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ISBN 86-17-06506-0
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· '!' , ','; • • .1 n
l'<' . l'IH
! \' , . .
\: • . . .
"'.\ , . .
. .
.•1. .
'";1
,
1926. zogue i' Peiiiopai'iia Kapaaiue goili.opa
( )
~

g 2 ll Kllll
, . .

(z)
,
, Stieltjes1 :
1. .f(z) z
f mod .f(z) dz z2
z = R ,
(1) _!_ ~ z.f(ti) dt = f(z) 7t t2 + z2
z . 2. /(z) z

mod f~z) dz,
, R ,
(2) _ -~ t.f(ti) dt = f(z) 7t. t2 + z2
* Generalisation de certains jormules de Stieltjes, Rendiconti del Ci1·colo Matematico di Palenno, Paleimo, 1903, t. XVII, .. 327-334.
1 Sur / developpement de log () (Journal de Mathematiques pures et appliquees, 1889, . 425-444). .
12
z.
. , ,
Stieltjes-a2 Hermite-a (z) . h Stieltjes-oe ,
, .
* 1. f(t) (z,t) z
t h
f(t)(z,t)
z t h t .
Tagaje
() f(z) = __ 1 f~(z,A+ti) f(A+ti) dt · 2n -~ (z, z) + ti- z
g . z g. h. og . ,
(4) = - 1-. (z, t) f(t) dt 21tl t- z
1 L = L.
R z ,
= f(z)(z,z).
= 1 + 2
, 1 , , , 2 , .
1
2 loc. cit.
S une exrensiun de / fumule de StiJ·!ing (Mathematische Annalen, Bd. XLI, 1895, . 581-590).

n n . -- -,

2 2
2 8i
1 = - 1 f (z, + Re8;)f(A + Re8;) Re . d8 2n + Re8
'- z
+R -R,
R 1 f . {(+ ti) 2 = -- (z,A+tz) · . dt.
2n +tz- z -R
R ,

Jl =,
limP = limQ = 1;
1 ~ . {( + ti) = -- (z A+tz) · dt 2 2 , '\ . , n ~~. +tz- z
(3).
13
2. f(t) (z, t) z t
f(t) (z,t)
14
z t t .
Tagaje
(5) f(z) = _1 (z,A+ti) f(A+ti) dt 2n_~ (z,z) A+ti-z
gi z g og . he
2 = .
. f(t) (z,t) t z ).1 _
f(t)(z,t)
z t ).1 t .
Tagaje
() f(z) = _1_ ~ (z,t + Jli) f(t + Jli) dt 2ni_ (z,z) t+Jli-z
gi z 'i g h og ).1.
he 1 3 = Jl .
4. f(t) (z, t) z t ).l

f(t)(z,t)
• z t ).1 t .
Tagaje
(7) f(z) = __ 1_ ~ (z,t + Jli) f(t + Jli) dt 2ni _ (z, z) t + Jli- z
gi z 'i g og ).1.
, , 1 4 = Jl .
15
5. () Stieltjes-oe . ,
= , 2z
(z,t) = -, t+ z
= , 2t
(z,t) = -, t+z
.
(), (5), (), (7) .
, , X(z) z
ezx(z)
z ' . , ,
f(z) = 1, (z,t) = e-1X(t)
(8) f~ ( + ti) e-ti ~t = -2ne.-zx(z) + tl- z
. x(z) z poga, g pegocill z g. h. og .
(8) = n z,
(9) f~ e-tix(ti) dt = (-1)11 2n d" [e-zx(z)], ~ (z- ti) 11 +1 1· 2 ·· .... n dz"
" Taylor-ooz pega
zge . g . g., .
(10) ( -1)'1 ~ e-ti .
~
16 )'
t -t, (9)
(11) = eti ( -ti) dt = ( -1)"-1 27t dn-1 [e-zx(z )] . _ (z + ti)" 1· 2 ·· ... · (n -1) dz"-1 '

.
(), (5), (6), (7). ,
.f(z)
=
< (z, t)dt,
< . , , < (z,t) t, .f(z)

.f(z) = l = z<p (t) dt 7t t2 + z2

Stieltjes ( < (t) .f(ti), t) .f(z), z
z
z 1t 1t
-- -. 2 2

Stieltjes-o
(13) .f(z) = lJ~ t\jf(t) dt n t2+_2 u L.
17
( \jf(t) .f(ti), t\jf(t)
) .f(z), z , z -
n -- -.
2 2 (13) z,
cl"f" 2~ /"( 1) -· =- t\jf(t)- clt. clz" n clz" t2 + z2

d" ( 1 ) 1·2·3· ·11 [ t] - 2 2
= ( -1)" .. · 1
t(t2 +z2) 2
(14)
f\jf(t) z dt = (-1)"-1 ~ .f "- () '
0 ~(t 2 +z2 )" 21·2·3· ... ·(n-1)
~t g f(z) . g . z . . g., g z paciiie . . z-
:. n n . . . .u. g. .,_" - - - ; z = tl.
2 2 , , : u An
l-2 pega
18
(18)
g~u ga 'i geo f(ti) og t pega iii 11 TayloY-oo'i iii (15) f(z).
, , .
/(z), , z =. \jf (t)
t =, t t = . - \jf (t) t
t , (18)
1\jf(t)l < Mt.

l l < 2 f~ t dt n n+l ' 7t -
o(t2+a2)2

(19) 2 IA 11 1 < 1 ( -l)na"- (n> 1).
u l-'i pega (15) , g, iii g og og'ioapajyhux l-'i pega -
2 ( z) --;- (z- ) log 1 - ~ .
(17)

ai"ctg !__ = ,
(20) ] 11 = --1 \jf ( tg ) sin (+ 1) cos"- 1 xdx, "

19
h
11 li- (15).

z, , ,

(1) F(z) = R(t, z) dz L
R z, t, L.
, Lt F(z) , L! 2 , Lt 2 (1).
, Caucl1y-e
L
(1) R, z, . .f(z)
< 1 ( z)' < 2 ( z), <, ( z), ...
( h
), Caucl1y-e
(1), he -
* , , . LXV, , . 25, 1903. . 79-1 62.
21
z, .

,
(1). Stieltjes . .1
F(z) , z,
, , z ,
F(z) , z z
(2) F(z) = 2 ~ z<p (t) clt, n t 2 +z 2

< (z) F(ti). F(z) , z
, z
() F(z) = 2 ~ t\jf (t) clt, n t 2 + z2

\jf (t) i F(ti). 2 ,
F(z), (1).
, : F(z), 1 z 1 < 1,
(4) 2:
0 1 - 2z cos t + z 2
(t) sin t, cos t,
l Sulc dcveloppcmcnt / log () (Jounal d. math. pues et appliquees 1889 . 425-444).
2 gl g/) z_,r ( LXIII, . . .).
3 Quclqucs .fonles geneales elarives azt calcul clcz inregales definies (Rendiconti del Cicolo mat. di Palemo 1897. . 252.).
22
F ( cos t + i sin t) .
, (2), (3), ( 4) .
F(z)
(5)
"
() "= A(t)[rCt)] 11
dt, L
A(t) r(t) t, .
.
~

( ),

" .,4 -~ '
~
dt . - 7t u
, 11 ,
. , F(z) z = , Caucl1y-e
= _1 F(t) (l)"dt " 2ni t t
L
4
4 . n. l: Lems sules.f'onctions cnriees . .
23
1 - 11 2Jt ( . )" "=~ [ (,t) -
1 -_ dt,
(1·, t) F( 1;), F(z).
.
., Stieltjes5, Borel Le Roy7. " (), h
(7) F(z) = A(t) dt L 1 - z(t)
(1). F(z) z =,
(5), z, h (z- ),
, F(z) .
h ,
(1). .
1
F(z) = log R(z), F(z) = arctg R(z),
R(z) ,
1
F(z) = (t, z) dt,
(t, z) z, , ,
(1). 8
5 Memoie sur les.fi·actions continues (Ann. de ! Faculte de Toulouse t. 8-9. 1894-1895).
6 Le~·ons SUJ' les seies dil•egentes (Paris 1901).
7 Sur les seies di,,agentes etc. (n. de ! Faculte de Tou1ouse 1900. . 17-4).
8 He1mite: Cous litlgapl1ie, 4-rne edition . 11 .
24
-~ At2 + 2Bt + -) - 2 '
, , t, h (1), .
, , ga g g F(z) 2 (1), 2 g ahu 2 2 2 Luua, he g .
, ,
R(t,z), z, , z, ,
f R(t, z) dt = . L
, , L , P(t,z) Q(t, z) .ru: z, t,
P(t,z)=oo, Q(t,z)=O
1 zl < ( ), t, L, h
f P(t, z) clt =, Q(t, z)
1~
(z) 1 zl <. <r L (0, ),
t, R(t, z)

1 z cos t - z _ 1 z t 2 ~ ( )" -- --- - COS/1 1 + t 2 2 - 2z cost + z2 1 + t 2 ~
( ), he ,
11,
JR(t,z)clt=O ()
\ 25
z, .
q:>(t), , n= , 1, 2 ...
q:>(t)t"clt = 9 ()
h R(t, z), he .
R L, goga jyhu , zt z g z ztz (1) xohe , uahe xohe (1) iu g g ucle gaLte F(z).
, , ,
(1), , z gt
t z. , t, , , Lt umte
zpau z geuucaLtu , Lt g
LigLt, h
. , ,
.
' 12

~ - 1 - - 1' 1 + ze-1' 1 (1 - e-at) dt
1- ze-ar t
( ),

9 \: Lc~:ons Slt lcs seics diPagcntcs (Pais, 1901. . 67-68).
26

g. h .e~uoga g F(z), g g. 2. (1), ga u g . . Lugu..
, , h , ,
F(z) (1).

