Applied Mathematics E-Notes, 13(2013), 36-50 c
Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/
ISSN 1607-2510
On A Generalization Of The Laurent Expansion
Theorem
Branko Sarićy
Received 19 March 2012
Abstract
Based on the properties of the non-univalent conformal mapping ez = s a
causal connection has been established between the Laurent expansion theorem
and the Fourier trigonometric series expansion of functions. This connection
combined with two highly signi…cant results proved in the form of lemmas is a
foundation stone of the theory. The main result is in the form of a theorem that
is a natural generalization of the Laurent expansion theorem. The paper ends
with a few examples that illustrate the theory.
1
Introduction
A trigonometric series is a series of the form
+1
a0 X
+
[ak cos (k ) + bk sin (k )]
2
(1)
k=1
where the real coe¢ cients a0 ; a1 ; :::; b1 ; b2 ; ::: are independent of the real variable .
Applying Euler’s formulae to cos (k ) and sin (k ), we may write (1) in the complex
form
+1
X
ck eik
(2)
k= 1
where 2ck = ak ibk and for k 2 @ (@ is the set of natural numbers) and 2c0 = a0 .
Here c k is conjugate to ck [8].
Let z = + i be a complex variable. Suppose that series (2) converges at all points
of the interval ( ; ) to a real valued point function f ( ). If f ( iz) is integrable
on the interval = = fz j = 0; 2 [ ; ]g, then the Fourier coe¢ cients ck of f ( )
are determined uniquely as
Z
1
ck =
f ( iz) e kz dz (k = 0; 1; 2; :::):
(3)
2 i =
Mathematics Subject Classi…cations: 15A15, 42A168,; 30B10, 30B50.
of Sciences, University of Novi Sad. Trg Dositeja Obradovica 2, 21000 Novi Sad, Serbia;
µ cak, Serbia
College of Technical Engineering Professional Studies, Svetog Save 65, 32 000 Caµ
y Faculty
36
B. Sarić
P+1
k
Similarly, if for = 0 a trigonometric series k= 1 c^k (ez ) converges in (
complex valued point function f (ez ), which is integrable on = , then
Z
1
c^k =
f (ez ) e kz dz (k = 0; 1; 2; :::).
2 i =
37
; ) to a
(4)
As is well-known, progress in Fourier analysis has gone hand in hand with progress
in theories of integration. Perhaps this can be best exempli…ed by using the so-called
total value of the generalized Riemann integrals [5, 6]. This brand new theory of
integration, which takes the notion of residues of real valued functions into account,
gives us the opportunity to integrate real valued functions that was not integrable in
any of the known integration methods until now. All of this leads to the following very
important problem. Let C be a simple closed contour in the z-plane, which consists of
= and any regular curve Q connecting the endpoints of = , inside and on which
a complex valued point function f (z) is analytic except for a …nite number of isolated
singular points z1 ; z2 ; :::; zn belonging to the interior (int) of = . Chose > 0 to be
and the semi-circumferences + = fz j jz z j = ; > 0g
small enough so that Q
and
= fz j jz z j = ; < 0g ( = 1; 2; :::; n) are disjoint. Having formed the
numbers c^k by means of
Z
1
vt
f (z) e kz dz
c^k =
2 i
=
" n Z
( Ry
#
n
X
X
f (z) e kz dz
1
kz
R
f (z) e
dz +
lim
(5)
=
x
kz
dz
+ f (z) e
2 i !0+ =0 l
=1
for k = 0; 1; 2; ::: where vt denotes the total value of an improper integral, l0 =
[ i ; z1 i ], l =P
[z + i ; z +1 i ] ( = 1; 2; :::; n 1) and ln = [zn + i ; i ], we
+1
may call the series k= 1 c^k eik the Fourier series of f (z) on = . This means that
the numbers c^k are connected with f (z) by the formula (5). Clearly, in this case, the
numbers c^k , which are the Fourier coe¢ cients of f (z), are not determined uniquely
since on account of Cauchy’s residue theorem
Z
Z y
0
Pn
vt
f (z) e kz dz =
f (z) e kz dz +
(6)
kz
2 i
]
=1 Resz=z [f (z) e
=
Q
whenever Q is in the left half plane of the z-plane. A question that arises is whether
this Fourier series of f (z) does in fact converge and if it does not whether it is
summable in any other sense and under what conditions is its sum equal to f (z)
in int= nfz1 ; z2 ; :::; zn g. As we shall see, in what follows, the answer to this question
is closely related to the Laurent expansion theorem.
