System
imaginary parts of the integral function F
(2.3) as
is satisfied, so the
derivative function is given by
e-J0(
which is euivalent to (4.6
o
We thus demonstrate for the generalized case the d[fferentiability of the
Fg
f)r
gff
0
Let
f
D
in
C
and
for every simple closed curve
(2.5)
In
PROOF.
(4.6)
is seen by
as necessary (as well as
F
sufflcient) that
TtlEOREM 8.
--r
g(r)
The analytic case
integral.
I.
satisfy Definition
g
in
Y
D
is g-analytic in
f
then
If
O
both the real and imaginary parts of the generalized integral
[..(P dr + 0 dO) where P and
"" we have independence of
hypothesis
are represented by real line integrals
Q
are both in
C
Since by the
Or
(l/g)(u0cos0
(I/)(vcos0
these integrals,
P0
path for
or"
+
v0sin0)
u0sin)
-VrCOS0
UrSin0
VrSin
urcos
(2.4) for
Al.ebraic manipulations of this system show
u+iv
f
Before showing some Weirstrass-type convergence results, we need a bounding
relation o the generalize integral which depends on inequalities for real line
integrals.
LEMMA 2.
When
y
is rectifiable
Ig/fl
.r
where
Kg
4
THEOREM
the sequence
PROOF.
(4.7)
dr/)(l+maxll/gl)), M maxlf
max((exD_1
zcy
zy
zcy
i
9. If fn is continuous on regular arc y
{fn]
converges uniformly to
Since
f
THEOREM I0.
If for all
For
z
A
f
y.
length of
n
such that for
n
and
then
<-- e ’KgL <
>
N
e
T
T
converges uniformly to
N
e
for each
y
on
(fn- f)]
f
{
by Lemma 2.
f
is continuous there is an
fn
PROOF
<KML
n
fn
C2
is -analytic on open set
on compact subsets of
A
there is a disk containing
then
z
f
A
and
{fn
is -analytic on
with its closure in
A.
A
We
FUNCTION THEORY FOR A BELTRAMI ALGEBRA
(D,g)
f
show
smple
A
closed curve in a disk contained in
then
that for
n
>
n
by
to
u
and
{v n}
fn
Un
converges uniformly to
n
where
v
lim(f )’
f’
lim(u
We now have
PROOF.
Let Y be any
for each
0
hence Theorem 8 gives R-analyticity in the disk.
0
converes uniformly
u+iv
A
z
If to the hypothesis of Theorem I0 is added, for
COROLLARY 2.
{u n}
glfn
Then
)
g[f-
By Theorem 9,
Theorem 6.
that
is g-analytic for
on such disks, hence
255
n-+
R
)r
lim(v )r
R
n
and
v
hence there
r
s
an
N
such
N
vers[o of the argument principle nd
available from the theory of Betram[’s
o Cauchys ntel
ormula
or completeness
euaton.
e sho the
atte fo -analtc functions.
THEORH 11.
D
i
and
[5],
See
.
f
is
z
n
(D,)
f
s
nd y
a simple cosed curve
the nte[o of
y
[
(g-r)
f(z)dz + 1
homoloou
to
0
then
et2Of()drd
2-25 nd p. 1 for discussion of ore enerl cses.
PPLIeTIONS
5.
One interesting exapIe of a g-analytic function ith
locally quasiconforl ith contnuou extension at the lated boundary
and -quasiconoraI on bounded ets, ps the finite pIane onto
(0,0)
fixed open
disc.
EPLE 3.
For fixed
f(z)
(20ei@tan-lr)/
(2.2)
for
0
<
define
0
z=O.
0
The function satisfies
z
<
0
real and such that
0
(l+r2)tan-lr
g
and mps the finite plane onto
[f(z){ < 0
Concentric annuli may be g-conformally mapped whatever the proportionm of their
the open disc
radii.
EXAMPLE 4.
For annuli:
A
0
A2=0
<
< I1 < 2
r
<1"1 <R2<
<R
let
f(z)
Then
g
f:A
r +
-->
A2
e
iO
(R2(r-rl)-R1 (r-r))/(r2-r
2
homeomorphically and
f
is g-analytic for
((Rlr2-R2rl)/(R2-R1))
The convergence properties of the g-integral allow a comparison to the general
quasiconformal situation. When the generalized functions satisfying the Beltrami
equation (2.2) are quasiconformal in a domain (see Section 3), they are of the type
called regular quasiconformal. Although the uniform convergence of a sequence of K-
256
B.A. CASE
quasiconformal mappings to a homeomorphism implies that the limit mapping is also Kquasiconformal, in the case of a sequence of regular K-quasiconformal mappings the
limit mapping need not be regular. (See [8], p. 17-19.) For g-analytic regular K-
quasiconformal functions, the limit mapping is g-analytic.
COROLLARY 3.
A Beltrami algebra
(D,g)
is uniformly closed.
REFERENCES
I.
PICARD, E.
Acad.
Sur un systeme d’quations aux drives partieIIes, C.R.
Sci. Paris 112 (1891) 685-689.
2.
BERS, L.
3.
CALDWELL, W.V. Some relationships between Bers and Beltrami systems and
linear elliptic systems of partial differential equations, Can. J. Math.
|7 (1965) 627-642.
4.
CALDWELL, W.V.
5.
VEKUA, I.N. Generalized Analytic Functions, Addison Wesley, Reading, 1962.
GILBERT, R.P., and BUCHANAN, J.L. First Order Elliptic Systems,
Academic Press, New York, 1983.
AHLFORS, L.V. Lectures on Quasiconformal Mappings, D. Van Nostrand, Pri,ceton,
6.
7.
Theory of Pseudo-analytic Functions, New York University
Lecture Notes, 1963.
The dilatation of harmonic quasi-conformal functions,
Proc. Amer. Math. Soc. 40 (1973) 205-208.
|966.
8.
LEHTO, O. and VIRTANEN, K.I. Quasiconformal Mappings in the Plane, Second
Edition, Springer-Verlag, New York, 1973.
9.
EGGLESTON, H.G. and URSELL, H.D. On the lightness and strong
interiority of analytic functions, J. London Math Soc. 27 (1952) 260271.
I0.
12.
WHYBURN, G.T. Analytic Topology, Amer. Math. Soc. Colloquium Publ.
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BERS, L. and GELBART, A. On generalized Laplace transformations, Annals
of Math. 48 (1947) 342-357.
AHLFORS, L.V. Complex Analysis, Second Edition, McGraw-Hill Book Company, New
13.
KAPLAN, W.
II.
York, 1966.
Advanced Calculus, Second Edition, Addison-Wesley
Reading, 1977.
Press, New York,
14.
AWRYNOWlC J.
15.
RUDIN, W. Functional Analysis, McGraw-Hill Book Company, New York, 1973.
CASE, B.A. On non-analytic functions related to a system of partial
16.
17.
18.
and KRY, J. Quasiconformal Mappings in the Plane:
Parametric Methods, Springer-Verlag, New York, 1983.
differential equations, University Microfilms # 70-9333 (Ph.D. thesis).
CALDWELL, W.V. An algebra of light interior functions, Duke Math J. 3
(1967) 225-230.
AGARD, S.B. and GEHRING, F.W. Angles and quasiconformal mappings, Proc. London
Math. Soc. 3 14A (1965) 1-21.