L
R z, . t.

R (z- )", ;
h ,
.
z = R(t, z), he z = t, h, , t = ~ 1 , ~ 2 , ~3 , ••• t. , h
t ( t = ~;) =
(8) R(t, z) = L \jf(n,t)z" =
he \jf(n,t)
(9) , (, t)[1; (t) ]"
1; (n, t) n,
,
27
t; ~-(n,t) z, R(t, z), z. ; n h ; -1, ; z = 1;(t) .
, \jf(n,t) he
(10)
Aj(t) 1;(t) n Ai(t) , 1;(t) R(t,z), z.
h, , L ~; , (10) h f(z)
(11)
" pega
, h, g, 2 2
n"J(n),
2ge J(n) gaiiia (12), u .
, ,
(12). ()
, f(z) z = . ()
, ,
n, . ogpef)u ae an g 2
28
{ 1 (t) [t (t) ]" + 2 (t) [2 (t) ]" + .. . }dt, L
zge g lz z.
z) =~[~- -r _, dt 1- z 1- ze t

, ," ; ,
11 = log(n + 2)
*
(13) .() = A(t)[(t) ]" clt L
u cogu ".
, ,
n. :
I. .
(14) = A(t) [(t) ]" dt
n t, , . . (14),
!' !'
/ + 1 = 11 - clt.
clt
h (13).
29
. . L . A(t) 1·(t) , .!()
n, , , f(z) . , , ,
A(t) (t) . 1·(t)
, A(t) . h .()
, he .() ca ciiiojaiii 22 , he u
(15) n(n -1) ... (n -11) ",
zge , ogn.
1·(t), t = q- ; he !(n)
(16) 1 (/ 11 9-
(nq -1)! dt'-

n:
1 (t) = -;
t
= , q = 1 (16)
1 d"- 1

. 11_ 1 , 11
A(t); , , .
,
A(t) (t), he () (15) (16).
. J(n) h
. h h
() :
f~ _,,r . dt sat- = actg
t n
~
1
f t2 " dt = n 1 ·· 5 · ... · (2n -1) ~ 2 2·4·6· ... ·211

f r(1)"d 2ni ~ t = 1 · 2 ·· ... · ( -1)
( ) .
31
, h
(n) , , ,
, ,
, ,
.
, f(z), u LUU zt g, h uuezpae
J(n) g g -t g. ,
,
. hy .
)
g , :

L
(18) IA(t)l ~ (t), l(t)l ~ lu(t)l,
h
(19) 1 J(n) 1< 1 (t) 1 [1 u(t) 1]" ds,
L
s . h (t) u(t),
A(t), t, h (19),
J(n). hy , .
, , he J(n),
(t) = const.
h , r(t) L. h () ;, h .
32
L', L h , minimum i. Darboux 10 h ; , g zog umuezpa
(20) = f A(t)dt L
. og g uhe
(21) J(n) <".
(t) minimum maximom, h , L, , (21) h .
.
1 [ ]" () = f A(t) t;l_-2 dt,
A(t) t, , (20) , , .
(t) = t(l- t) t-
11 , L, t = t = 1,
(22) - 1 - 1 - 2 ± 2~( -1)
1 Memoie S/11' /'appoximation c/es .f'onctions de res-ands nom!Jl'r!S er Sl/1' l/11r! classe crencluc / dc\•eloppcmcnrs scie (Jounal clc mat/1. pues ct uppliquees 1878. . 5-56 i 377-416).
11 Daboux: . cit.
33

h 1. he
() <",
(20), (22). ()
, f
t. .
~
(23) J(n) = A(t)[<t)]"clt,
A(t) (t) t , , , k, 11, :
.
(24)
O(a2+t2)2
,
. .f'(z) z,
: 1. z; 2. z ; . z . 12
12 g g. 2-"r ( LXIII, . . ., . 225-,-226).
34
.f(ti) = < (t) + i'l' (t)
,
~ "
(25)
.f(z) z =
(26)
F(t) '1' (t). .f(z)
z, z =, \/ (0) = ; , , - z z = ,
\/ (z) z . - z
h N, h
F(t) < Nt,
(26) (24), :
(ll g An ( n> 1) pega (25) og g
2N (n -l)rra"- 1 •
(27) () = A(t)[Ct)]"clt L
L , he , t = . he he , . R1 R2 •

t =+ 8;,
2Jt
= modA(a + 8;)
35
() !!() , 8, 27t, R1 R2
~ (), S ~!!().
(29) I.J(n) < 2()(!!)]",
R1 R2 •
(30)
R1<p< <R2 • he n, \jf(n), h
(31) I.J(n) 1 < 27t\jf(n).
: 1. ()
( R1, R2 ) = R1 = R2
. he
(32)
(33)
2. ( ) , R1 R2 •

'() =; he n ro(n), h
36
R1 < ro(n) < R2 •
R2 = ( he , A(t) (t) ), he (34) n h R1•
og , g 2 2 g .g 2 J(n) h 27f (n), 2ge g \jf (11) g g. og (32), (33), (34).
. A(t) J'(t) : ty k, !1, ,
1
(t) 1 ~ }2__ "
( t), t .
he, , A(t) gu 2 2g g 2og pog (gene) : i ~- h 1 +, A(t).
he
- + 1 =
~ IO(n) ~ (; 1 ); ,
() .
np-i
(; 1 )"'~-'
37
, R, h i5() > R n > ] ]
, og z n = 11 h u
" 1 () 1 < ---~~,-,_-,--1 '
(- 1)
zge uu, , zu, og g
= 2kt 1 '
( )
h A(t) (t), , .J(n).

g 2
.!(). . z- z = z = , A(t) (t) , (t) f.l(t) , < t <
(t) ::; A(t), u(t)::; ,-(t)

(35), h ().
) ()
, Ossian Bonet 1 3 . .
:
< (z) , (, ) , .f(z) ,
1 Picad: Analyse t. 1. . 20 !.
38
,,
.f(z) < (z) clz =< () .f(z) dz,
, (, ). < (z) ,
, h h
.f( z) < ( z) dz = < ( ) ( z) dz.
J(n), (t)
(, ) ( )
'(t) ,
,, (36) = A(t)dt
, t
(, ),
. (37) J(n) = ",
(36), , 1·(); , , '(t) , (37), he
t
= A(t) dt < t < ,
he (). , g, N u g z g g 'i gL
zuao g Lz , h
(38) NB" < () < ".
) L , A(t) (t) , , . .() Scl1waz-oy , < \lf, , 14
14 . . i<~: !!i du poretiele\\'fonie (P<~ris 1899) . 345.
39
(39)
, h
.

< = A(t), \jf = [(t) ]"
z g uhe
(40) .J(n) < ~(),
zge u, og n g
(41)
(42) Y(n)= f[Ct)] 2 "dt. L
.
~
.()= J(t)e-/11 2 dt,
A(t) , l),
, h

)
40
J(n)< ~fij 1·3·5· ... ·(4n-3) -vab 2 · 4 · 6 · ... · ( 4n- 2)
.
) J(n) h , ,
.
< (z) , z (, )
, ,
, h ,
1d15 ,
(43) _....1 <.'.......:.( z..2.1c...) _+_._· ._+_....:'"'-" <.!.......-'.( z-"""-'-' ) > < ( 1 z 1 + 2 z 2 + ... + 111 z"' ) . 1 +2 + ... +111 1 +2 + ... +am
A(t) (t) : 1. , (, )
(
); 2. 1·"(t)
, (t) (, ). .i ( 43)
ZJ = t, z2 = t + dt, z, = t + 2dt
1 = A(t)dt, 2 = A(t + clt)clt, 3 = A(t + 2dt)clt ...
( ,
, ),
: g g z z
,, (n) = A(t) [(t) ]" clt

15 . . Bull. des Sciences matl1ematiCjues 1890. . 96.
I 41
uhe gt " ige Lt, og n gmue o paculYta ,, ,,
= A(t) clt, L = A(t)tclt. (1 (1
, A(t) ', h

, , R(t,z), .f(z),
(44)
11 h ,
P(n, t) [t) ]",
1·(t) t, n z, t; P(n, t) n, R(t, z), z.
(, t) , ,
(, t) = 0 (t) + nA1 (t) + n(n -l)A2 (t) + ... ,
he Ai(t) t,

42
P(O,t) = 0 P(l,t) = 1 + 1 (2, t) = 0 + 21 + 22 P(3,t) = 0 + 31 + 62 + 63

11 ( 44) ,
(45) n (n -1) ... (n- k)A(t) [(t) ]",
"
F(z) = R(t,z)dt = 0 +a1z+a2z 2 + ... L
F(z) ,
n ( -1) .. . (n- k)J(n),

() l(n) = A(t) [(t) ]" dt. L
A(t), , n, A(t) L{Ul-, (t) z R(t, z), z.
, , F(z)
~

lk-1 ~
z ~ :L.,.r(n)z", clz 0
,
F(z):
uiUezpau u (), u g
F(z),
43
8(z) = I J(n)z" uuuo ogzoapajy Lu z, he F(z) u u z 22
81 (z), /8 1 2 d281
7- z -- ' - clz ' dz 2
82(z), /8 2 2 d282 z-- 7--
clz ' - clz 2 '
,
.f(z) .f(z) , R(t,z) , 8;(z) : , h
F(z).
, . .
.f(z),
A(t) = e-at 2
, (t) = -!/ 2
,
L ,
z.
~ 11
A(t) = sinat, (t) = -1, t
L ,

(t)=-, t
44
L , , h .f(z) z : .