2
Main Results
By the Laurent expansion theorem, a single-valued complex function f (s) of a complex
variable s, analytic in an arbitrary punctured disc fs j 0 < jsj < Rg of the s-plane,
38
On a Generalization of the Laurent Expansion Theorem
P+1
has a Laurent series expansion f (s) = k= 1 c^k sk . The Laurent coe¢ cients c^k are
determined uniquely as
Z
f (s)
1
ds (k = 0; 1; 2; :::)
(7)
c^k =
2 i Cr sk+1
where Cr = fs : jsj = r; 0 < r < Rg [3]. On the other hand, it is well-known that
the horizontal strip D = fz = + i : 2 ( 1; ) ; 2 [ ; ]g of the z-plane is
mapped onto the punctured disc D? = fs : 0 < jsj < e g in the s-plane by the nonunivalent conformal mapping s = ez . Accordingly, if a single-valued complex function
f (s) is analytic in D? and > 0, then, on account of the Laurent expansion theorem,
P+1
k
the Fourier series k= 1 c^k (ez ) with the Fourier coe¢ cients (4) converges in D to
f (ez ). Furthermore, it follows from (7) that for any real number <
Z +i
1
f (ez ) e kz dz (k = 0; 1; 2; :::).
(8)
c^k =
2 i
i
Here a single-valued complex function f (s) is taken to be the function f (ez ) of period
2 with respect to in the z-plane.
Generally, a complex valued point function f (z) is said to be integrable, of bounded
variation and the like, with respect to either or , in the strip D, if its real and
imaginary parts satisfy separately the aforementioned properties for all …xed values of
the second variable in D.
As is well known, the …rst theorem, chronologically, in the theory of Fourier trigonometric series, is as follows [8].
THEOREM 1. If a real valued point function f is of bounded variation, then the
Fourier series of f converges at every point to the value
1
lim [f ( + ) + f (
2 !0+
)] .
(9)
If f is in addition continuous at every point on an interval I = [a; b], then the Fourier
series of f is uniformly convergent in I.
The result of the preceding theorem implies the following result.
THEOREM 2. If a single valued complex function f (s) is of bounded variation
with respect to argP
s in a punctured disc D = fs : 0 < jsj < e g of the s-plane, then
+1
the Laurent series k= 1 c^k sk , with the coe¢ cients
c^k =
1
2 i
Z
C
f (s)
ds (k = 0; 1; 2; :::),
sk+1
(10)
where C = fs : jsj = e ; < g, converges at every point s 2 C to the value
1
lim f sei
2 !0+
+ f se
i
.
(11)
B. Sarić
39
If f is in addition continuous on C , then the Laurent series of f is uniformly convergent
on C .
This result strongly suggests that single valued complex functions having isolated
singularities inside =
may be expanded in the Fourier trigonometric series, too.
Let’s show this works. Let w be a complex parameter independent of z = + i . As is
well known, if a real valued point function f ( ) is of bounded variation in the interval
[ ; ], then for ^ 2 ( ; ) and Re(w) 0 there exist …nite limits, [3],
Z
^
lim w ^ eiw(i z) f ( iz) dz = i lim+ f (^
)
jwj!+1
and
lim
jwj!+1
w
!0
=
Z
e
iw(i^ z)
f ( iz) dz =
=^
i lim+ f (^ + )
!0
(12)
^
where = = fz = + i : = 0; 2 [ ; ^]g and =^ = fz = + i : = 0; 2 [^; ]g.
