f 1917 I
ON A CERTAIN GENERAL CLASS OF FUNCTIONAL EQUATIONS
BY
WILLIAM HAROLD WILSON
A.
B.
Albion College, 1913.
A. M. University of
Illinois,
1914.
THESIS
Submitted
in Partial Fulfillment of the
Requirements
Degree of
DOCTOR OF PHILOSOPHY
IN
MATHEMATICS
IN
THE GRADUATE SCHOOL
OF THE
UNIVERSITY OF ILLINOIS
1917
for the
UNIVERSITY OF ILLINOIS
THE GRADUATE SCHOOL
JUZ
I
il
IQI 7
HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER MY SUPERWilliam Harold Wilson
VISION BY
ENTITLED
_
Qn a Cftrtain fieneral Class ef Functional Equations
BE ACCEPTED AS FULFILLING THIS PART OF THE REQUIREMENTS FOR THE
DEGREE
OF.
_
Doctor of Philosophy
.......
i^^l^yi^r
^ ^^'
/
4f
In Charge of Thesis
Head
Recommendation concurred
of Department
in :*
Committee
on
Final Examination*
^Required for doctor's degree but not for master's.
376690
UJUC
Table
§
I-
S
II.
§
III.
Introduction and General Considerations
Reduction to
IV.
V.
§
VI.
§
VII.
§
VIII.
§
IX.
X.
§
XI.
6
a
Network
S
Determination of the Normal Solution at
•
•
•
•
•
.
..
a
Dense
..
12
Solutions of the Normal Equation
15
Solutions of the Original Equation
18
The Converse Theorem
•
Discontinuous Solutions
23
£5
Certain Tyves of Equations having Variable
Coefficients
§
1
Determination of the Normal Solution at the
Set of Points
§
Page
Normal form
a
Vertices of
§
Contents.
of
•
•
•
•
•
.........
Avolicat ion to Binomial Equations
•••••••••
Equations involving more than one Function
27
3C
32
.
ON A CERTAIN GENERAL CLASS OF FUNCTIONAL EQUATIONS.
§
Introduction and General Considerations.
I.
Addition formulae of the general type
ilfix
where 3 is
+
o,
oolynomial in its three arguments, play
a
the theory of elliptic functions.
in
=
u) t fix) , r{ii)]
a
prominent role
A
natural generalization of such
•
•
formulae is
rtx,9,r(x),r(y),r(* ± x
+
-;fi* n x
+
p a »)]
= o
(i)
where
denotes
a
polynomial in its
argument involving
f
is
(i
P
)
+
n
4
arguments such that every
explicitly present;-
(ii) x and y are independent variables;-
(iii)
(iv)
oc
i
9
3^
nd
i
=
2,
1,
•
•
n,
*,
are given constants;-* and,
fix) is an unknown single-valued function to be determined
equation (T) shall be identically satisfied.^
so that
For the purposes of this paper
is
it
convenient to carry certain
a
statement of these hypoth-
hypotheses in regard to the
3
and 3» 3
eses is to be found below.
^ A theorem of some interest in the general theory of these functional equations is that every solution fix) of equation (I) is a
solution of a similar equation in which x and y occur only in the arguments Of the function f.
To prove this arrange ? as a polynomial in
X and U
The substitutions X = S +
^t , y - t, where ko = C and
0C '
,
fe
.
d
1
3
fe
A
~
<x
3.
3
and
+
dl
1
v
kA
+
L£ _ _i
a,.
h
h
a
o
transform (I) into equa
tions similar to (i) such that the highest degree in S and t is the
A
=
O,
1,
*
*
*',
i
-
1,
h,j
=
1,
2,
*
*
',
n,
Digitized by the Internet Archive
in
2013
http://archive.org/details/oncertaingeneralOOwils
5.
Equation il)
function
said to be of order
n.
The decree
said to be the decree of equation
is
f
is
m
of P in the
(I).
Cauchy* discussed two special cases of (i) and two related equanamely,
tions,
fix
+
w)
* fix)
f(xu) = fix)
>(»),
+
fix
+
y) = fix)fiy),
fixy) = fix)fiu).
fiu),
One or the other of the first two of these eauations has since been
treated*
Darboux,
by
E.
B.
Wilson,
Carroichael** has 5iven
and Harael.
a
Vallee Poussin,
Schimmack
generalization of the Cauchy
eauations while Jensen** has discussed several applications of them.
same for all of them.
It is easily seen that a finite number of nons
may be employed such that the variables S and t may be elimzero
inated from these equations, insofar as they occur as coefficients, b;,
Sylvester's dial y tic method of elimination.
The result of this elimination is an equation (il) which states that a polynomial
in the
function f has the value zero.
The arguments of f are linear combinations of two independent variables S and £.
Hence (il) is similar to
(i) although in general its order and degree will differ from those of
Furthermore, if no two arguments of f in (i) are proportional
(i).
then no two arguments of f in (il) are proportional.
Cauchy tr eated the last twcl
COUrS d' Analyse, (1821), Chapter V.
equations by transforming them into the first two equations.
It is
obvious that similar transformations may be applied to reduce more
general equations to the form of those considered in this paper.
*Darboux,
M at
h e m at
s
h e
A n n a 1 e n, vol. 17,
i
c
S. B. Wilson,
(1880), p. 56.
of Mat
ser. 2, vol. i, (1899), p. 47.
Vaiiee Poussin, Oours d' Analyse infinitesimal e , (1903), p. 30.
Schimmack,
A c t a, vol. 90, p.
N o v a
Hamel,
vol.60, ( 1 90 5 , p 4 5 9
'
Annals
hematic
Mathematische Annalen,
American Mathematical Monthly,
(1911
198.
**T idsskrift
for Mathematik, vol.
)
18,
)
,
p.
vol.
p.
2,
(1878),
.
149.
Ser. 4,
1
I
Cauchy* has treated the equation
?(x
+
*
u)
-
<P(*
2f(x)f(v).
)
jf
Carraichael* has considered the equations
h(x
y)h(x - y)
+
ih(x)]
y)t{x - y)
+
f(*
=
=
2
[h(y)]
4
[6(x)]
2
2
-
c
2
,
- Eg(»)3 8 -
Van Vleck and H'Doubler** have discussed the equation
V(x
+
y)y(x - y)
=
h'(x)v(.v
)]
2
.
Other related equations have been considered by several writers and
systems of functional equations have also been treated.
It
seems that no systematic account of
tions of the form
has ever been undertaken.
(!)
signed to contribute to such an account.
the principal part
general theory for equa-|
a
(§§
II
This paper is de-
The equations considered in
to VIII) of the paper are linear homogeneous
equations with constant coefficients.
They may be written in the forn
n
J^i f(a
It will be
are
i
x +
zero and no ratio
a /Q
±
±
+
Y n+1 f(*)
shown that if some a's and
all the remaining ratios,
al
+
£
'
s
Yn
+ 2
f
M
=
0.
(1)
having different subscripts
of non-zero a's and S»s is distinct from
the equation is exceptional.
case receives mention only in §§ VI and XI.
generality in assuming that no
is
oc
The exception-
There is no loss of
zero in the non-exceptional case.
For ^convenience in exDosition the hyoothesis will be carried in the
*Gours d'Analyse, (1881), P
114.
.
American Mathematical Monthly, vol. 16,
(1909
180.
Transactions of the American Mathef}matical Society, vol. 17, (1916),
$
) ,
p
.
p.