: .f(z), R(t, z) (z). :
.f(z), :: ::, , :: -

,
(z). , :: F(z) , (z),
, .f(z) .
(z) paHCl!el--lgerle . , , :: ::
z. (t), (z) cogu t, h
() = ",
, (z) h
~
1- az
he g t2 L , ltlt og2oapajyha (t) , A(t) 2 . , , he () ,
n (n -1) ... (n- k + 1)HK"-k,
n. (z)
A---clk ( 1 ) clzk 1- Bz '
he (z) .
45
8 (z)
.f(z) , ,

8(z) = .J(O)+J(l)z+.l(2)z2 + ...
(47)
1 , 82 , ,, ... .f(z) R 2 2 ca.Nto2a pega (46) uhe og (47). ,
1 , 2 , 3 , ...
( 47), og Lliux gLt gLi h g g 2 2 R.
, , 8 (z). , .()
n, 8 (z): he g gpy2oj og gt
(48) 1 . ()
()
(),
Ltt gLi
n, . ()
, n=
lim () = 1. U(n)
Lt he 2 2 pega (z) Li g gpy2oj og pegoauu
(49) 1 . U(n)
46
, h h r J(n) h J(n), , h :
q> () , n
q> ()+ q> (l)z + q> (2)z 2 + ...
r , g , g , h g g 'i .
, L

(50) = 1 A(t) 1 cls L
( s ) , h (z) h
J(n) . -
, h
r , ..!. h 1.1
.
oLuLu g ..!. g- 1.1
g 'i .
, he u g 'iog A(t) (t) ,
g, uue'ipauje L cogu g 'i 'i geo , . (, ) pegoauu.
, (t)
t = t = (
47
), . (37) h
() = ", ,,
= A(t) dt, = (), < t < .
, , , h
, 1·(!) h (, ) (),
, ,
fl = l'(b),
1 =-.
fl
'(t) (, ),
() = ",

1 --
- 1·() ·
h (t) (, ) (),
1 =-.
(t) '(t)
(, ).
, .
h J(n) " () he
48
. () h h .
h h (t) (, ), he
1 =-
!1
(t) .
J.l ,
, : g 2og 2 L cogu g geo (2 ) g A(t) (t) g u2
f A(t) dt L
og g og2oapajyha (z) u 2{Li z.
h he . 1 .
-,
!l 1·(t) -
.
L , ; t = , L : 1. '() =; 2. h (t) .
' +" --6- 1 1 aga ne =-.
!l Daboux-a 16

, n h
16 Loc. cit.
_ [()]" J(n)-N ;; (l+E11 ),
n
N , n, 11 , n .
= lim (n) = _1_ ~n + 1 1 + " = J(n+1) () n l+E11 +1 1·()'
*
, 8 (z), , , .f(z), J(n), -
-
n.
. Laplace, , h A(t) (t) L, , 17
n, <() , 11 n, n .
D~boux Laplace-o
,
, 18 . A(t), h L t = , , '() = r(t)
17 ·s t. . 18 Loc. cit.
50
, . he gu u2 ()
(51) [() ]" N,
N = -2n () . "()

, t =, , ,
l(t)l=l(a)l,
(51). .Lu he g 2 J(n) <t.
.
~
.J(n) = tk(te-')''dt,
(t) (0, ) t = 1,
1 (l) = -.
1. {27t, ~--;-
==2,7182 ...

1 [ ]" () = J.f(t) t;l_-2 dt, ()
.f(t) t, , ,

Af(a)
".[;{'
= l_~(l- B2)n 2
= 1 - 2 + 2~( - 1) ;
h 8(z) h
= 1- 2 + 2~( -1).
51
t,
.f(t) t"dt, .f(t) (7 clt
Flamme19 Hamy20 .()

F(t), .
Le Roy21
~
(52) .J(n) = A(t)e"1dt,
A(t) . :
<p(t) = -logA(t),
1. <p(t) , t , ;
19 Rececl1es des expessions appoclu!es etc. (These de doctorat, Paris 1887).
20 S / developpement l de lajonction petubatice etc. (Journ. de math. pures et appl. 1894).
21 Valeus asymtotiques de certaines seies etc. (Bull. des Sciences Math. t. 24. 1900. . 245- 268).
52
t

t . . he g 'i J(n) g
(53) --1/-<()~ <"()'
'ige g
(54) <p'(c)=n.
(54)

( /1)p~l = - ;
An1'B"k,
, , !1, k , n. (52) ().
. (t) t = , , t = .
he ,,
(55) () = A(t)[(t)]"dt
(52). hy h

~

A(t) ( ) z poga. A(t)
(57)
t . h (57) e-mczt h (, ) ( n, n = 0),

= f tk ,-111 dt = 1 . 2. ..... k k k+l n

n - + 2 + ' + ... n n -
A(t) , he in5-
lik!ak = k =,
he (58) - 1 .
-,. n. n 0 , h n =
li () = 1· ,
li () = 1 1 ' . " "+l
: g t = g A(t) =, -
- - - - - ( ) - () . - t pegocutu uzuezpaa n --; , , = -
54
-2 pega .l g, .ll he g l- 2 l
2 og2oapajyhe2 pega (z) jegumu.
.
A(t) = sin Jt Jt
; () = 1,
J(n) = siftft e-1/(dt
h ..!_, h (z) 11
h 1. (56)
J(n). , (t) , t = ,
, , t = , (t) = -=
(55) (56);
h (z) h _!_.
, ,
A(t) (t),
().
; .
, ,
F(z) . - . .

,
*
(59) (z) = .() + J(l)z + J(2)z2 + ...
55

z, (59) . :
A(t) 1·(t),
() = A(t) [t) ]" clt, L
h (z)' (59), z.
,

:
(z) , u g ga g g gpy2a og 2g
.
lim~.J(n) =, lim .( + 1) =. .()
A(t) = e-ar', 1·(t) = 1;, > ,
L (-, ), h
-n' -
lim~.l(n) =,
, , h e(z) . h .(), h
n, 2 g g g J(n)
~uu pegoULhy. . (z)
( ) _ ~ sin -fi -nt t .. n- -fi ,

h , - - - 1
-. n
, h ,
(n), -
56
, :
A(t) (t) g ~i ga i'i
Y(n) = LR"dt,
, ~i, g z,~eo'ia 'i
8 (z), z,i og'ioapa 'i J(n), uhe .
h :
Kag 'iog i'i
.J(n)= A(t) [(t) ' clt L
'ih geopucai:Uu uue'ipmtje L Li ga - oauajyhu Li zi - g ; ga g (t) g 'i g'i g
ga, g z,
= Jl A(t) lds L
og g og'ioapajyha 8 (z) uhe z.
h (t) , ,
, h,

h 8 (z) h
1 . . h - . , , -

57
, = , h , 8(z) . ,
.
J(n) = J~A(t)(-1-.) 11
clt, +t1
, A(t) t,
.
,
, , he (-,),
A(t) (t) .
(t) = - 1 -.
+t1
, he ogzoapajyha (z) l . .
A(t) =< (t)- 2
,
, < (t) , ,
t , . he
, .
h ,
mt 8(z) ; hy , , . 46 :
58
(z) . zuuu u- 2 g g, 2 2 geo , A(t) I(t) ., g2h 2 . og g.
(z)

(61) J(n) = A(t)[(t)]"dt, L
F(z), , ,
,
,
.
h ,
, h h
, - he l) -
, . .

~ z" e(z)=L.. , 0 ~+ bn
he .

,
1\, 8 (z) h - h
, -
.
(z)
(60), :
\ I 59
(62) e(z) = A(t) clt L l-z(t)
(60) (61).
(z) , ; z,
.
(z), . .

, ,
, :
,
; z, , . .
(z) ,
(z) z, ;
,
.
) e(z)
(z), (60).
, A(t) (t), h (), e(z). h
, A(t) (t), .,. ,
e(z).
*
h , A(t) 1·(t), h (z) -
60
z z.
,
ro(n), - n
\.J(n) 1 ~ ro(n),
, ,

z) = .L )z", z, ~(I) g h g z ' \ (z) \ gu z Lu z z pega ~(z).
,
.J(n) l) , ro(n), ;
(z). , h ,
(t) L ,
= fjA(t)jds L
, h
ro(n) ro(n) ="
ga he
~(z)=--
1-Mz
z g, gu g z zr Atogya (z) g z tu
-- ----- 1 z, tz uuu .
A(t) (t) t , , k, !1,
. . A(t) ~ kt
2 + t2
k 2 ('1)"-2 . . J(n) < -- -
n- 2
~(z) = J(O)+J(l)z+J(2)z 2 + f(z),
gu
61
gh g 2 2 .g .zu (z)
- - -- ··-- gz z Lu , -.
11 h ,
(), A(t) 1·(t)
h ,
(z) .
h h
(z) .