Since the above conditions play a key role in the proof of Theorem 1 based on the
Cauchy calculus of residues, see [3], it follows that all functions that satisfy these
conditions have a Fourier series expansion
f( + )+f(
lim+
2
!0
)
=
+1
X
c^k eik ,
k= 1
with the Fourier coe¢ cients (4). Furthermore, pursuing the matter in the proof of
Theorem 1, one easily comes to a conclusion that conditions (12) cannot be satis…ed
for Re(w) 0 but only for Re(w) > 0. This is an immediate consequence of the result
presented in the following form.
LEMMA 1. For w 2 C, let a single valued complex function f (w) be analytic
in the w-plane with the exception of an in…nite but countable number of isolated
singularities wk (k = 1; 2; :::) on the half-straight lines fw = jwj ei : jwj > 0; = ^g and
fw = jwj ei : jwj > 0; = + ^g by which the w-plane is divided into two open adjacent
regions R = fw = jwj ei : jwj > 0;
+ ^ < < ^g and L = fw = jwj ei : jwj > 0,
^ < < + ^g. If limjwj!+1 wf (w) = CR , whenever w 2 R and limjwj!+1 wf (w) =
CL , whenever w 2 L, then
+1
X
Resw=wk f (w) =
k=1
CR + CL
.
2
(13)
PROOF. By de…nition, the residue of f (w) at the point at in…nity is equal to the
residue of the function f (1=w) =w2 at the point w = 0. In symbols,
Resjwj=+1 f (w) =
Resw=0 f (1=w) =w2 .
40
Since
On a Generalization of the Laurent Expansion Theorem
f (1=w) =w2 has no singularity at in…nity, it follows from
+1
X
Resw=wk f (1=w) =w2 =
Resw=0 f (1=w) =w2
k=1
that
+1
X
Resw=wk f (w) =
Resjwj=+1 f (w) .
k=1
(14)
For R (w) = wf (w) CR and L (w) = wf (w) CL let MR = maxfj R (w)j j
w 2 Rg and ML = maxfj L (w)j j w 2 Lg, respectively. By means of an arc length
parametrization for the circular arcs
+^+
C R = fw = jwj ei : jwj = ;
^
( )
( )g
and
C L = fw = jwj ei : jwj = ; ^ +
+^
( )
( )g
where ( ) is an angular function, which can be made as small as one desires, such
that lim !+1 ( ) = 0, we obtain
Z x
Z x
f (w) dw +
f (w) dw
CR
=
=
=
i
i
"Z
(Z
i[
CL
^
( )
i
e f
e
i
d +
+^+ ( )
^
Z
+^
i
e f
CR +
R
ei
e
i
d
^+ ( )
( )
+^+ ( )
( )
d +
Z
+^
#
( )
CL +
^+ ( )
L
ei
)
d g
2 ( )] (CR + CL ) + GR + GL
R +^ ( )
R^ ( )
ei d , that is,
ei d and GL = i ^+ ( )
where GR = i
L
+^+ ( ) R
jGR j
[
2 ( )]MR and jGL j
[
2 ( )]ML . Therefore, the lemma conditions: limjwj!+1 wf (w) = CR (w 2 R) and limjwj!+1 wf (w) = CL (w 2 L), lead us
Rx
to GR ! 0 and GL ! 0, as ! +1: This implies that lim !+1 C R f (w) dw = i CR
Rx
and lim !+1 C L f (w) dw = i CL so that
1
2 i
Z
1
f (w) dw =
lim
2 i !+1
C1
"Z
x
CR
f (w) dw +
Z
x
CL
#
f (w) dw =
CR + CL
2
R
where C1 = fw j jwj = +1g. Since C1 f (w) dw = 2 iResjwj=+1 f (w), it follows
from (14) that
+1
X
CR + CL
Resw=wk f (w) =
.
2
k=1
B. Sarić
41
REMARK: Clearly, the circular arcs C R and C L form the circular path not until at
R
in…nity. Thus, if the limit lim !+1 C f (w) dw; where C = fw : jwj = g is a circumference that encloses the n singularities of f (w) in its
R xinterior, does not exist, then
Rx
it is more convenient to use the partial limits lim !+1 C R f (w) dw and lim !+1 C L
f (w) dw; as was done above.