9.
text that in addition to no
os otj/yj are equal.
a
being zero no
zero and no two rati
is
3
The additional argumentation for the remaining
non-exceptional cases is supplied in footnotes.
normal equation of order
A
n
is derived
(in
II) which is satis-
§
fied by every solution of any non-exceptional equation
This normal equation forms
a
(1)
of order
n.
foundation uoon which the entire develop-
ment of the theory of equation (1) is based.
Any normal solution may
be uniquely determined at each point of a dense set covering the com-
plex plane if it is given at the vertices of
work (§§ III,
lytic
in the
IV).
is
It
shown in
§
mal solution continuous
u
and
is
in
n.
It
is
is
an
also shown that the nor-
the neighborhood of the point zero of the
an arbitrary polynomial
are real and x = u
v
V that the normal solution ana-
neighborhood of the point zero of the comvlex plane
arbitrary polynomial in x of degree
complex plane
certain triangular net-
a
+
y/-~T.
in u
and
v
of degree n where
The normal solution analytic
along any line in the finite complex plane is also an arbitrary polynomial in x of degree
n
and the normal solution continuous along any
line in the finite complex plane is an arbitrary polynomial in
degree
n
of
if the line is not parallel to the axis of imaginaries and an
arbitrary polynomial in
the axis of reals.
u
of degree n if the line is not parallel to
The analytic and continuous solutions of
found (§ VI) from the normal solutions.
§
u
(1)
are
Examples are exhibited in
VI which show that equations of type (1) may have non-trivial contin
uous solutions but no non-trivial analytic solutions while other exam-
ples show that equations of type
(1)
may have analytic solutions which
are also the most general continuous solutions.
briefly considered in
§
VII.
It
is shown
(in
§
A
converse theorem is
VIII) that if
a
func-
.
,
tion fix) satisfying an equation of type
nuity
the finite complex plane
in
plex Plane]
has a point
it
of the plane
has
(1)
[or on any
point of disconti-
a
line
the finite com-
in
of discont inuity in every region
[interval]
however small.
[line],
Equation (1) is employed
IX)
(§
solve certain equations of the
to
type
n
i
y)f{a i x
^ i ix f
ll
where the
(§
X)
<p's
to find
3
+
lV
)
^^ 1 {x
+
t
y)f{x)
are known functions.
*
V n . 2 (x r9 Jf(g)
Equation
(l)
*
<P
nt8 (x, V
)
=
also employed
is
analytic solutions and in some cases the continuous
all
solutions of binomial equations of the type
[f(<x
B
1=1
where no
a
*
3. v
+
Y
)]
i
i
±s
zero,
is
a
V
=
[
= k+ 1
f (a
1
+
x
8 ,y)]
1
Yi
[f(y)]
Y * +2
constant and the Y's are constants of which
the real parts are positive.
Pexider* used the first Cauchy equation
solve
to
fix)
In
+
<p(z/)
=
w(x
+
y)
XI the method of obtaining the solutions of
§
is used to
(1)
solve
the equation
J/iM
f
a
i*
+
M>
+
Y
n+1 r n+1
U)
+
Y n+2 f n + 2
of which the equation considered by Pexider is
proved that when no two arguments of
greater than
h
y
s
i
k,
)
=
special case.
f
is
a
vol.
14,
(1903),
fu
p.
r
Mathematik und
293.
It is
polynomial of degree not
n.
Monatshefte
P
V
in the foregoing equation are
f
prooortional each continuous solution
a
(
.
6
Reduction to
II-
§
Normal form.
a
The solution fix) of the general nth order equation
J/ifOV
y^ 8 r<*1
rt^rix)
-
=
o
(i)
is contained in that of an equation of the
same form and same order
[equation (6) below]
is
in which each a,
3,
y
a
given integer.
The
derivation of (6) from (1) is accomplished by elimination.
If
placing
i
the result is
n+1
Y a+2 [f(.v +
t
n+1
)
-
= 0.
(2)
subtracted from the equation derived from (2) by replacing
is
(3)
r'
+
+
x by x -
+
by y
y
from the equation derived from (1) by re-
u
•j
If
is subtracted
(l)
3
i
t
+
m+t [r(»
and
A
<Vi
?/
-
by
+
<x
- f(* *
t
+
t
?/
n+1
1
t
1
the result is
,
3
3
+
i-y
itn+1
- r(» +
)
+
)
r (« lJC + p iif )]
- r(» +
*
n+1
)
+
r(*)i
=
o,
o)
where
A..
J i
It
= a. 3.
3
-
a .8
i
i
.
J
is easily seen that this is true because the given
leave
%
r
In
1
x
+
3
±
y
unchanged but replace
+
P t»
general,
+
tJitt
oc.3
.x
ltl =
<x
+
3.y,
i
4 1,
by
iX k 3 iV + A litl .
equation (a) results after such eliminations,
if an
then each argument of
-
ot
substitutions
f
in
(a)
is
a
linear expression in
x,
y
and
certain t's, the subscripts of the
eliminated.
terras
The substitution of
jf-8£
forx
gives rise to an equation
<*
±
x
3
+
(b
and
+
v
*.£.forjy
which differs
)
(a) replaced by * x + P » + A
t
4
^
of
iV
corresDondi ng to those in the
fc's
±
by having each
frono
(a)
£ ^
Since
.
since (4) does not affect the t's that are found in
that the difference formed by subtracting
for which
terra
is eaual
i
from
(a)
the fixed integer
to
(4)
=
(a),
(b
it follows
contains no
)
Since
.
and
n
is finite
these eliminations may be continued until only terms having the coef-
ficients
+
remain.
Y n+2
terms for which
=
i
is obvious
It
Moreover, the order of elimination of the
2,
1,
•
•
Since
assumption that (1) is of order
dividing by
Y n
n + P
2
n,
,
is
immaterial.
that the equation resulting from these elimination
linear and homogeneous.
is
'
It
.
Y
n+2
is
not equal to
the equation may be simplified by
n,
is easily seen that there are
fej
distinct terras in which the coefficient of
the argument of f is obtained by omitting
y
+
5t
1
£
1
+
*
2
are possible.
£
+
2
•
•
+
ot^t^
+
This is true for
tions fx for « 1 t ,
1
to
•
t
2
for
a
2
£
3
,
£
zero by the
= 0,
•
•
•,
(- l)
is
1,
•
t
for
n
k
ft)!
and in which
terms after the first froi
ft
Moreover,
n+1 .
ft
f
(n ,+ 1 -
•
•
no other such terms
+
n
,
1
t
The substitu-
.
n+2
for
9j
serve
completely determine an equation independent of the original x's,
P's and Y's.
If
2.
11
denotes the sum of the
—
r
ft!
which the arguments are formed by omitting
the equation which is satisfied by f (x) is
ft
{n
+
1
-
t's from
—
r
terms in
ft)!
t if
then
liquation
involves
(5)
+
a
independent variables.
3
In order
normalized equation having the same form and order as
a
+1 = x and
t
1)*
f[(n
t
--~-™-~
-ft)!?
fe!
•••
+
+
let
(1),
t
x
-
whence (5) becomes*
n+2 = y,
(- l) n+1 k
+ y]
to obtain
+
(n
(- l) n f(x + v)
v)
1
+
(- l) n+1 f(y)
C
=
(6)
From the foregoing considerations we see that every solution fix) of
equation (1)
is
solution of equation (5) and of the normal equation
a
(6).
While the solutions fix) of (1) are included among those of (8),
is not necessarily true that the solutions of
it
Equation (6) for
n
= 2
f(3x
+
y) - 3f(2x + y)
be shown in
is
V,
§
*The results in (5)
oial cases.
are included in
The following examole suffices to orove this statement
those of (1).