(z) z. ; , Hadamal'd-ooj , ~.J(n) n . he
() =[!)]",
f(n) n, n .
<() : 1. ,
62
; 2. n = ; . ,
(63)
< () , , ;
(64) <p(n) = z

n=\jf(z)
,
z ( , (64) n z).
(65) "" z" u(z) = LJ [ ]"; < (n)
,
Hadamaid22
p:z)d: -
1 z 1 . (63) :
. h g u , ga g z g
(66)
g g z g. , < (n) ;
(65) h < (n).
.
22 Etucle su les pOfnieres des joctios etiees etc. (Joun. de math. pures et 1., 1893. . 171-215).
63
, h
< (11) = ", 1 \jf(z) = -logz

g ogzoapajyhe (z) h og
2ge uu.
A(t) = e-ar, (t) = 1;
( , ), (-, ).
, .
, 2
, goux
(z), . , J(n) n,
()
.() ~ (),

~cz) = I )z",
~(z) gh g g 2 e(z) pegoca:tu z, peg ~(z) z.
(),
;
(z). , , A(t)
(t) , , ,
()
co(n) =",
, n. Taga he
64
~(z)=-p- l-Mz
- - - 1 g z . og -.

(0, ) A(t) (t) , k, h,
A(t) ~ kt,
log (1- ':) z2
g g g ipamy (z)
- g z . og - .
11
*
(z) z, z he (z).
h < (n), (63)
n= \j/(z)
<p(n) = z;
dn·d- , g 1 z 1 paciiie, he (z) paciiiu og
(67) 1 <=
(z) = ---;-d=
z . .g.. . ,
1 .() 1 ~ -"2 '
h 8 (z)

:
1. n , ;
2. n , ;
. n2"(n) n, n= .
, Le Roy23 ,
g . z g pegoauu, he 8(z) .u zu
(68) e"ro'(u)-ffi(u)
2 -" ' zge u g
'() = log z.
h , ,
8 (z), z . .
() = e_"r 1 << 2,
h
w(n)=n~', = ( lopg z );::1
~
8(z) = Ie_"rz"
23 Valeu;·s asymtotiques de cetaines series (Bull. Sc. math. 1900. . 262).
66
a(log z )k e~(logz)'' '
; , ~, k, !1 , . · 8 (z) '. he 8(z), he z, , .
r Le Roy24 r:
e(z) . hy , . . . , .
, (z) , , z az , n ;
~eh e(z). he z, , 1?
n () -
n · n . n= ' ~I;I, , 8 (z)
~ ;r::··~;, he - 1- z .· '·;.~·-"·rj; -~-: ·
() - . = ; :-)~t., , e(z)·.>;~~·:..
, Le Roy25 ,
F(n) , g n .i gLt 'i () n = , g - g z gLt 1 - he (z) t
(69) cF(-1 ) 1-z
('ige u), he . u u[u
g.

24 Loc. cit.
25 \la/errs asymtotiques de cetaines seies (Bull. Sc. matl1. 1900. . 250).
67
8 (z), z 1,
cln ~-=-;.
*
8 (z) , Le Roy ,
,
F(z). ()
(t) = , 1·(t) = 1. ()

(70)
J(n) = < () " /,
. A(t) < () -,- t
(t)
(71) 1

(71) Le Roy ; , , ; , , <p(z) < z < 1, (71) z = 1 z = .
26
26 Comptes rendus t. 127. . 654-657; Annales de ! Faculte des Sciences de Toulouse t. 1900. . 317-430.
68
(z),
,, (72) J(n) = A(t)[(t)]"dt

:
Kag zog z (72) I (t) , ga g og lz z g gu , gpyzy z gu l, ogzoapajyha (z) . . gpyzx
zLl z = 1 gLl h og jegzm-
e. , g 'i, A(t), . I'(t)
I (t) = z, g zf . z, . < z < 1, (z) . . gpyzx 'iu z = 1 z = .
(72)
(73)
1·(t) =-=,
J(n) = (z)e-mdz.
(z)
li'i poga, (), (. . 53),
1 -, -

e(z) . , Le Roy 27 :

<p(z) = 0 +a 1z+a2z2 + ...
~ , n, n
27 Comptes rendus t. 127. 1898. . 655.
69
l, <p(z)
, z = 1. 8(z)
:
Kag 2og 2 (72) ! (t) (u ga g og 2 2 g g , gpyzy 2 g 1,
g, g z, A(t) (t) = e-z '(t)
2 poga z, ogzoapajyha (z) uahe
g 21, { pegociii z = 1. .
e(z) = J(l)z +J(2)z2 +()z3 + ... ,
·R1 IS og- J(n) = t t"dt
~log~ , z = 1.
, (t) ,
k.
(t) = ku(t), kz = , e(z)
81 ()= L U(n)J",
,, U(n) = A(t)[Ct)]"clt.

u(t) , 1, :
Kag 2og 2 J(n) r (t) ga g og - 2 2 ociiiaje , gpyzy gouja og g k, og'ioapajyha (z)
. - - - -- 1 gpyzux z z = k
g ehux og l. , , ~(t) k /' (t)
70
(t) = z z 1 zl < k,
8 (z) . .ru gpyzux 'i z = _!_ k
z = . (t) = -= , he 8 (z) .
- - - - 1 cuzyapu~ueut, Luo z = -.
k h,

8 (z), . ,
A(t) (t), .
h ga ~u . g (t) g og z 'i u , gpy'iy , g g u z 'i.
h 8(z) , Desaint28 h a(t), a(n) -
F(z) = I, a(n)z"
F(z), . h , .

8 (z) = () + J(l)z + J(2)z 2 + ...
. , g . t
g cLupae
*
h .!()
, 8 (z), h z,
28comptesenclust.l32, !90l.p.!I02-1104.
; ,
,
.
, 29 .
. >- 0 <:: " , - .
, 0 , 1 ,2 , ... ,
(74)
, , , h

. z , , h
(75) 1-L = z~ ~z) h , (z). , h he J.I, (75)
: = z,
~z) , he z
1-L = ,.~ z = /' ~(~)< (76)
() dz z2
~ ~(~)>. z= ~-t=a dz z2
Mef)yiliu, zii.ao g pegociliu en gue g, g uiliezpae J(n) z uiliezpaa: gu en uhe
L( n, agpmuy f) z z g uuezpaa J(n) uu h og z agpailia.
29 z g ( LXIII, . . ., . 73-114).
72
, ";
J.l. .
A(t) (t), (z), z . he, ,
. . J(n) < ~(2),
,
1 = f[(t)] 1 dt.
L
Y(n) = f[(t) ]" dt. L
g , g, ro(n) , , i ga ~~ pegoauu n
(77)
(78) (z) = w(O) + ro(l)z + ro(2)z 2 + ...
lu ; g g z g g
z, (z), pegoCLu Jl, g (76), zge (z) ~ u (78).

~

, , h ~(l- ),
+ 3(1- ) log(l- )=
1.
73
, (), , 8 (z).
8 (z)- , ,
z, 8(z) .
z, 8 (z) , h h,
.
, 8 (z) Hadama<;l ,
,
,
8(z), ; . h
8 (z) h , J(n), Hadam:d-a
, , .
) 8(z)
O(z)
(79) O(z) = A(t) dt 1- z(t)
L
, . ,
, , .
.
O(z) = L_z_ 0 a+n
z, h , ; , , ,
.
.
, (z) , ,
h . h
, e(z) .
,, (n) = A(t) [(t) ]" dt
(/
.(), e(z). A(t) (t) (
).
,, (z) = f A(t) clt,
1- z(t)
(z) z , ga he (z) 'i g z
u ga Lueu g z u ., z ; he { uauo Lu
pegoauu z og _!__, 'ige g h
pegocLu . . r (t) l'i-
'i.

30 . . t: Su / summatiun de cataines sries com•egentes (Comptes Rendus t. 118, 1894. . 239-241).
75
A(t) 1·(t)
z , V . , , 8 (z) z = ~ + 11i, 11 = , ~~ 8 (z) .z u gu
. z.
~ < l_, ga g, 8 (z),
- - h 1 , t og -.

8(z)=I~ 0 +bn
: z, z > 1; , z , , ; 8(z) = ;
= 8(z) z = 1 .

8 (z), , .
Le Roy,
, ,
8(z)
.
, : (t)
, -
k; , , ~(t) . (t)
76
(t) = z z. (z)
1 z = k z = . he ?

(z)
U(n)= J<p(z)z"dz,
< (z) A(t) '(t)
u(t) = z. ,(~), (80), Le Roy31
< (z) ( he , ), ,(~) , ~ ~ = 1 , h
2ri.!. <(.!.) ~ ~ .
, (z) u
z, z z, g g , g z z
z = .!. , h k
2mri ( 1 ) --<-' kz kz
zge ; uzu z = .!. uhe, g, ozapu k
u zu.
.
31 n. de Fac. des Scieces de Toulouse 1900. . 328.

0 n+ 1 kz
77
< (z) = 1, - - he
. ( ) 1 2mrti z z =- --, - k kz
(z).
(z) = ~ k" z"' -7' (n+ 1)1'
,
1 1 1( 1 )"-l -=-- log- z"- 1dz
" () z
() z
- - he (z), 1
z = -, k
2mrti (1 k 1 )"-1 -- og '+ ogz , kz
.
* , h
h
F(z). h . . . ,
, ,
,
. , ,
h ,
.
*
F(z) . () F(z) z , ,. .
, ,
()-·r+l
N . ga : 1. g 'i 'i N og'ioapa g gu ; 2. gu F(z) z = ;
3. poga F(z), goyuta ga g 2 g F(z) , Ltu g z F(z)yua g gmuy gu .
,
(2)

* rfl Remaque su1·/es u!os desfoncrions enriees,. Bulletin de ! Societe mathematique de France, Paris, 1904, t. !, . 1-3.