So, if the limits (12) do not exist on the straight-line Re(w) = 0, but only strongly
in the open right half-plane Re(w) > 0, then the in…nite sum of all residues of g(w; ^) =
R
^
e iw(i z) f ( iz) dz=(1 e 2 w ), which is a meromorphic function with countably
=
in…nite set of isolated simple poles wk = ik (k = 0; 1; 2; :::) in the w-plane, does
exist in some sense more general than the Cauchy one. In symbols, for any ^ 2 ( ; ),
for which the conditions (12) are satis…ed in the half plane Re(w) > 0,
+1
X
k= 1
Z
1
2 i
ek(i
^ z)
f ( iz) dz = lim+
!0
=
f (^ + ) + f (^
2
)
.
(15)
Clearly, if f ( ) satis…es the Riemann-Lebesgue lemma condition, more precisely if the
Fourier coe¢ cients (3) for k ! +1 tend to 0, the region of convergence of (12) extends
to Re(w) 0. Now, to answer a¢ rmatively the question from the introductory part
of the paper, it is su¢ cient to prove the result in the following form.
LEMMA2. For z = + i let a single valued complex function f (z) be analytic in
the z-plane with the exception of a …nite number of isolated poles z1 ; z2 ; :::; zn inside
is not a singular point of f (z), then for Re(w) > 0 there holds
= . If ^ 2 int=
" Z
#
iw(i^ z)
lim w vt
e
f (z) dz = if (i^)
jwj!+1
where =
^
= fz :
=
= 0; 2 [
^
; ^]g.
PROOF. Suppose, without loss of generality, that ^ > 0 and all poles z1 ; :::; zn
of f (z) lie in the interval ( i ; i^). Chose > 0 to be small enough so that the
^
endpoints of =
and the semi-circumferences + = fz : jz z j = ; > 0g and
= fz : jz z j = ; < 0g ( = 1; 2; :::; n) are disjoint. If we apply the rule
R
^
for integration by parts to the integrals l eiw(i z) df (z), where l0 = [ i ; z1 i ],
l = [z + i ; z +1 i ] ( = 1; 2; :::; n 1) and ln = [zn + i ; i^], as well as to the
R y iw(i^ z)
Rx
^
integrals
e
df (z) and + eiw(i z) df (z), then (5) yields
lim+
!0
=
"
[f (i^)
n Z
X
=0
e
x
e
iw(i^ z)
l
w(^+ )
df (z) +
n
X
"
=1
f ( i )] + iw vt
( Ry
Rx
+
Z
=
^
^ z)
df (z)
^ z)
df (z)
#
eiw(i
eiw(i
eiw(i
^ z)
#
f (z) dz .
(16)
42
On a Generalization of the Laurent Expansion Theorem
It is easy to prove, by the complex mean value theorem [1], that
lim
jwj!+1
n Z
X
=0
x
eiw(i
^ z)
df (z) = 0
l
whenever Re(w) > 0, and
lim
jwj!+1
n
X
=1
( Rx
Rx
+
^ z)
df (z)
iw(i^ z)
df (z)
eiw(i
e
=0
in the region of w-plane in which Re(w) 6= 0 and
Re[iw(i^
ei )] =
Re(w) (^
sin )
Im(w)
cos
Re(w)
< 0.
In view of the fact that ^ > > 0 there exists a positive real number k such that
^ = (1 + k) . Hence, the previous condition reduces to the condition
Re(w)[(1
sin ) + k
tan ' cos ] > 0
where tan ' = Im (w)=Re (w). As jsin j 1 and jcos j 1, then Re(w)[(1 sin ) +
k tan ' cos ] > 0, whenever ' 2 ( =2; arctan k). Since k tends to +1 (arctan k
tends to =2) as tends to 0+ , it follows …nally that both sums of integrals converge
to zero in the region Re(w) > 0 of the w-plane. This last result combined with (16)
implies that for Re(w) > 0 there holds
" Z
#
^
lim w vt
eiw(i z) f (z) dz = if (i^).
jwj!+1
=
^
By using the results of Lemmas 1 and 2 we are now able to extend an existing class
of functions which can be represented by a Fourier series, as follows.