As will
(6)
3^
If
a
3 h
_
a,
.
+
3f(x
+
y) - f(y)
= 0.
(7)
the most general continuous solution of (7)
and
(6)
can be rendered more precise
A..h
—
and the
substitution of
X
—
y -t
3
ape-
in
J
•
for
J
+ 3 yf
+ 3 y
leaves
as well as
unchanged
Hence the elimination of terms for which i = J also eliminates all
terms for which i = h where h is any value for which & ^ = 0.
But
X
and y
+
OC.t
for
.
Z/
f
...
.
wnen
a
'-j
•
u
h
—
- o
U,
fi
j_
a
S
a
J
h
and
therefore this proportionality
is
a
necessar
h
sufficient condition for the simultaneous elimination of terms for
more than one value of i.
Hence the solution of any mth order equation (1) in which no
is zero is contained in the solution of equation (-5) or of equation (o) where n is the number of distinct ratios
S./a. in (1).
and
ot
V
over the finite complex xr-Dlane is
fix)
where x
=
u
u/"T,
+
trary constants.
au 2
=
and
(u
buv
+
y
+
cv 2
real),
+
rtu
and a,
+
eu
b,
+
c,
f,
^
(8)
e
and r are arbi
From the above considerations, we see that the solu
tion of any second order equation of form (l) is included among
those
of
However,
(8).
substitution shows that for the eouation
f(?x
+
p) - %f{x
we have d = e = f = 0.
+
fact,
In
9) - 2 f(x)
such an equation need have no other
than the trivial solution fix) 5
fi2x
§
+
- fix
y)
+
fin) =
+
as
y)
is
shown by the equation
- 2 f(x)
+
fin) = 0.
Determination of the Normal Solution
III.
Vertices of
a
the
at
Network.
The function fix) to be considered in this section and in
§§ IV,
is the
solution of the normal equation developed in
observed that if
nx
'*
fe
is given for the
then
v o'
-
is determined by
f
positive integer.
- 2, -
- 1,
mx o
+
(6)
it
where
k Vo>
+
(5)
is
+
It
is to be
xQ
+
ky Qt
2x Q
in
+
l)x Q
+
ky Q ,
2,
where
3,
m
ky
+
ky
•
is
by
•
of
•
any
Similarly, by successively giving to j the values
determined by (6) for any argument
f
is
where m is
a
negative integer.
•
argument
for any argument mx
Q
,
3,
,
II.
ky Q for successive values 1,
•
•
is easily seen that f
ft
arguments ky
is knowD for the
f
By putting y = jx
Q
(6).
.7,
+
o
f
§
is known for
any integer or zero,
By interchanging x and y in
all
arguments mx Q
if it is given for
+
fc#
Q,
the arguments mx
,
—
mx o
+
^—
mx o
Uo>
arguments mx
•
*?.y
k\j
m,k
mx Q
+
Sy
=
1,
0,
where
,
+
ni/
•
if f is given for the
then it is known by (6) for
n,
any integer whatever and
is known by
it
•
=
ft
0,
for all arguments
(6)
f
known for the arguments
is
are zero or positive integers and m
ft
ky
+
+
where
,
<
ft
m,
-
ft
The figure illustrates the case for which n = 4.
n.
*
=^
are any integers whatever.
ft
where m and
,
•,
•
is
m
Hence,
o*
then it is also known for the arguments mx
Q
1,
1,
+
where m and
,
ky
+
,
mx o
will now be proved that if
It
0,
*
*
—
—
'»
*
''Vo'
and finally,
n,
*,
*
mx Q
n
+
arguments mx
all
1,
^—
The
points designated by the small circles represent the arguments for
which
supposed known.
is
f
The
parallelogram formed by the lines
+
n
Vo
.joining the
nx
ny
+
joints
0,
nx
,
and ny Q will be de-
,
noted by Sj
The parallelogram h
contains, on and within its
boundary,
gig
mx Q
and no other points as vertices.
Equation (5) furnishes
problem.
t
Let
is reauired,
=0,
1,
•
•
•-,
n,
therefore, to find
t
=
x
t 2
=
•
•
a
simple means of solution of the present
•
=
t
q
= x
mx Q
+
ky Q where m and
n
1
except in the first term in which
m = g and
qx
m,k
+
(n
t
,
q + 1
=
•
•
•
=
£
= y
Q
n+1
and
3y these substitutions every argument in (5) is of the form
n+2 = 0.
+
ky Q ,
+
the vertices of K not designated by small circles.
at
f
It
all the points
=
ft
+
1
n
+
1
ft
- a;
are zero or positive integers and m
hence
f
is
m
+
ft
- n
+
1.
+
ft
<
Moreover,
determined for the argument
- q0y o in terras of linear combinations of its values for
1
1
arguments reoresented by vertices within or on the boundary of the
parallelogram bounded by lines joining the four ooints
qx Q
(n
+
+
- q)y
1
dotted lines for
succession,
q
+
(n
,
=
2)
- a)y
1
is determined
f
line /joining the points x
the values
a
at each vertex of «
and nx
ny
+
+
yQ,
1,
2,
'
'
•
(the dot-and-dash line
figure).
from
linear equation in one unknown with unit coefficient.
a
is to be noticed
in
n,
,
which lies on the
in the
It
,
(represented in the figure by
,
By giving
.
qx
0,
that each determination is made
Hence
the functional values so found are unique.
Having determined
m
+
k
=
n
+
1,
m
+
ft
=
n
+
2.
£
n+2
=
ky Q in
+
h
for which
it is easy to determine f at the vertices for which
In
Tnus
v o'
for the arguments mx
Q
f
let
(5)
=
tP
•
•
-
«
•
t
=
x.and
determined for the argument qx Q
is
f
=
t±
t
.A
=
=
- q)y
(b + 2
+
•'-•-
•
in terms of arguments represented by vertices within and on the
boundary of the parallelogram formed by the four lines joining the
points
is
0,
ax Q ,
qx Q
+
+
(n
- q)u
2
,
+
arguments in
and nx Q
+
2y Q
n
arguments of
*,
•
n,
f
is determined for
+
ny Q
.
by giving
n
successively,
£
is determined for all
f
the values 2y Q ,
n+2
^or each value hy
arguments qx Q
before giving
•
3,
which lie on the line joining the points 2x
Q
Proceeding in this manner
is
This determination
Hence no indeterminat ion can be introduced.
1,
By giving q the successive values 2,
all
- o)y .
Q
2
also made by means of a linear equation involving but one unknown
with the coefficient
all
+
(n
£
n+ g
+
(n
+
h
+
the value
1
(h
of
t
l)y Q
(n
•
- l)y
a+ 2,
f
must be determined for
=
h
+
- q)y Q>
+
3y Q ,
the remaining
q
.
As
1,
h
+
2,
••
•
*,
n,
before each determination
made by a linear equation in one unknown with coefficient
+
1.
~
'
1?