(4) = _1 ~ = ()'". + 1'
79
,
(Bulletin de ! Societi matblmatique cle Fmnce, t. XXIX, 1901, . 303-312) : (2)
(5)
h
()
.
() 1 , 2 , 3 , ...
(3),
(7) 1 1
u(1·) = - 0 -+-" ill_L. ,.2 /'2 L..,; 2 '
1
() h -
1 1·. h = ~ h
: g g z g F(z) g
(8) 1
zge g z z N, g F(O); g g
1
8(t)
80
8(t) = I,-"1 • 1
F(z) . he h:
= , = = 2, 7183; = 8(2) = 1,0639;
·= 1,
= V3e = 2,0128; =(%)= 1,5383; ...
h (8) \ F(O)- \. z F(z) .
*

(1)
, k z, k (1), . Lg- Su les equations algehiques ( ) 1 .
, h , g .Q (z) (1) . h
. ~~, ~ 2 , ... , ~~~

_!_ = ~1 + ~2 + ... + ~11.
~; ,

.. rE: ~ .!. _!_' · ~ 0 n 0
* Su une classe de seies entiees, Comptes rendus, Paris, 1906, t. CXLIII, 4, . 208-210.
1 1's, t. l, . 33-36, 199-206.
82
: an 2 pega .Q (z) g
~(El__)" n" 0
, peg .Q(z) g z poga g.

z g .Q (z) og
2ge = 1 z 1 2ge G (z) g g
= 11
1 1
.

gaje g 2 2 g .Q (z). 2 I,.a 11 z"

g h
2 Borel, Le~·ons sules jonctions entiei·es, . 34-36.
83
h " Q(z) , , .
Q (z) h z. , '- Laplace-oe
,, u(t) [(t) ]" clt,
n
, . peg Q (z) og
zge . g
_ ~2roa0 -, '
1
, , h : . 2 pega Q (z) g h og
2ge .
1 = ft, --:= = , 696 ...
L,-2"
, , g g m pega Q (z) h og
. Petrovitch, Bull de la Soc. math. de France, t. XXIV. 1901, . 303-312.- . Landau, Bull. de la Soc. math. de France, t. XXXIII, 1905, . 251-261.
84
l-
Q (z) .
*
(1)
, z-,
() .f(z), .t;Jz) n + 1 .
u g (1) (), h : 11 , he h .
h 0 = ( h) 1 = ( .f2 ),
0 = 1, 1 = 1. h
(2) rn (7) = 711 +zn-I + 7n-2 + + 't"ll - ..... 2.... .. /
l .f11 (z) =, <p 11 (Z) z