THEOREM 3. For z 2 C, let a single valued complex function f (z) be analytic
in the strip D = fz = + i : 2 ( 1; +1) ; 2 [ ; ]g of the z-plane, with the
exception of a …nite number of isolated poles inside = . If ^ 2 int= is not a singular
P+1
^
point of f (z), then its Fourier series k= 1 c^k eik , with the coe¢ cients
8 R
^+i
Z
1
1 < ^ i f (z) e kz dz for any ^ < 0
c^k =
vt
f (z) e kz dz =
(17)
R
2 i
2 i : ~+i f (z) e kz dz for any ~ > 0
=
~
i
for k = 0; 1; 2; :::, is summable and has a sum equal to the function value at the
point ^. In symbols,
+1
X
^
c^k eik = f (^).
k= 1
B. Sarić
43
Furthermore, this still holds for all points ^ 2 intD on either the right or the left
of = , in addition to the fact that if ^ lies to the right of = , then
c^k =
Z
~+i
f (z) e
and if ^ lies to the left of =
c^k =
kz
dz (k = 0; 1; 2; :::) for any ~ > 0,
(18)
dz (k = 0; 1; 2; :::) for any ^ < 0.
(19)
~ i
Z
, then
^+i
f (z) e
kz
^ i
The theorem proof, see [4], is omitted because it is an immediate consequence of
Lemmas 1 and 2 as well as a formula for the expansion of functions into an in…nite
series derived from the Cauchy calculus of residues, see [3]. In view of the non-univalent
conformal mapping ez = s the result of the preceding theorem implies the result in the
form of the following theorem representing the extended Laurent expansion theorem.
THEOREM 4. Suppose that a single valued complex function f (s) is analytic in the
s-plane with the exception of a …nite number of isolated poles on the unit circumference
C1 . Then f (s) has a Laurent series expansion
f (s) =
+1
X
c^k sk ,
k= 1
at all regular points of f (s) lying on the unit circumference with the coe¢ cients
1
vt
c^k =
2 i
Z
C1
1
f (s)
ds =
sk+1
2 i
( R
RCR
f (s)
ds
sk+1
f (s)
ds
Cr sk+1
for any R > 1
(k = 0; 1; 2; :::) (20)
for any r < 1
where Cr and CR are circumferences with center at the origin and radius r and R,
respectively.
Furthermore, this still holds for all points s inside and outside the unit circle, in
addition to the fact that if s lies inside C1 , then
c^k =
1
2 i
Z
Cr
f (s)
ds (k = 0; 1; 2; :::) for any r < 1,
sk+1
(21)
f (s)
ds (k = 0; 1; 2; :::) for any R > 1.
sk+1
(22)
and if s lies outside C1 , then
1
c^k =
2 i
Z
CR
44
3
On a Generalization of the Laurent Expansion Theorem
Examples
We consider an expansion of sin = (1 cos ) in a Fourier trigonometric series on the
interval [ ; ].
R As f ( ) = sin = (1 cos ) is an odd function, the Cauchy principal value of
f ( ) cos (k ) d is equal to 0, for each k 2 @0 , where @0 is the set of natural
numbers plus the number
zero.
R
R
In addition, since 0 [sin [(k + 1=2) ] = sin ( =2)]d = and
cos (k ) d = 0, as
well as limz!0 zf (z) cos (kz) = 2, for each k 2 @0 , it follows that
( Rx
Z
Z
1
1
1
i
+ f (z) dz
Ry
vt
f ( )d =
vp
f ( )d +
lim
=
(23)
a0 =
+
i
f (z) dz
2
2
2 !0
and
ak
=
=
1
1
vt
vp
Z
f ( ) cos (k ) d
Z
f ( ) cos (k ) d +
1
( Rx
+ f (z) cos (kz) dz
Ry
lim+
=
f (z) cos (kz) dz
!0
2i
(24)
2i
for k 2 @ where + and
are semi-circumferences centered at the origin and of a
small enough radius in the right and the left half plane of the z-plane, respectively,
as well as that
Z
Z
1
1
sin sin (k )
d =
sin (k ) d = 2 (k 2 @).