Hence
f
uniquely determined at all the vertices of
is
and we have
R
the following result:
Eueru solution f(x) of the normal eauation
points mx Q
ky
+
where
Q
and
m
iiven at the ooints mx
Q
+
or zero and m
1
8
+
ft
<
+
n
for which
Q
are oositive
ft
+
integers
If
and
k
m,
combinations of
2
s.
s
±,ky
f
where
ky
Q
and finally for 2' s mx
are integers then it
ft
+
in
2
-8
III-
§
ft.z/
,
Hence if it is most
m,k = 0, 1,
shown that
is known
f
when it is known at mx
Q
that
and.
f
yQ
at §mXg
for the successive values
is known at 2
B
mx
2' B ky
+
|<fe#
e
Q
= 2,
s
3,
21
*'•
since the Doints 2' 3 mx
Q
+
ever,
form
sufficient to show that
£mx
+
£fc.y
points of
a
a
dense set,
,
m,k = 0,
it
1,
•
is
•
Zn,
•,
~s
•
fore,
,
to
m,
ft
Furthermore,
ft.
and
show that
s
f
and
x
it
,
for all integers m,
2~ s ky
2n,
for all integers m and
then by regarding
ky Q ,
+
+
• •,
•
for f will
III,
§
then be known at these points for all integers m and
if it is
for all
Q
then it is evident that 2' e x
and
Q
sufficient, in view of the argument of
is also
2' 3 ky
+
Q
found for certain appropriate linear
is
at 2 ~ s mx
f
and
m
may be regarded as were x
and u
Q
as x
Q
integers
Dense Set of Points.
a
+
x Q and 2~ 8 y ,
Q
convenient to determine
ft
is
it
-
points mx Q
all
may be found for ±mx
Q
z/
and
m
if
will now be shown that if the solution of the normal equation
is known at
it
are any integers or zero
ft
Determination of the Normal Solution at
IV.
It
2'- 3
ky
known for the
is
(6)
is seen
and
k
~
21 3y
s.
There-
any integers whatis
f
known at
known at all
is
dense set.
Supnosing
f
known at
|cbjc
+
£fti/
,
m,
ft
= 0,
2:,
••
•
•
,
2 b,
it
is
13
only required to learn
Replacing x by j*
tained by giving
linear in
3,
•
*
*,
f
at
j"ix
and u by
f->zx
Q
f*u
,
+
*?y
in
the values n -
h
m,k = 1,
+
1,
(6),
The eauations are
0.
Alternate terras involve the arguments ^mx Q
f.
2n -
1,
for which
sidered as linear eauations in
n
unknowns.
n/2 or (n
l)/2 according as
+
n
fe.v
=
m
,
1,
is easily seen that the
It
determinant of the coefficients of the unknowns is (is
+
The n eauations may be con-
is unknown.
f
3n - 1.
*,
•
'
eauations are ob-
n
1,
*
'
3,
is even or odd
1)
and A
v
£
n
is
n
where
,
v
determi-
a
nant of binomial coefficients such that the element in the ith row and
,/th
column is
- I) I (n - 2j
(2./
unless Zj
the
n
(i
- 1,
<
i
or 2j
n +
>•
i
+
in order,
+
i
1)!
in which case
1
l)th row from the ith row,
+
+
i
having the values
the element in the ith row and
Subtracting
is zero.
it
1,
column,
j'th
%
2,
••
•
i
/
n,
be-
comes
(n -
unless
the
n,
./th
-
in the
2.7
<
i
+
in
+
4./
or 2j >
+
2i
n
+
i
1)!
2) T T-7
(2j - i)! (n -
+
in which case
2
+
2j
+
i
is
it
- T
2)!
zero.
.Adding
to
column the sum of the preceding columns, giving j the values
1,
•
•
*
,
2,
in order,
AQ
ith row and ./th column,
7
assumes
i
4 n,
is
-
4ft
+
2i
a
form in which the element
+
2) (n
h=i j2h~-~i)]Jn~-~2h~+~i~+~2)T'
If
for anv given value of j.
this sum is
-r(2.7
-
nl
—
i)l(n t
7
—
r
2.;
+
i)!
,
un-
14
less
<
2,/
or 2j
i
that for J
or
=
>
2.7
n
it
1
+
>
is
-2
i
n
in which case it is
£
?2~T~3)l (''-
in which case
,
.
»!
Since 7 -
respectively.
or n,
U " 1SSS
-
2,/
=
i
-
i
and eaual to n for 2J -
=
=
i
*J
* *
or
2J
+
2.7
-
3
i
1
for
1
!
follows by induction that
it
1,
t)
"
that is,
*~ is eoual to
r-
- 1)1 7(» -
(2,7
S
For
easily shown
is
the value of S is merely the first non-vanishing terra,
1
2.7
it
- fll'
t
zero.
is
it
*J *
zero,
has the value
_n!__
unless
or
2.7
>
n
column the value of
S
is
i
=
n,
2,7
it
<
i
+
is clear that
in which case it is
i
always zero for
¥
i
For the nth
since 2j
n
>
+
n
the last term in the nth column is 2 n
formed that the principal
Eence
.
l) n+1
involved,
(1
+
A
has been so trans-
is
- l)-rowed minor found by deleting the
(n
last row and last column of
A
A
is
n
and the last column consists
n —l
exclusively of zeros with the single exceDtion of the element
the nth row.
(A
A
is
Therefore
trivially 2).
= 2 nA
A
Therefore
It
A
is evident that
= 3I+8 +
n
V '*b
distinct from zero for all finite values of
follows immediately that
jmx Q
+
kif
Q,
m - 1,
3,
Having determined
2n,
it
letting
is
h
For
i.
the elements are affected only by the process
addition and since every alternate term of
of
zero.
•
•
is
f
f
%
at
n
in
= 21+2
A
,
= 2 £n(n+i)
w hiefe is
Since (-
V
1)
^ 0, it
A
uniauely determined at the points
2n
—
and hence for m =
1
the points ?mx Q
only necessary to let x
take the values
n.
2 n
- 1,
= |i/
•
•
•
+
ky Q ,
and y = jhy
,
1,
0,
m
1,
2,
=
1,
+
|.«
•
2,
•,
•
•
in
•-,
(6),
to determine f at the
2n.
15
points jmx
£-*t/
*,k = 0,
,
1,
•
2n,
•
D
for (-
v*
n
again the
is
determinant of the coefficients of the unknown terms.
Combining the results of §§ ITI and
is
IV,
may bs stated that
it
determined at each point of the dense set 2~ B mx Q
known at the voints mx
for which m and
ku
+
?-*ku
+
if
it
f
is
are positive integers
k
or zero and
§
Solutions of the tiormal Equation,
V.
Eauation (o) may be written in the form
n+1
j=o
where x
seek
a
+
(n
(- l)j
2
1
j\ (n
=
u
+
——-t1
- j
+
= s
y
t/~I,
+
in which the coefficients
in
(6
)
s) + /-~T(.; y
+
*)]
=0
and u,
s,
u,
are real.
t
Let us
(n
i
h
I
q=0
^"yV"
(S)
qh
are constants.
Setting this value of
The terms involving u*
j
~p
- j
+
1)!
spy h
h - f'- r
<!Lli2l
ji (»
+
h
N
l)l
+
j] in - j
J-0
i)
h
N
J
h=0
we have
n+1
u
i-o
+
solution fix) of this equation in the form
fix) =
f (x)
f[(.iu
1)\
,-
1)1
h =o
~r
lr
lc
t
q
r
=o
ih
are
«»;nr^7f rUi~~T-~}Tu
Since no term outside of the brackets involves
the given expression vanishes,
and only when
- o)!
(ft
o!
for non-zero
./,
h
,
it
u,
"
evident that
is
s,
3
u
and t,
when
1
f qh a
vanishes.
[(-1)
2
--7
.7
Except for sign the quantity
B
~
ence of x h p
~r
.7!