(4) = <pn(z)
(5)
* Tlu!oi!me SUJ' les seies de l, Comptes endus, Pais, 1908, t. CXLYI, , . 272-274.
86
Oz-oca " .
h ( 4) h h
(6)
<p 1(z) = z + 1, : 1. (6) Oz-ocy n -1 , , ( 4) Oz n , -" h
~~~
(5); 2. 11 <~"' Oz n ; . 11 = ~~~, , .
,
(7)
(2), ~~~ he (
) ,
(8)
= " < 11 () . , , h :
g peg . .
(9)
(), t g ga g . g og .'i 'i g (8), cuo g n~ 2. he , , he 11 < ~~~ 11 = ~~~-
(9) () : zto peg
(10)

87
= ,
.
__ 1 3 - 54'
z2 1 .f2 ( z) = 1 + z + 4 = 4 ( z + 2)
2 '
- 4 54 54 2 . z2 z z4
14 (z) = 1 + z +- + - + = . 4 54 2379,423
= 1 (z + 19, 1172)2(z + 4,3225)(z + 1,5064). 2379,423
(10) z -=. . . h "
< 1 " n! ( 2 )11(11-I)
, , z = 8; (1·2), (z)
-./12
(z) = L_e_z" n!
=_!_ log 2, , -1, 2
, , n
(n+ 1)( 2 )11 •
I. -
1. (t) t, (, )
,, (1) g" = cp(t)cosntclt

, n.
,, (2) !()= cp(t)log(1-2xcost+x2 )dt,

log(l- 2 cost + 2 ) = -2 ,L; , l 11
( ) , 1 1
(4)
* /r! r!nentaie cl' application dcs intr!,r;ales cinics r!elles mtx ti af.r;ictes er ranscenclantes, Nouvel\es anna\es cle mathcmutiques, Paris, \908, 4 scie, t. VIII, . 1-15.
...
(5)

()
89
R n () ( (1) n). , (3) 1 1 ~ 1, :
1. R ~ 1, /() he .() 1 1 < 1; ,
log (1- 2 cos t + 2 ) = 2log + log (1-l cos t + ~)
he , lxl > 1, I(x)
(7) Q(x) = 2g0 logx+.(~).
2. R < 1, /() .( ) 1 1 < R Q(x)
lxl>l. R uz I(x) , g, g og ,
.( ), .Q (), og z ga -iii z 2
2 u . gu. .
2. < (t) /() (1) 11 ,
11 ~ , . h
, h , g [<, , , ]. , < = cost.
[<, , 2kn, ].
[<, , , k -1].
90
(11 = )
[<, , , 211 -1]. < (t) 2n
t = [<, , 2n, ], .
() -

(8) ()=-2[g 1+'~2 + ... +'; "}
R =. , , < [<,,,], () =
(9) I(x)={O l1~1, 2g0 log 1 1 > 1.
3. 2, < (t),
2n,
~ ~
(10) < (t) =+ L 11 sinnt + L 11 cosnt, 1 1

()
(12)

(13)

() = L""".
, < (t) \jf ( 1;), \jf (z) lzl < 1, 0 = "=
,r;0 =, () =.
... 91
< (t) '1f(C1i),
.1"

R he 1. h, ,
(15) (z)=--1 zf'(z), '~"' 2n ·
.f(z) 1 z 1 < 1,
,~0 =, () = .f'(x),
, , h :
uz I(x) i z og go 2n, ogpeguiu < (t) ogzoapa gaioj (), \\< 1.
< (t)
-- 1 - z'(z) z = e1
i. < (t), h !l (t), 2n
1 . h -j..L(e1
').
n I(x)
, n.
.-

.f(z) ,
z z =. 1 , 2 , ... ,"
, ~~, ~ 2 , ... , ~m ,
h h
.

,, (16) () = <p(t)F(,t)dt,

F(, t) zf'(z) z = 1i •
.f(z)
(17)
X(z) ,
(19)

2 'i -+ e-ri - = 2 ' ()2 1--cost+ -,. ,.

1 ak 1 ~k 11 - - cos t /1 - - cos t
(20) F(,t)= L ,. 2 -I ,.
2 +\"(,t),
1 1 2ak (ak) 1 1 2~k (~k) ---cost+ - --cost+ -,. ,. ,. ,.
\"(,t) zx(z) z = eti.
()
(22)
(/
"
1- cos 1 = 1- ~ .!{_ 1og (1- 2 cos t + 2 ) 1 - 2 cos t + 2 2 clx
I3 ... 93
he, h 1 1 = 1,
cll () = i~o --2-l .

, !(), he 1 ( (5) (7))
()= iio + !l(X),
he 2
() = -11(~}
!l(X) (); R1 R2 1 2
R1 = 1 R, 1
R2 = h 1 -, R
R (). < (t)
[<,,,], h ll(x)=O, R==,
1 2 , h

B(x)=g0 lxl<1, ()= lx1>1;
~( ~-k = ng0 ,
~(~~ = mgo.
, x(z) 1 zl :::;

h
~
\jf ,., t) = 2 I. ",.ll+l cos + 1) t l
94
~
(25) () = 2"" ,-n+l = ,L..J n~..')n+l ' 1
() (n-m)g0 •
:

. f(x) g g z
,, - 1 ~ (t)F(, t) dt,
gO
z F(l', t) geo zf'(z) z = l'e ti, ~ (t) o f(z)
g [~,, , ].
, ,
_1 sintF(,t)dt, ,fio t

21t
-1 ~(t)F(,t)dt, gO
~ (t) 2, t = Foiel'-OB ;
g0 =n~ (0).
~ (t) = 1, -
21t 1t

... 95
. -
f(z)
(26) f"(z) " ( z ) ( z ) log-· - = Llog 1-- - Llog 1-- +~(z), ) 1 ak 1 ~k
z)
z = .
,, (27) () = ~(t)M(,t)dt,

(29)
(30)
1 [ . . ] (, t) = l log '') + log (-'') ,
1' -" 2 ( . ) ( . ) [ ( )21 log 1-~ + log 1-~ = log 1 - ~ cos t +
/1 [ ( )2] 1 2 (,t) =- Llog 1--cost+ - 2 1 ak ak
[ ( )2] 1 21' --Llog 1--cost+ ~
2 1 ~k 1-'k
~
~
96
(33)
() () =-II _ --II _!_ 2 1 ak 2 1 ~k
" h + log .f(O) (t) clt + (t) \jf (1·, t) dt.
(t) [, , , ; (9) .
(34)
l! ~
(35) cp(t)\jf(,t)clt = Ic11 g11 1·" =, 1
()
(36) (1·) = g0 log [.r)"- 111 ~ 1 ~ 2 · · · ~"' ]·
1 2 ... "
(1·) (36) h :
cp(t) g [,,,], h (, t) 'izi g f(z) g 'i
1·, ogpebeu uiii.e'ipa
~

27!
q>(t)M(,t)dt,
q> (t) 27t
q>(t) = 0 +1 sint+A2 sin2t+ ....
h
~> = 1, 1 = 2 = 3 = ... =,
. Jensena f(z) z .
IV.-
1. .f(z) z n. -
h
(29), (30)

(39) 1 /1 ( ) K(r)=-Ll __ +g0 logf(O). 2 I ak
, /(), a 1,a2 , ... ,am f(z) r, am+J,am+2• ... , " , h k = 1, 2, ... ,
(40)

L L . (1) (2)
g g ai h 2 K(I").
;, L (2)
( 42)
(43) () = L (l + 8 log .f(O)' ak
h ai h K(I").
2. ; h
,, () = q>(t)F(r,t)dt.
(44)
hh
R h
() = g0 + !l(x) 1 xl < 1,
()=-!1(~) alxl>l,

g Lt ; h Ltz ().
= , Lt L~
(46)
<p(t) [<,,
,], , j..l.(x) ,
Lll(_!__) ai, k
.f(z) Lt
pacpocLtpLta ; LtLt z , H(r) .f(z).
3. ai
" (47) L(r) = < (t)(,t) dt,
(,t) f(z) ;

100
(48)
g . ai .h L(r). h
(49)
g . ai .h L(r), h <p(t)
[<,,,], g 2
(50) 1 --L(r)
go
h g ai 2 ; h
z"t(~} g og2oapajyhe2 (50) h ai 2 .
, , h
.f(z) , ,
.
*

1. . .Q(z) z
. , ,
(1)
(2)
n . .
(3) ( ) 11 11-1 + - <" z = a0 z + a1z + ... + a11 _ 1z 11 -
_!_ .fr,(z) = ot. <p 11 (z) - z

< 11 (z) = Z(/) 11 _ 1 (z) + an (/)o(z) =.
= <pn(z)
* Su une c/asse ql de seies entie1·es, Atti de\ IV Congresso intenazionale dei Maternatici, Rorna, 1908, Sezione 1, vo\. 11, . 36-43.
1 <p 11 (z}=z"f11 (~) z'I'O, < 11 (0)= 11 (. .)
102
() = z<p 11 _ 1(z)
z-oca -11 11 . (5)
(7) = <-1 (z),
:

(7) z-ocy n-1 , (5) z-ocy n , 11 ~~~ - 11 (6);
2. 11 < ~"' (5) n ; " = ~~~, ;
3. - 11 _ 1 , -" ,
(7) z- ;
4. z-oca (7) , , ,
-" . , L':l 11 (a 0 ,a1,a2 , ... , 11 )
(3)
(8)
-"
- (7) (5). , n = 3 n = 4
= ,
27 + ( 4~ -180 1 2 ) =
103
" (8) f-1 11 ( ) , h ~" 11 ~" f-1 11 •
, , h :
fP (z) .Q(z), g 1. ga g
2 <-1_.
2 4 '
2. ga an (2 <n~ ) g g g ogzoapajyhx g n fln.
" f-1 11
: g an = g g .
.Q(z) ; ,
; , ,

(9)
h n- 2 /,,(z) .
2. ; .
h :
fP (z) Q(z), g ga an g g og z z n z g (8), g 2 ~n~ .
, .~,(z)
Q(z): fP (z) he z g peg
(10)
z z, h , t< , zu, gh, -
104
, ga 'i g, g g. 'i 'i'i g.'i , 'i.
, , (10) ~ z-a; , ,
(11) < (_!_)" 11 1l . n 0
~ 1 ,~ 2 , ... ,~ 11 , , (3),

3. (11) Q(z) 2. Lageue-oe
.Q(z).
"(z) = + A1z + A2z2 + ...
n ; n
. , n , 11 (z) F(z) ; n 11 (z) = ; F(z)
eaz+t>, 2. , Laguee,
:
1. n , 11 (z) (z); 2. 11 (z), ,
, . .
2 Lague·e, Oeu\les, t. , . 174.
105
1 . ,
-=- 1m- n= .
h Q(z), h :
Q.(z) g g 'i poga g "' 'ige u g
= 0 , = El_
h h
ai Q.(z).
(n -l)a~_ 1 - 2""_ 2 >

, ,

'2 2 n.
an zg Q.(z) g g
(13)

(14)

(13) , , · z = 'i Q(z) 08 (~1·), 8 (z)
(15) -an2
8(z) = L,-e-z11 •
n!
(14) , ,
Q(z) h ~' al
11
n 2 2 n . Q(z) h

(13). he, ,
Besel-oa
L ~ z" -- .
(n! )2
4. Q(z). La guee

107
:
,
ffi 11 , , h n: 1. ()
;
2. G(n) n, , ;
3. e-an 2
, ;
4. 1°, 2°, ; 5.
1 () 2 () 1°, 2°, 3°.
O(z) = 0 +a1z+a 2z2 + ...

.Q (z) = aoffio + ai(J)IZ + a2ffi2z2 + ...
ffi 0 ,ffi 1,ffi 2 ... n. Q (z)
. G. . Haidy4, , , h :
() , h
,
( +~)~ ( ),