(25)
bk = vt
cot
1 cos
2
Accordingly, a Fourier trigonometric series of f ( ) can be expressed by the following
functional form
#
"
+1
+1
X
X
sin
cos (k )
sin (k ) i 1 + 2
=2
1 cos
k=1
k=1
for every 2 [ ; ] and 6= 0. Separating the real and imaginary parts in the
preceding equation, we …nally obtain that for every 2 [ ; ] and 6= 0
sin
1 cos
=2
+1
X
sin (k ) and
+1
X
cos (k ) = 0.
(26)
k= 1
k=1
In the complex form, for every
At the endpoints of [
2 [ ; ] and 6= 0,
P+1
ei + 1
1 + 2 k=1 e ik
P+1 ik = i
.
e
1
1 2 k=1 e
P+1
; ] there holds k= 1 cos (k ) = 0. This implies that
+1
X
k=1
k
( 1) =
1=2.
(27)
B. Sarić
45
The complex form (27) points out the fact that we may use the Laurent expansion
of the single valued complex function f (s) = (s + 1) = (s 1) to get the same result.
Namely, since
8
2
>
>
; if k < 0
>
>
0
>
>
Z
<
1
s + 1 ds
1
=
vt
; if k = 0 ,
k+1
1
>
2 i
1s
C1 s
>
>
>
0
>
>
; if k > 0
:
2
where C1 is the unit circumference in the s-plane, it follows from Theorem 4 that for
every jsj = 1 and s 6= 1
P+1
s+1
1 + 2 k=1 s k
P+1
.
=
s 1
1 2 k=1 sk
P+1
Clearly, this is the same as (27). Furthermore, k=1 s k = 1= (s 1) for jsj > 1 and
P+1 k
s) for jsj < 1.
k=1 s = s= (1
An expansion of 1= (1 cos ) in a Fourier trigonometric series on the interval
[ ; ].
On the one hand,
Z
d
a0 = vt
1 cos
( Rx
Z
Z
dz
+
d
d
0 (1 cos z)
Ry
+
+
]
= lim+ [
dz
1 cos
1 cos
!0
(1 cos z)
0
=
sin
lim+ [
cos
!0 1
2 sin
1 cos
+
1
sin
]=0
cos
(28)
i
where +
: jzj = and 2 [ ; 0]g and 0 = fz = jzj ei : jzj = and
0 = fz = jzj e
2 [0; ]g are circular arcs bypassing the second order pole at the point z = 0 as a
singularity of f (z) = 1= (1 cos z).
On the other hand, for each k 2 @ the function f (s) = sk = (s 1) is a meromorphic function having at the point s = 1 a simple pole as singularity. Since
lims!1 (s 1) f (s) = 1, it follows from the Jordan lemmas that for any k 2 @
Z
Z
2
sk
2
ieik
vp
ds =
vp
d
i
1
i
1 e i
C1 s
Z
Z
cos (k )
cos [(k + 1) ]
1
vp[
d
d ]
=
1 cos
1 cos
= 2
where C1 is a unit circumference in the s-plane. Similarly, as the functions z cos (kz)
and z sin (kz) sin z= (1 cos z) tend to 0 as z ! 0, then for any k 2 @
( R x cos(kz) cos[(k+1)z]
( Rx
sin(kz) sin z
dz
+
+ [cos (kz)
1
1
1 cos z
1 cos z ]dz
0
0
R y cos(kz) cos[(k+1)z]
Ry
lim+
=
lim+
=0
sin(kz) sin z
2 i !0
2 i !0
dz
[cos (kz)
1 cos z
1 cos z ]dz
0
0
46
On a Generalization of the Laurent Expansion Theorem
From the preceding two results it follows further that
Z
Z
1
cos [(k + 1) ]
cos (k )
vt[
d
d ]=
1 cos
1 cos
Since
1
vt
Z
cos
d =
1 cos
2+
1
vt
Z
d
1 cos
2.