(n
-
p
- r = n.
putting
p =
The particular value
was taken as the largest value of
satisfies (S) if
(9),
the
is
h
l)th differ-
+
(n
Therefore
B
q
obtained from
7z
.V
=
it
h,
Moreover,
n.
h
ft
- p - r by
B
+
1)
= f<Ca
IV it has been shown that when f is known at the 1 + 2
and
m
+
l)(n
+
k
<
+
n
points mx Q
2),
+
1,
+
ki/
,
ft
then it is known over
Since
? (x)
necessary to prove that each
f
at
the given points mx Q
+
qhv
ki/
+
Q
is
l)(n
+
2)
By §§ III and
+
•
a
•
+
•
(n
+1
dense set of points covery
are not collinear with
given by (9) is continuous,
as
given by
as
and k positive integers or zero
ing the entire finite plane provided x Q and
the point zero.
N
the value of C qh is arbitrary.
arbitrary constants G qh which may be assigned at will.
^{n
Since
.
n
q k
follows that fix),
Equation (8) shows that there are 2?
=
zero when
is
^
must be included in the discussion of
-
r
]
.7
1)!
The degree of each difference is one less
for x = 0.
than that of the preceding difference.
h
777
:
-
j-o
it
is only
.7.0
uniauely determined by assigning
to know that f (x)
is
the most general
continuous solution of (6) over the finite complex plane.
This can be
done by direct substitution* but inasmuch as the determinant so formed
is
unwieldy it is more easily accomplished by observing that the prop-
erties sought for any desired oblioue network are readily deduced by
a
projective transformation from similar properties of
a
square array
on the axes of reals and imaginaries with the units 1 and -/-"I,
pro-
vided only that x Q and y Q do not lie on the same straight line through
The value af the determinant of the coefficient s may be shovrn to
n(n+l) (n+2) n -1
2 OU l
(
- u t ) 7B
be (y s
[(n - a)!]
.
v
•
(
)
17
Confining attention to the rectangular array men-
the zero-point.
tioned,
write
f(xY
=
n
h
£
2
A
h=o q-0
*
u
v
h
<*>u
U-«>
where
u
U>
=
- 1)
u (u
For any integral value of
•
•
•
less than
u
(
- q
u
+
=
a,
1)
.
0.
Beginning with x
=
and proceeding outward through a triangular network similar to that
employed in
of the net*
III it is possible to determine an A^ h with each point
§
Having determined the
4
h
's
is only necessary to ex-
it
Dand and collect the terms of the expression for fix) and compare with
the expression involving the
completely determine each
to
's
Thus we see that the most General solution fix)
of eauation
continuous over the finite comolex x-vlane
(o)
trary polynomial in
and
u
of degree
u
i>/- 1)
+
f iu
is
an arbi-
n.
The analytic solution of id) over the finite complex plane is
that sDecial case of the general continuous solution for which
fix) =
Replacing x by
u
+
for k > n and hence
aQ
y/~~~T,
a
k
=
a ±x
+
is
it
0,
+
ft
>
•
*
*
of degree n,
a
k
xk
seen at once that a,u k must be zero
Moreover,
b.
the most general analytic solution of
in x
+
(6)
is
ak
= C
k0 ,
fe
=
n.
Hence
an arbitrary polynomial
*
solution fix), if it exists, is a polynomial
of degree not greater than n is readily seen by direct differentiation
If one differentiation is made with respect to each of the
of (1).
variables y, « .X + 6 y f j = 1, 2,
n, it results that fi^iy) =
>
whenever
n because the arguments are independent in pairs which
are not proportional.
That the analytic
•
ft
'
To obtain the most g'ensral continuous solution of
line in the finite complex plane,
it
is only
along any
(•:)
necessary to observe that
by the argument of §8 III and IV it was proved that
f
dense set of points on the line if it is known at
* 2
line which are separated by some convenient unit.
ft
known at
is
a
points of the
The argument at the
beginning of this section shows that along any line not parallel to
the axis of imaginaries
B
fix}where each
it
arbitrary, satisfies
is
a
a .u i,
2
j =
3
(6).
Since fix) is continuous,
remains only to show that the o's are determined by the functional
values at the
+
n
points on the line to know that fix) is the most
1
general continuous solution of (6) along the line.
Substitution shows
immediately that the a's are uniquely determined by the
of f on the line.
Hence the most General solution of
n
+
1
values
(5) continuous
aloni any line not parallel to the axis of imaGinaries is an arbitrary
polynomial in
of deiree
u
n.
Similarly the most General solution of (6) continuous alone any
line not parallel to the axis of reals
v
of defree
is
an arbitrary polynomial
in
n.
§
VI.
Solutions of the Original Equation.
The determination of the existence of any solution fix) of the
equat io n
Jj.fCoc.x
*
3^)
and the determination of fix)
+
y
n+1 f(x)
Y n + 2 f iy)
if it exists is
=
(1)
accomplished by substi-
—————————
1
II
tuting the corresponding solution of the normal eauation (6) in (l).
It
has been shown in
V that
§
the general
solution of (6) analytic
over the finite complex #-plane or along any line in the finite complex jf-plane is an arbitrary polynomial in x of degree
Pence the
n.
determination of the general solution of (1) analytic over the finite
complex x-plane or along
exists,
is
line in the finite complex x-plane, if it
a
accomplished by substituting
f(x) = c~
o
in
(l).
+
c
A
l
x
+
*
+
•
•
c
n
xn
Since the result of this substitution is an identity the
coefficient of any power of the variables must vanish.
It
is convenient to consider the terms of degree n independently
of the remaining terms.
Now
c
q
These terms are given by
is necessarily zero
C!i
Yi
(
unless
V M
+
)Q
+
Y
n+1 *
n
+
Y a+3 sr
tt
l
m,
0.
Placing the coefficients of this identity eaual to zero we have
n
y
n
Y
ol
I1
i
i
1
+
Y n+
i
'
kp
1 «i"
i= 1
i= 1
Since only the ratios of the
generality to assume
Y
n+2
i
i
f
Y'
s
- 1.
=
1
=0V
'
= 0,
iYi
±
1
fe
=
1,
•
•
n
- 1>(10)
n+ 2
are significant it
implies no loss of
Under this assumption equations (10)
may be employed to express the remaining y's in terms of the «*s and
3's provided fix) contains
a
non-vanishing term of degree
n.
If we
20
write
r
a
=
i
i
^
i
then for
»
<
i
= —
y
1
n
--.
'?
1
where the prine indicates that
+
1
_i— s.
r
a_
if'
W-i
ft
(r
h
- r
1
)
does not take the value
Solution
i.
also gives
Therefore
(1),
if
a
x n,
c
n
/
is
0,
s
terw
o<*
analytic solution fix) of
tfte
(l) way be written
necessary and sufficient that
is
it
c
in the
form
i=1
i
where
r
i
r
If
™
h= 1
'
=
i
r
h
a
//
i
~
r
v i
-
i'
the solution of
(1)
includes the term
c
m
xm,
<
m
n
and
c
•> *
-
m
^
0,
then
I «?Y.
1
i»l
+
*
Y n 411
k8
Yi
11
i
2
1
i= l «i"
2 bTyi-
Since there are but m
to express m
the first m
+
+
1
y
+P
= °>
=
= 1, 2,
fe
*
'
1,
o.
linear eouations in the Y's they may be used
Y's in terms of the
1
1
•+
+
= 0,
a 's,
y's and remaining Y's.
Y's in terms of the remaining Quantities,
For
solution-
gives
(»)
—
>
-
Y
fl
(r
— r
)
—
Y
+
(—1
)
r
r
'
'
'
r
r
'
' *-r
Y
c
it
xB
<
m
.
and c. / 0,
n
is
term of the analytic solution fix) of
a
necessaru and sufficient that each of the first
is
the value iiuen
in
+
Y's
1
(l)
/zas
(ll).