4 t!J.e zeoes ' class ' integal.fimctions (The Messengei of Mathematics, Nov. 1904 . 97-101)
108
; ,
.

1 q-:;.- 3
, .
5. , h ,
(16)
Q(z) : peg (16) . . .
(17)
2ge bk g h .o2ylty gu. Ak

(18)
109
, . , , 2 ,3 , 4

- 1 4 - 2379,423 .
bk .Q(z) -
z2 1 = 1 + z +- = - (z + 2)2
4 4
= 1 + z + i__ + _i_ = l_(z +6)2 (z +l) 4 54 54 2
z2 z z4 =1+z+-+-+ =
4 54 2379' 423
= 1 (z+19,1172)2 (z+4,3225)(z+1,5064). 2379,423
(17) , , e-z, , , , h
1 n ( 2 )". z = 'f'; 8(-2), 8(z) (15). .Q(z) .
*

4 27 256
" a11 z"
1 = n"
,
: ,
z, h 11 •

he , h, .

,

. he
.
* , . LXXVII, , . 31, , 1909, . 1-44.
111
1.

e(z)=L~ n"
e(z), (z) z.
.
(z)
(2)
(3) e(z)=l+z(z).

l
(z) = f 21 dt
ga g (z) z . . gz illezpaa
112

, he :
( 4) (5)
()
(7)
v = -.
h ( z) . ( 4)
(8)
L'l(k)(z) = ukezv dt'
'= Ll + z<'l' " = 2<'1' + z<'l" em = 3<'1" + Zll 111
1
1
113
k-
(11) 1
e<k) = f uk-l(k + zu)ezudt.
he .
, , k-
(12)
(13)

= (n+k)! " n!(n+k+l)"+k+I ·
.
Ll z
pog (genre) og 2.1
. 1 1 = t og-
t
1 , . r 1 z 1 .
114
t 1 ,
(16) 1 =-= 0,36788,
1 1
:" dt < Jl z 1 dt = ~,
1· z. (3) ( 4) :
, .'i g z .g L! L1 . og .g
" '
.g . og .g
- 1 + ze".
L/e'dt < 1 : IJ ukdt
(8) ga , .'i gu z .g k-o'i g . og .g
k! -; (k+1/+1
'
(9),
1 1 < k 1 U-IJ 1 + 1 zl1 1,
ga , .'i g z g k-o'i g . og g22 g
J.;I -+ . - " [
115

l l
uke"'dt <Mk e'"clt
( = 1 z 1), ga he pegocLYiu z g k-iuoza g . og pegocLYiu
g k-LYioza g -t og g
. () -1 (ke+)e-k ,

z , :
l
l
.(k) z + ) = uk ezu " dt,

(8) ga
pegocLYiu z

(20)
(18) (19), , _Ck)(z)
(21) ':!:__k

, , (17) (19) (3) h
(23)
(24)
z

(9), (18), (20), (21), (22) (z).
2 2 f::. .
(n+ 1)" " > ..._______
n!
(n+ 1)"+1 (+ 1)!
g 2 :
g z uhe
e(z) > 1+(" -1).
z :
he
117

e<k\z) > k![-1 + z ]~ Ck + l)k+l
.
h , gaje g l . g 'i 'i /}. g z . gl.

(29)
n h n = 5, n
(30)
R11 (z),
(31) ( z) = " ( z) + R",
P"(z) -
(32) z2 zn
h ~ k
(30) h n > 5
(33) ~ (~)k < R < ~ (f)k. k.. k' 11 k.. k' n+l · 11+! '
118
, h
(33)
(34) i (~)k k!
=
(35)
.f(z)
n+i z f(n+IJ(coz)
(n+l)!' '

(36) (n+ 1)! e2(n+IJ '
(35)
(37) (n+ 1)! 22+I '
1 2 1. :
g z h
(38) 8(z) = P"(z) + R11 ,
'ige Pn n-o'i , gaLu 'i , 'ige Rn n 2: 5 g g'i g
(39)
119
he

z
W]Z W2Z

. 8
1
z) = ezudt
8(z) = 1 + z(z),
< t < 1, :
1. z , 8 ;
2. z , ,
8 1. , z
, he 8
120
, he ~ , 1, z .
( z ) ~ , .

( z ). Laplace-oe
:
h
() = f(t)[<p(t) dt
f(t) < (t) , t ,
< (t) t =, , , h
[<() () = -{; (1 +),
.
= /() -2n < () <"()
( t = < , <" ) .
(z).

=?.· : z
(z) h
121
z
(40) [21t " ~(z) = {--;- .[; (1 +),
z . : .i g ~(z) u g z g
(41)
~ z
(42) {2m~
(z) =~----;-" (l +)
, , g (z) g z g
(43)
1
:
1. z , ;
2. z , ;
. z ,
, , z . ; Laplace-oe
z .

122
1
K=z

: z , he t:..<k)
(44)
z
2
(45)
8,
f:.. z

(46)
acuiiioiiia pegociii k-iiioz g 8 (z)
123
(47) -J2rz ~ --,-.
k+ e 2
IV. . 8
h ga og . 8 z . h Hadamal"d-oe 2 , :

(48) P(z)eG(z),
P(z) z , G(z) z, 11
, n
,
G(z) z. , , 8(z) ,
. P(z)
8 (z)
P(z)
z z
8(z) = eG(z)
P(z) '
2 Hadamard: Etude s· les popl"ietes des.f"onctions entiees etc. (Journal de math. pures et appliquees 1893).
124
G(z) z. , , e(z)
(z)=P(z) eG(z),

1 "<-,
n!
Hadamard-oa G(z) z, l (z). , ,
1 < _!_ (n+1) 11+1 n!
(z)
.
ezudt
, ,
< t < 1 , z, . .l g .
: ga g og .'i l
. z = + yi l .!: g
= -1te, = 1te.
1 1
( + yi) = cos uy dt + i sin uy dt
z =+ yi (z)
(49)
(50)
1
1
125
, ( 49) h
(51) -n< uy < 7t
. , , (51) h
-n< < n,
h ,
__ - '
he (51)
-1te << 1te
, , h

. .

"= 2nni,
- n. L1 g n- l og 'i 'i n. Hadamard-a "
n,
- h
(1-E)<p(n),
n .
!!!.
126
1 " 1 > (1 +) (n+ 1)
1 " 1 n.

~(z)
= -ne = 1te z = + yi; , g n- h n.
,

=
()
(n+ k)! <(n+ k + 1)"+k+1
1 n -, -
! . , Mz), : ; = -7te = ne g n .
8 (z), h .
: Mz). h g , g ociuae .
,
. :
127
1. z , \! (z) , l 0 ;
2. z - ,
\! (z) , h -N, , h .
N ga h og g.
z =-
z , h z
(53)
1
. .
1 < u < -,

he - h

1 l

(54) -\jf(z)>xe .
, \jf (z) z h h

. , , = = 1, N > 1, -
.
(55) = ~()
128
!_
= -
. = -1 , , (55) , 1, ,
() = 1 + Ll(x),
,
, .
: (z) g , 'i, 'i 'i
g og 'i 'i. z .il(z)
ga g e'(z) g
1 + z .il(z),
'i g , 'i ; , 'i
, 'i.
g.
: 1.
(z) -30 -40,
-1,405 -1,406;
2. e'(z)
-5,718 -5,719;
3. N,
z .il(z),
129
V.
=~() =8()
~ h
()
()
= ~()
=8()
, h . (), 1,
= -; - +
0 ,
.
h ,
h = 1.

= "
() = 1 ,
. , ,
- = -( -1) .
(). ,
130
, (),
.
(), 2, = 1 = -; - +, , , , 1-N [ -N z ~(z)] , , ,
= 1.
.2.

(), (= , = 1) ;
.

21U" =~-----;-- .

~(z) +=
8(z)+a=O,
.
() () = -, :
\
131
- > 1 - < 1 -= 1;
2. (57) : ) > N -1 >, ; ~) = N -1 >, r
;
) < < N -1, , ;
8) -1 << , , - ;
) = -1, z = ; ) < -1, . (56) (57) , , -

.
VI. .

,
.(z), e(z) . , ,
.
,
. , ,
:
1y - /0 ()- + f..(x)--
xlogx =
, ga og = go = 1, 'i . . 2 2 'i, .h g ()
g.
132
, he
, k , r
(60)
h (58) h, ,
log- 1 - log-1 ( )k 1 Ce log- ,
he

h . .
dy +(+ logx)y =, dx

-
= ~. =
, = = , ,

.

(58), ,
133

(61) d"Y d"- 1Y
0 --+1 --1 + ... +11 =, dX" dX"-
.h g (z)
g.
, (61) ,

~ ~

C~(r) C~CkJ(r),
, ,
8(z) .
III.
(62) dy +nxy = dt dz -+my=O dt
( , n, ),
(63)
= 3 ,
5 = plog-1
4=-~ mC1
: g . 2 (63) t = 1 , , . z, ., . (63)
=-, = __ 2_
!
IV. , ,
L1 . . t R L, i , ,

-
at logl 1

135
~ t = t = 1
Q = Ct-.(-a) = 1- (-).
*

!-. ,
. .
, , .
he .
1. .f(z) n 11
n
g n
" la"l<-. n"
he g g n. ,
.f(z) z = ,
(64)
, r z, Pk ak,
3 . . . Borel: Le~·ons sul·les.f"onctions entiees, . 62.
136

~ z
(65)

,
fi (1 +__) < lim[1 +__ f -1 ]m 1 Pm 1 Pm
= , , (65) (64), r, M(r) .f(z) r
I
'
.1 ,
(66) ~ = i-1. 1 Pk
1 1
(67) !
1 an 1 < !!..__
/"11
. , ,
(67) ,
137
n =-
(~1 n h
(68) 1 1 < (!le)" " "
:
Mogyo g f(z) yiiioz poga z z = z, og g u (!ler), zge r z og r pegociii gaiiia ( 66).
(z) , , , h ' h
. ,
~'", ,
, z
z
(69)
, (69) .4

4 (Bulletin de \ Societe mathem. de France t. 34. . 165-177), , he , F(z). 8 (z) i !. (z) , .
138
~~·~2·····~11 (
)
.!..= ~] + ~2 + ... + ~~~-

< (.!..)" "- n 0
F(z) h h
[ ( ) ( ) 2 ] 1 a1z 1 a1z
1 + 1 - + -2 - + ... ' 1 0 2 0

(70)
z, :
2 g z g g F(z) og g (70) z ; g.
h h
8(z) F(z). F(z)
(z), .
139
. 11 h (70) ga F(z) z ga pog l.
8 (z) z
8 ( :~z) <:~:,
F(z), , :
2 g z g g F(z) og g
,
z F(z)
8(z),
:
F(z) z
. , h , ,
8 (z) F(z) , , .
140
. h, ,
(z) I . , , \jf(z)
1 , 2 ,3 , ...
h h
1 , 2 ,3 , •.•
,
Ja"J<I-1 1· "_

he , he \jf (z) . ,
h z

~1 !l= ~-1 ·1 ,
= z . , f(z) , h ,
(. )" " <-, n"
. f(z),
s < (.)
141
11 < ()
:
Mogyo g g h og g g ogzoapajyhx f(z) yLuoz poga og
zge r = 1 z 1 , ogr, g g g f(z).
,
(z), . , z
z
e(z)<l+zee.
, g z '1' (z) h, g z, og g ry, zge og r g
= 1 + +I.
*
(z) .

*

1 4 27
,
,
.