=
2,
we obtain …nally that
ak =
1
Z
vt
cos (k )
d =
1 cos
2k (k 2 @) .
(29)
As sin (k ) = (1R cos ) is an odd function, the Cauchy principal value (vp) of the
improper integral
[sin (k ) = (1 cos )]dt vanishes, so that for each k 2 @;
vt
Z
sin (k )
d = lim+
1 cos
!0
( Rx
+
0
Ry
0
Since limz!0 sin (kz) = (1
sin(kz)
1 cos z dz
sin(kz)
1 cos z dz
.
cos z) = k for each k 2 @, it follows …nally that
Z
1
sin (k )
ik
d =
(k 2 @) .
bk = vt
ik
1 cos
[
Therefore, a Fourier trigonometric series expansion of 1= (1
; ] ( 6= 0) is as follows
1
1
cos
=
+1
X
2
k cos (k )
k=1
i
+1
X
(30)
cos ) on the segment
k sin (k ) .
(31)
k=1
Separating the real and imaginary parts in the preceding equation, we …nally obtain
that for each 2 [ ; ] and 6= 0
1
1
cos
=
2
+1
X
k cos (k ) and
2[
+1
X
; ] and
ke
ik
=
k=1
At the endpoints of [
k sin (k ) = 0.
k= 1
k=1
In the complex form, for each
+1
X
; ] there holds
+1
X
k=1
ei
.
(1 ei )2
P+1
k=1
k ( 1)
6= 0,
k cos (k ) =
k+1
= 1=4.
1=4. Therefore,
(32)
B. Sarić
47
As in the previous examples, the Laurent expansion of the single valued complex func2
tion s= (1 s) comes to the same result. Namely, since
8
k
>
>
; if k < 0
>
>
Z
0
<
ds
1
1
0; if k = 0
=
vt
,
2
>
2 i
s) sk
C1 (1
>
0
>
>
; if k > 0
:
k
where C1 is the unit circumference in the s-plane, it follows from Theorem 4 that for
every jsj = 1 and s 6= 1
+1
X
s
ks k =
2.
(1 s)
k=1
P+1
P+1 k
2
2
k
Furthermore,
= s= (1 s) for jsj > 1 and
s) for
k=1 ks
k=1 ks = s= (1
jsj < 1.
4
Remark
Fourier expansions of real-valued functions
sin
1 cos
; if
0; if j j <
f (t) =
8
< b; if 0
0; if j j <
and g (t) =
:
a; if
j j
0
0
0
where 0 > 0, satisfying Dirichlet’s conditions in the closed interval [
follows. For every j j 2 ( 0 ; ) ;
Z
+1
X
1
f( )=2
[
k=1
0
sin
1 cos
,
0
; ], are as
sin (k ) d ] sin (k )
and
g( )
=
1
2
+
Z
Z
0
ad +
b sin (k ) d
0
Z
bd
0
+
+1
X
1
Z
k=1
sin (k ) +
Z
0
a sin (k ) d
0
a cos (k ) d +
Z
b cos (k ) d
cos (k ) .
0
Using of the fundamental trigonometric identities we obtain the following recurrent
formula for the Fourier coe¢ cients of f ( )
Z
Z
sin sin [(k + 1) ]
1
sin sin [(k 1) ]
2 sin (k 0 )
1
d
=
d
1 cos
1 cos
k
0
0
sin [(k + 1) 0 ] sin [(k 1) 0 ]
(k 2 @) .