The coraoutation involved in finding the most general solution of
(l) continuous over the entire finite plane is
it
so tedious
to make
as
exoedient to give results only for the general second order equaThe problem for any given equation is merely
tion.
tution and algebraic comoutation.
+
Y 2 f[(a 21
where
a llf
a
a 22 i)x
+
b ±1>
±2 ,
b
+
(b
+
21
a 21 ,
12 ,
b
matter of substi
a
For the second order equation
22 i)u]
a 22 ,
b
Y 8 f(?)
*
b
21 ,
= 0,
Y 4 f(&)
are real and
22
/- 1,
=
i
the normal solution with which substitution must be made is
C
00
+
C
+
10"
C
+
01 V
c
when and only when Y-
Y3
Y2
+
20 U2
+
C
+
11 UU
C
02
y2
«
nn may be assigned different from zero
This substitution shows that
+
C
+
Y4
= 0.
It
also shows that if we
write
Ai =
Ti«n
+
=
Y i a !2
+
and
c
91
then for
However,
0.
I?
c 10
58
- 1.
A
=
Yiojj.
S
=
^12
c
Y 2 a 2i
B
Y3,
Y
Dl
Y 2 a 22*
+
Y
Y l & 12
+
Y 2 b 22>
Q1 to be independently arbitrary A
10 = G
i^i if
4
=
i
+
b
b
Y
%
ci =
D i k i>
B
1
31 =
£7*
or °i ^ °'
i
Furthermore if we write
+
Ygali
Y 2 a| 2 ,
+
Y3
,
5
=
Y^!^^
=
Yibjj.
+
Y 2 a 2i a 22>
Ya&lj.
+
Y4 ,
=
and
=
32
+
Y 2 b 2i b 22>
F
=
y ± bl 2
Y l a ll b 12
+
Y 2 a 21 b 22'
H
=
Y i a il b ll
+
Y 2 fl 21°21'
Y i a i2 b 12
+
Y 2 a 22 b 22'
*
=
Y i a i2 b ll
+
Y 2 fl 2 2 b 2 1'
i8
3
Y2
»2V
j
J
c on
=
=
and
*?c_
(i)
=
..
1
and
(a)
A
+
(b)
9,
(c)
1
=
k
(ii)
-
c
rto ,
=
+
2
- §t
P,
or
-7
- H / 0,
K
—
D
and » =
4*
(b)
B
or | ^ 0,
(c)
C
=
k
= - 1
E,
F
P
=
or H
Em.,
7 ¥ 0,
+
and 8(f
3) =
+
( -7
- H)m.
if
or 3
(a)
4,
if
i
+
- G -
D,
then it is easy to show that
*
D
0,
5,
= - 1
ft
«c
= ©,
0,
7 =
ff,
and 3
K =
+
C.
I
(iii)
!
(a)
4,
(b)
§
and m is arbitrary if
D,
B,
or 3 ^ 0,
= £ = 0,
G =
A,
P = D,
7 = B,
and Q
+
K
=
1
(iv)
k
=
and m =
1
(a)
^2 + ^2^
(b)
= - 4,
if
^2
P
+
?2}
= - D,
or 5 2 +
K = G,
ff
^
2
0,
and J = -
H.
The following equations constructed on the basis of this infoT'tation answer interesting questions.
3f[(l
+
2i) x
+
(3
+
i)y]
- f[(l
+
The eauation
3f)x
- of
*T he
no
attennt
+
+
(8
(jc)
+
3i)y]
llf (p) =
allow Cgg to be differ e nt from zero and
is made to discuss the possibilities in ca s e C 02 = °-
res ults desired
here
23
the general continuous solution
as
f
(x)
=
c
^{u'
an eouation may have a continuous solution but
v2
+
,i
Q
showing that
)
no non-trivial
analytic
solution.
The general solution of
i)f[3x
+
(1
+
(1
- 3f[x
- i)y]
- (6
+
+
9i)f(*)
2y1
+
(10
+
3f)f(y)
continuous over the finite complex x- plane is
c
2Q x
2
=
c on (u 2
2£a» -
+
y2
=
)
showing that an equation may have its analytic solution as the
most General continuous solution.
The elimination outlined in the footnote of
solution of an eouation of form (1) having no
§
II
shows that every
equal to zero is in-
oc
cluded in the corresponding solution of the normal equation whose
order is eoual to the number of distinct ratios 3. A.,
n,
in
the eouation of form
which each ratio
(1).
least one other such ratio and at least one
subscriots,
namely,
different from zero,
and
,
One case remains,
a
- 1.-2.
i
that in
is equal to
and one 3,
at
of different
That there are equations of this exceptional
are zero.
type which have infinite series solutions is proved by the equation
Hx
+
y)
- f(ix
+
- f(- ix) - ft- iy)
iy)
which is satisfied by
a
+
fix)
+
f (y)
=
C,
series in positive integral powers of
i
=
havin
arbitrary coefficients.
Til-
§
It
was shown in
§
The Converse Theorem,
V that
any polynomial in x of degree
fies the normal equation of order
ms'*
It will
satis-
now be proved that any
j
polynomial p(x) in x of degree
order
not greater than the number of non-vanishing terms of p(x)
is
n
satisfies an eauation (1) whose
plus the decrees of such terms and whose «'s. 3's and
may be assigned at will provided
a
Y
certain determinant
.
and
1
of the
A
Y
^ o
oc
'
s
and 3's is not zero as a consequence.
Suppose that
pU)
=
1
a±x
a2 x
+
The substitution of v(x) for fix)
t*l*J t
- - Y n+1
a
in
+
•
*
•
a^x
+
J
.
gives an identity from which
(1)
,
(12)
2
i
= l
n
? 3
mb
=
Y
for the values h - 1, 2,
-
v
-
•
"-,
equations is not greater than
The total number of independent
./.
+
j
2. ^ji,.
ber of independent equations we have
ar
equations in
unknowns Y.,
1 7
n
Y2
>
7
*
dition that the unknown
a's,
3
!
s
fj*s
oc '
equal to the num
*
7
Y n 7,
provided
Y
and Y n+2
^
n+1
-,
necessary and sufficient con-
A
are uniauely determined in terms of the
and two assigned Y's is that the determinant A of the coef-
ficients be different from zero.
in the
n
system of non-homogeneous line
a
are not both assigned equal to zero.
Setting
s
Furthermore,
and 3's.
It
is obvious
it
is
at once
that
A
is
a
polynomial
evident that the term
formed by the product of the elements in the principal diagonal is
unique.
tically.
Hence the polynomial in the
Therefore the
that the polynomial
If Y n+1
and
Y
A
n+2
« '* s
°c
'
s
and 3's does not vanish iden-
and 3's may be assigned in any way such
has a value different from zero.
are both assigned eaual to zero equations
(12)
system of
n
linear homogeneous eauations in
non-vanishing of
A
is
a
of the unknown y's
is
zero.
form
a
satisfied by vix)
infinity of equations
unknowns and the
necessary and sufficient condition that each
In this case the equation
In general,
.
n
is
trivially
anu oolynomial vix) satisfies an
then,
whose orders do not exceed the number of
(l)
non-vanishing terms of d(x) dIus the sum of the degrees of such terms
and whose «'s,
3's,
Y
a+1
certain
only that a oolynomial
A
's
and
of the
Y
n+2 s ma V be assigned at will provided
'
and P's does not vanish for the
<*'s
A
assigned values.
§
VIII.
discontinuous Solutions,
satisfies the normal equa-
Any solution fix) of an equation (1)
tion whose order
is
fix)
n
is determined by
Suppose that the domain of
(1).
any line in the finite complex x-plane and that fix) is con-
tinuous in some interval of length
&
of the line.
>
seen from the normal eauation satisfied by
f
It
is readily
that if the first
n
+
1
arguments are so chosen that they represent points in the interval
while the remaining argument reoresents
a
Doint outside the interval,
fix) is determined at the last named point as the sum of continuous
functions.
way
f
Therefore
f
is
continuous at the outside point.
may be shown to be continuous at all points in the two intervals
of length 6/b which lie at the ends of the given interval.
f
is
In this
continuous in an interval of length [in
+
2)&l/n.
interval of length o may be reached in this manner by
of extensions of the interval of length
that if fix) has
a
.Any
a
Therefore
finite
finite number
We may therefore state
finite voint of discontinuity on anv line in the
'
26
comolex x-plane
the line,
has
it
ooint of discontinuity in every interval of
a
however small.
Suppose fix) is continuous in
plane.
A
a
region of the finite complex x-
circle may be inscribed in the region such that fix)
is con-
tinuous in the closed res'ion of which the circle is the boundary.
Suppose the radius of this circle is
cle of radius
[(r?
+
6
and consider
a
concentric cir-
8y means of the normal eauation
l)&]/n.
f
may be
determined at any point in the area between the circles as the sum of
n
+
1
continuous functions, namely,
Therefore
radius 5.
f at
+ 1
n
points in the circle of
at the point between the circles
f
is
continuous.
This is true for every Doint of the area between the circles and hence
continuous in the circle of radius
[
may obviously be repeated to prove that
f
is
a
point of discont inuity in
f
is
u ence
region of the plane.
the finite complex x-olane
f(x) has
if
it
(n
This process
+
continuous in any finite
has a point of discontinuity in every
finite region of the Diane,
G.
Basel
(loc.
cit,) has exhibited a discontinuous sol ution
f (x)
of the Cauchy equation
fix
+
From the treatment in
§
II
f(2x
+
y) -
Replacing
y
by hx
+
y)
=
f(x)
+
f(y)
it is clear that f
Zflx
+
y)
+
.
also satisfies
fiy) = 0.
multiplving the eauation by (-l)
y,
h
— 7 --h\ \n
for successive values
ft
=
0,
1,
tions so formed it is easily seen that
n
h
- 1)!
- 1 and adding the n equa-
27
n* 1
I (_
(n
^
k
i)
1)!
ill
+
r(jk ,
v
.
)
o.
(6)
We therefore have a discontinuous solution of the normal
each order
§
equation for
n,
Certain TyDes of Equations having Variable Coefficients.
IX.
The functional eouations that have been discussed may be employed
solve certain eauations of the form
to
(13)
2
.
_^
<P.
i
(x,
3. v
i
y)f (« i x
where the
9
'
s
+
)
cp
n*J.
(
x ,y)f(x)
are known functions.
+
9
^(x
n+o
A general
f
y)f(y)
+
9
_(x,y)
n+o'
statement and
a
=
few ex-
i
amples suffice to indicate some of the eauations that may be solved.
Suppose there are
transformations
ft
WW
x = y.x'
+
5
equation is multiplied by
A
u
WW
= \ x*
+
\i
,y'
(13) to obtain new equations such that if each
which may be applied to
form (1).
y'
a
non-zero constant the sum of them is of
solution of (13),
if
it exists,
is
included in the corre-
sponding solution of the auxiliary equation of form (1) deduced from
In order to
(13).
find
a
solution of (13) it is sufficient to substi-
tute the solution of the auxiliary eauation and compute the coeffi-
cients of the variables.
Equation (13) includes the non-homogeneous equation in which
9
i
(x, y)
,
i
- 1,
2,
•
•
•
•-,
n + 2,
is further restricted to be a constant.
In this case it is obvious from equation
polynomial in x and
y
if
(13)
that
(
P
an analytic solution exists
n + 3 (%,
y)
is
a
and a polynomial
in
and
s
v,
a,
if
t
continuous solution exists.
a
Such
a
non-homo-
geneous equation is
fix
+
=
y)
fix)
solving of which was DroDOsed by
I.
+
M.
fiy)
2xy
+
(14)
American Mathematical
De Long,
Transformations which may be used to
Monthly, vol. XXIII, page 300.
solve this eauation are
x = x'
= x'
y
,
- y'
x = x
and
t
1
=
y
,
- x'%
y'
after dropDing the crimes,
whence,
fit* - y) - fix - y) - fi- x
The arguments x -
y
and - x
+
tion is therefore of order 2.
+
y)
- 2fix)
= 0.
fiu)
+
are proDortional
and the normal equa-
Hence the general
solution of (14) ana
y
=
lytic over the finite comolex x-plane is readily seen to be fix)
a xx
2
+
x
where
,
ax
is
arbitrary.
The general
solution of (14) con-
tinuous over the finite complex jc-olane is fix)'
where
-
+
a 10 u
*
a 01 u
x
,
and a 01 are arbitrary.
a 10
Supoose all the ?'s are constant and
9
A o
£ 0.
n+ S
If
2._.?. / 0,
then fid) is finite and uniquely determined and the transformation
fix)' n
v,
&ix)
fiO) may be employed to obtain an equation (l) of order
+
Hence fix)
in g(x).
is
a
oolynomial (in
as the case may be) of degree
x,
in u and u,
not greater than n.
in
u
or in
The transforma-
tion used in this case has the advantage of furnishing an auxiliary
equation
(1)
whose order is not greater than that of the original
eauation (13).
An equation which illustrates reduction by interchanging argu-
ments of
f
is
29
(cos 8 x)f(x
If
u
is
(sin 2 x)f(* -
+
v)
replaced by -
(sin 2 x)fix
+
- 2(cos 2x}xy = 0.
the eauation becomes
u
(cos
+
v)
- f(x) -
t/)
2
x)f(jc - y) - fix)
- fi- y)
The sum of these eauations is of form (l).
2(c©8 2*)*»
+
=
0.
The solution analytic over
the finite complex x-plane of the eauation having variable coeffi-
cients is fix)
- x2
.
It
is easily seen that this is the most general
(as
+
continuous solution.
The eauation
2fi2x
+
y)
+
y)fix - y)
may be reduced by reolacing x by 2x
2fibx
4y)
+
+
3(*
+
y)fix - y)
+
+
+
3fix) - 3fiy) =
y
and y by x
3fi2x
+
+
2y,
- 3f(x
gr)
+
whence
2y) =
0,
and subtracting 3 times the original eauation from the transformed
epuation.
The reduced eauation is
ZftSx
It
+
4y) - 3f(2x
+
y) - 3fix + 2y) - Bf(x)
+
9f(y) = 0.
is readily seen that the eauation with variable coefficient has no
continuous solution.
If one of the first n +
maining
fiy-
±
x +
cp's
3
i?/ )
2
q> •
s
of
(13)
are constant the product of that
is a polynomial.
function whose denominator is
The variable
a
factor of
variable while the re-
is
and the corresponding
q>
f i*
therefore
is
<p
±
3.y).
x
+
+
2f(z/)
/\n
illustrating this point is
(x
2[*—!]*f
x -
y
- y) - fix
+
2y) -
<*(*)
=
0.
a
rational
eauation
The equation obtained by replacing x by 2x
3L8C—
—]V(*
x -
- y) - f(4*
+
5V
- r(2*
)
and
+
y
+
v)
by x
y
+
2f(*
+
2y is
3f) = 0.