(k + 1)
(k 1)
48
On a Generalization of the Laurent Expansion Theorem
Since
1
Z
Z
2
(sin )
1
d =
1 cos
0
sin
0
(1 + cos ) d = 1
0
0
and
1
Z
0
Z
sin (2 ) sin
2
d =
1 cos
0
(1 + cos ) cos d = 1
2
sin
sin (2 0 )
2
0
0
it follows …nally that for every j j 2 ( 0 ; )
"
+1
k
X
X1 sin (
0
f( )=2
1+
2
sin (k
k
0)
=0
k=1
0)
#
sin (k )
and
g( )
=
"
+1
a + b 1 X sin (k 0 )
+
cos (k )]
2
2
k 0
k=1
#
+1
aX
k sin (k )
[cos (k 0 ) ( 1)
:
k
a+b
2
+
b
0
(33)
k=1
Based on the expansion of g ( ) if a = b, then for every j j 2 ( 0 ; ) and
+1
X
sin (k
k=1
Thus, for every
2 ( 0 ; ) and
+1
X
0 ) cos (k )
=
k 0
[cos (k
k
0)
( 1) ]
P+1
k=1
+1
X
cos (k
k
( 1) sin (k ) =k =
"
lim
k!+1
(34)
sin (k )
= .
k
2
(35)
2 ( 0; )
0)
k=1
since
>0
>0
0
k=1
This implies that for every
1
.
2
0
1+
it follows …nally that for every
sin (k )
=
k
2
=2 for every
0
2
k
X1
sin (
2(
0)
=0
0
2
(36)
; ). In view of the fact that
#
sin (k 0 )
= 0,
k
2 (0; )
+1
X
sin (k
k=0
k
0)
=
2
+
0
2
.
(37)
B. Sarić
49
Since 0 2 (0; ), it would be natural to ask whether functional expressions (34), (35)
and (36), hold in the limit as 0 ! 0+ ? In other words, is the limit of a sum equal to a
sum of the limit, as 0 ! 0+ . Based on the results ofPthe previous examples, as well as
+1
on the well-known result of the series theory =4 = k=1 sin [(2k 1) ] = (2k 1) for
2 (0; ), we may give an a¢ rmative answer to the former questions. However, the
problem of generalization of the preceding conclusion stays open and can be a subject
of a separate analysis.
Similarly, if we already know that d [sin = (1 cos )] =d = 1= (1 cos ) and
d ln (1 cos ) =d = sin = (1 cos ) for j j 2 (0; ), then closely related to results
(26) and (32) of the P
paper, as well as to the well-known result of the series theory
+1
ln (1 cos ) =2 =
k=1 cos (k ) =k for j j 2 (0; 2 ) [7], is the following question,
is the derivative of a sum of Fourier series equal to a sum of the derivative of series
members, separately. This question also stays open for a separate analysis.
Some of the paper results have been predictable. So, the alternative numerical series
P+1
k
k=0 ( 1) has a sum, more precisely it is (C; 1) summable and its sum is equal to 1=2
[2], just as it has been assumed yet by Euler and Leibniz. By using this assumption they
obtained absolutely exact results. It is nothing other to be left than to prove the validity
of this assumption. As for result (32) from ExampleP
2, one can say that it is causality
+1
related to result (26) from Example 1. Namely, since k=1 k sin (k ) = 0 for = =2, it
P+1
P
P+1
k
+1
k
k
follows from k=0 (2k + 1) ( 1) = 0 that 2 k=0 k ( 1) =
1=2.
k=0 ( 1) =
Acknowledgment. The author’s research is supported by the Ministry of Science,
Technology and Development, Republic of Serbia (Project 174024).
References
[1] J. Evard and F. Jafari, A complex Rolle’s theorem, Amer. Math. Monthly,
99(9)(1992), 858–861.
[2] J. Marcinkiewicz and A. Zygmund, Two theorems on trigonometrical series, Rec.
Math. [Mat. Sbornik] N.S., 2(44)No.4(1937), 733–737.
[3] D. Mitrinović and J. Keµckić, The Cauchy Method of Residues, Vol. 2: Theory and
Applications, Dordrecht: Kluwer Academic Publishers, 1993.
[4] B. Sarić, The Fourier series of one class of functions with discontinuities, Doctoral
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of Science, Department of Mathematics and Informathics.
[5] B. Sarić, Cauchy’s residue theorem for a class of real valued functions, Czech. Math.
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50
On a Generalization of the Laurent Expansion Theorem
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