I
The eauation obtained by subtracting the transformed eauabion from
times the original one gives an eauation (1) of order 3-
solution f(x) of the eauation with
easily seen to be ax 2 , where
§
is
a
a
9
The analytic
variable coefficient is then
arbitrary.
Aoulication to Binomial Equations.
X.
An application of linear functional eauations having constant
coefficients may also be made to certain eauations of the form
J
if{« ± x
where G is
is
a
H 1 f)]
Y*
= C. =
l +1
[f^ ± x
constant, the real oart of each
some point x
-
a.
Let y eaual a in
5
i
for all values of
x.
an infinity of zeros.
[fioL
=i
1
x
positive and no
is
The case of
difficulty.
It
a
it
is
oc
analytic at
a
zero
Then
a
finite region in which fix) has
But this is impossible since
f
ix)
is
analytic
Therefore when fix) is analytic throughidentically zero
it
is
never
continuous solution fix) of (15) presents more
is evident that
if
one member of
(15) has either no
factor or only one factor involving fix), then fix)
less
(15)
Suppose that fix) has
(1-5).
(15)
)]
1
the finite complex Diane and not
zero.
Y
[f( y
=
+
Hence there is
throughout the finite olane.
out
3..v )]
Let us first consider the solution f(x) of
zero.
all ooints in the finite complex olane.
at
Yn+2
Yi
*
identically so.
is
never zero un-
The function
=
<p(x)
where it is understood that the
log fix),
principal determination of the logarithm is employed,
continuous with
when the latter has no zeros.
f
analytic or
is
Therefore in each of
the cases considered above vix) satisfies the eauation
n* 1
k
+
2 Y.<P(« .x
1
1
3
1
i-i
where
K
and
is
s
is
u
in
2
i=k+i
Y,?^,^
1
1
+
3
1
u) - y
+
n+2
j{
+
3sM
=
the principal determination of log C with its sign changed
For any given
any integer.
not greater than
a
-
)
variation in
s
s
<p(x)
is a polynomial of degree
Furthermore the argumentation of
n.
§
IX shows that
affects only the constant term of v(x).
Therefore
each of the cases considered above fix) is an exponential function
of the form
fix) =
where
?
is
a
p
*
k(8)
polynomial of decree not ireater than
constant depending on
(15)
s
s.
(16)
n
and kis) is
a
Thus we see that in general the solutions of
are given by (13) for the various possible values of kis).
The same result may be stated for the continuous solution fix) of
(15) when each member involves at
vided
f (C)
in
For
a
f
pro-
For ®ix) as defined is continuous in some region
4 0.
about the ooint x
region.
least two factors containing
since fix)
-
given s,
is different from
zero in some such
therefore, ?(*) must be everywhere continuous
the finite complex plane because it satisfies a non-exceptional
eauation of type (1).
The eauation
y(x
mentioned in
§
I
+
u)y(x - y) = [y(x)v(9j]
is included
in the last case considered.
For sup-
32
pose there is at least one Doint x
x =
-
u
b
-
b
at
which y(x) is not zero.
Let
Then
-
and v(0) / 0.
It
is easily
analytic over
seen that the solutions
the finite complex Diane are
and the solutions y(x) continuous over the finite complex plane are
-t
=
x\
where
e
°20 u2
*
°11 UV
°02 t2
*
'
3in
= +
s
e
20 u2
°ll uv
+
+
c
02 v2
and the c's are arbitrary constants.
a
Equations involving more than one Function,
XI-
§
Consider the eauation
JjiM*!*
where no
Y
+
s
i
v)
+
Y^xf.^C*)
+
r
n+2 f n+2
(.v )
(17)
= o,
zero and the f's are unknown continuous single-valued
is
functions to be determined if possible so that (17) shall be identi-
cally satisfied by them.
tinct.
(17).
The functions f. may or may not all be dis-
The method of elimination employed in
If no «
is
zero it is evident then that
normal equation of order
not greater than
n.
n.
Therefore
zero.
u
by
a
In this case,
is
f
a
Under the assumption that no
zero and no two of the ratios ^/oc
the argument
f n+2
is
II
§
.
„
applicable to
satisfies the
polynomial of degree
and no 3.
are
are eaual any term may be given
linear transformation which makes no
therefore, every function f. of (17) is
and no
a
3
voly-
t
33
nomial of decree not greater than
To find any necessary restric-
n.
tion on the coefficients of these polynomials it is sufficient to
substitute the nth decree polynomials having general coefficients in
(17) and to eauate to
It
zero the resulting coefficients of the variables.
obvious that eauation (1) is
is
some of the ratios
If
are eaual
loss of generality that eauation
having arguments of
a
special case of
a
(17)
is
(17).
may be assumed without
it
arranged that functions
so
common ratio are placed consecutively.
If
the functions having arguments of a comracn ratio have subscripts
such that
=
$
i
=
h
all
i
then we may write
h
hh
h
.
_„
i
i
l
i
Equation (17) may new be written
2
where no
a
denote by
F
is
a
M«
hh r
and no
+
q
2
8
+
3
h
y
)
+
v
n+ 1
(x)
+
F
are zero and no two ratios
a
h
are equal.
h
a.
if the f'*s of F
Each
f 's
If the f's of any F
(1) which have
F
no
oc
P
'
(.J/)-
Each
therefore
having arguments
are identical,
f.
a
non-
The equations
equal to zero but have some ratios Pj/a i
equal or some 3's zero are soecial cases of (15) given by
Y n+ 2
We
(1c).
are the same function F determines
homogeneous mixed g-difference equation satisfied by
of type
(16)
0,
/a
8
non-homogeneous equation in certain
differing by constant factors.
that is,
=
the number of terms in the first member of
polynomial of degree not greater than
determines
Ay)
n+2
F
n+ 2
(y)
=
The equations of the exceptional case noted in the last
paragraph of
§
VI are equations of form
stant multiple of
a
single
f.
In this
(16) which have no F a con-
connection it is interesting tc
34
note that the function
f
of the example in the paragraph cited satis-
fies the equations
f(x) - f(- ix) = f(x) - f(ix) =
whence
f(- ix)
=
f(ix)
=
f{- x)
=
fix).
VITA.
William Harold Wilson was born on
County, Michigan, November 17, 1882.
a
farm near Sdwardsburg, Cass
He attended the Public Schools
of Sdwardsburg for three years after which he attended those of Miles,
Michigan,
from which he graduated in June,
1908.
The collegiate work of the writer has been done at Albion College
Albion, Michigan,
A.
3.
and at the University of Illinois.
degree from Albion College in June, 1913,
from the University of Illinois in June,
the University of Illinois in 1813-14,
1815-15 and
a
fellow in 1816-17.
1814.
He received the
and the A. M. degree
He was
a
scholar in
an assistant in 1814-15 and
He taught
a
course in Differential
Calculus in the University of Illinois in the summer of 1915.
While in attendance at Albion College the writer was elected to
AHZ (local), the only honor society at Albion.
He was Salutatorian
of the Class of 1813 of Albion College.
While in attendance at the
a member
of
University of Illinois he has been made Phi Beta Kappa.
The writer wishes to express his gratitude for the helpful sug-
gestions and kindly aid of Professor R.
D.
supervision this thesis has been prepared.
Carmichael under whose
UNIVERSFTY OF ILLINOIS-URBANA
3 011 2 079097363