, , - h
= = 1

log = t
(), h (z), ().
e(z) , , h :
1. f(z) (.l.), = 1 zl 11
f(z).
* //· d' une transcendante entiere, Comptes rendus, Paris, 1912, t. CLIV, 8, . 499-501.
143
2. < (z) h
, < (z)
1·8()
'
= 1 z 1 < (z).

; e(z) .
(z) z; hy . ;
h
h
(2) k 1
(1) (2)

k < + k ' dz
,, = 1 zl; e(z +), ek(z +) (z).
z , (z), , h , , 8(z) 1, .
z h , 8(z)

~ :: - [21t ~-te -v z 11 - ~ ~ .
(z) , , -1,405 -1,406, -39 -40; , ,
.
= 8() = 1 = -; - +, , -,
= -0,68772 ... , , , -, -
= 1 h

*
(1) "= _"_a_h __
udt
1' t, , t (, ),

(2)
()
, r = const.,
* Sur des trenscendantes entieres generaeisant les fonctions nentielles et trigonQmetriques, Compies rendus, Paris, 1913, t. CLVI, 16, . 1213-1215.
146
!()= , !1 () = cosrx, /2 () = sinrx
, r , .2, , . .
,

2 () = ~ sin rx dt,
h /,/1,/2 , h :
1. , g. 2. !()
. Laguee-oa
. h , /() h ,
.
!() .
. /1 () /2 () h , , ,
, .
, ,
, ,
h ,
.
/1 /2 , . 4. , s, sinx
.
, h N h (, ), () h
:
. !() ,
. . k 2n 2n . 1 .; - . , -
,. (, ), = !()
... 147
- + , ,
!(). I(x)
.
1 -log ( )
N .
. h
-, 1- h 1 + l,
I(x)
I(x1 + 2 + ... + 11 ) = I(x1 ), I(x2 ) 2 ... I(x,Y"
.
I(xlx2) = (l22.
. h /1 /2 + 1 -1 . h . Moivre-oe :

( 1 2 ), h
[1 ()+i2 () = H1(mA 1x)+iH2 (mA 1x),
H 1(mx) + i H2 (mx) = [1 (2 ) + i 2 (2),
.
148
/, /1 /2 (
,
, ,
h .),
.
S des tanscendantes entiees gem?alisant les fonctions exponentielles et tl'igonometiques (Comptes rendus, 21 avril 1913, . 1213)
{ ()= 1-- 2 2 +~4 - ... 1
2! 4! '
()= - + as xs - ... 2 1 3! 5! '
(1)
h
udt
( t, (, ))
cosax sinax.

~ ~
.

(4) + L " cosnx + L 11 sinnx 1 1
* Seies hypetigonometques, Comptes rendus, Paris, 1913, t. CLVI, 24, . 1823-1825.
150
, 2n, ( ) , .
ga peg (3) g g g 'i .
,

(t) = ~ ~1
f "'(~) cos 9 d~ ~
. '1' (~) (~ 0 ,~ 1 ) (',') .
, ,
Ossian-Bonnet-oo .
, n .!_
.
h
f cos dt = ) f dt'
(5)
h h
six dt = 2 () dt,

) < r, n
n
8 n. , ; ;, peg (3) 'i g u
()
,, f() dt
()= _"_-,,---
udt
151
< < 2,
t . f(x) 2 ,
.
= const. (3) , , (4). t (, ), (3) (), . ,
- + ,
1· t . (3) (6)

.
,
= 1 1 L -l2 (nx) = -(- 1 ), 1 n 2
~ (-1)"+ 1 ( ) _ l 1(px) __ 1_ L.J 1 nx- ,
1 n2
- 2 2 sin 2 2
= (-1)"+ 1n L 2 2 l 2 (nx) = . 12 (),
1 n - 2s
, cospx sinpx;
2 < < - .

*
1. v , , .
(1)

, :
1. v , , (- v) 2 (, );
(5)
* lu!ou!mes de / moyenne sans estictions, Nouve\les annales de mathematiques, Paris, 1913, 4 serie, t. XIII, 4-9, . 400--406.
153

;
2. v , (- v) 2 , . Darboux-a,
() - · 2 8 = -2-()/() ,
m : 1, 2.

g g g - gh 2
- g u v g geo, geo. 2

,, (8) vdx
{/
. . . v ,
u v ga . v , (8)
V _ - 2 V _ - N2 2 2 '
N - v . , , (8)
(9)
154
4 4

(12) v dx = i (+ v) 2 dx- ~,

g ~
(14)
z (), . gz z, v .
v , h
/
(12)
W_ - 2 W_ - N2 4 4 '
(12)
(16) W-(b-a)M2 ;N2'
(17)
3.
h ,,
(19) uv dx < ± 1 + v 12 dx,
~ ,
(21) J1u2 .•. U 11dx < n1"J(I1 1+ ... +1"l)"dx, L L
L u1, u2 , ... , 11 .

.
h h .
, , , t,
t (D) t.
(22)
"
(23) /1 = f ulu2 ... undt' L
L (D).
(24)
(25)
(26) Sfln
. . s .
: peg (22) g g og z, poga g, g, g z, og
(27)
(28)

h z,

(31) ) < ,
: g og f(z) , g z, og
(32)
, , Jensen-o
157
f(z),
f(z)
z = t.
*
I.-
, , , , ~. .
~ sin 2 + sin 2 ~ - 2 sin · sin ~ · cos <p(a,f-',y) = . . •
sa+sl-' (1)
h, h ,
sina = ~. sin~ =,

* Su quelquesjonctions des cotes et des ang/es d'un tiangle, L' En seignement mathematique, Geneve, 1916, t. XVIII, 3-4, . 153-163.

l::;::;1, 2
159
1 - ~ = 11, 1
2 ~ 11 .

- cos _ < ,~--'' _ ,
rr-y< ( )<1 cos-- _< a,l-',y _ . 2
rr-y 1+cos--
4
(3) he : 1. ~=); 2. ~ 11 .
rr-y rr-y rr-y <(,~,) = cos2 -
4 --sin2
-- = 1. 4 4
(4) rr-y
4
160
:
tg2 n- · 4 '
' = ~, g tg2 n- g 4
g og g 'i ~ .

. h < (,~' ) g
' g g g
tg 2 n = 171 8 '
' g g y'iao 180°.
> 120° < 0,070 > 140° < 0,040
(5) > 150° < 0,018 > 160° < 0,007 > 170° < 0,002 > 175° < 0,0003.
n-y cos2 --
,
, ' ,
n, h
n-4arctg.JE' , '
> n-4-JE'.
h , , ,
h, pefta jego'i 'i
+ g y'iao . , l1 ;
+ = h,
161
= ~ 2 + 2 - 2 · cosy = h<p (,~, ).
Ogage ga
= ( +b)cos2 --(1 ±), 4
2ge 2 < .

(7) = (+ ) cos2 1t- 4
140° 4%, 150° 1,8%, 160° 0,7%, 170° 0,2% .
. -
(8)
, , ,
,
(9)
. 1, .
= = , 1 .
zge , , jegoz z,
z z 1 ~ .
162
, , ,
, . ,
l) ,
s , - s s +;
=, 2 +2 =, a+b=n,
.. + 2
2 + 2
- n - + ' (11)
- +,

1 t_. '11 --:} = .
....
n
Jl (12)
(13) ~ 2 +2 +(-)2
1 () 2
, ~ s 1, h. ~2
: , , jego'i 'i, g -

g g
1 3 = 0,5774 ... '
1 ..fi =, 7071 ....
(14) ~ 2 +2 +2 = (+)( ++),
zge h g
(15) 1 = 3 = 0,5774 ... ,
1 1 = ..fi - 3 =, 1297 ... ,
zge g g 1.
163
( = ) -
; ~ ( = 1)

I.
2 + n2 + 2 = l (2 + 2 + 2) 4
, n, , , , ,
(16)
! = 0,5000 ... , ! fi=0,6124 .... 2 2~2
L , , S
0,5000 0,6124.
164

L S : 26°30' 31 °30', 58°30' 63°30'.
11. S, 'AS, 0,5774 ... 0,7071 ...
.
f1(x), fz(X), f()
= =

h
(15) 1.
IV. , h , , , z .
h g g , ogl.oapajy og jegol. go gpy2o2, . . .. . . 0,5774 ... 0,7071 ...
, ,
f(x,y,z) =, <p(x,y,z) =

~ dy - dx ~ dz ~ dy + dx
( he h , , z), h
:
(19) < - <Q<P+T - -- '

(20) = z Q=
= z
< < , < < , < <
z z
.-
f(x) ==
(21)
ai , 0 1 .
:
(22) (x+y+z)l' +!'+ zP ,
, , z, , 1 3~'- 1 ; 1 = 1 , , z ; 3~'- 1 x=y=z.
= 2, 3, 4, ...
(23)
(24) (k = 2,3,4, ... )
+ + z (21), (23)
(25) f(x) + f(y) + .f(z) ~ f(x ++ z) + 2/(0),
(24), , , z , , z,
ak(x+~+z)k ~a;(xk+/+zk)

(27) 3f (+~+ z) ~( )+ f(y) + f(z) ~( + + z) + 2/(0),
h
(28) ( )+ ( )+ f(z) = F(x ++ z) + (+ + z),

(29) F(t) = 3/(~ (t) = f(t)- 3/(~) + 2/()
1. 1 f(t) , , z (= 1), = = z(e = 0). ·
h :
: , , z jego'l z. h s , (28)
(30) f(a) + f(b) +( )= F(s) + (s),
F (29). = ; = 1 .
'l : , , z 'l jego'l 'l g og n. (28)
(31) ( )+ (~)+ ( )= F(n) + (n),
F (29). = , = 1 n.
(30) (31) gajy 'l 'l.
167
f(t) t =
(32)
; ,
0 1 . (30) (32) t = s, (31) t = 1t.
, ,
~ ~
= 3 = 8,54896 ... , N = 2+e1t -3 3 = 16,59134 ...
, , (27) (28)
.

*
1. f(x) =
(1)
; ,
0 1 , , .
1 ,2 , ... ,11 (1).

n n
).1 ;
h f(x). ga og ).1 g -
g g.
n ;
(2) F( ) - -1( 1') ( )1' 1 ,2 , ... ,11 -n 1 + ... +" - 1 + ... +"
, h,
, h:
1. > 1,
() 1 = 2 = ... = 11 ;
* Tlu!on?me su / moyenne aitlmu!tique de quantiufs positi1•es, L'Enseignement mathematique, Geneve, 1916, t. XVIII, 3-4, . 163-176.
169
,
;;
2. ? < 1, h (3); , ;;
3. = 1 . he , h
(4)

,
~ 1 > 1; ~ 1 < 1
he ;
.
: 1 n"-1 ;
;
, ; = 1; ; .
h , = 2, 3, 4, ...
(5) ( ) > ( + ) 1 + ... +11 _1 + ... 11
() = = 1. (5)
.f(x1 + ... +11 ) ~ .f(x1)+ ... + .f(x11 )-(n-1).f(O)

( 6) , h nx; ;,
( 1 + ... +11 ) < ( + + ) _ 1 ... "
n n
170 ':!

(9) f(Jl) ~~ f(nJl) +(n -l)f(O) n
gaje .o'iyhe 2 . g . g .
.f(x).
(9) .
. - . g
.
(11) (l) = f(Jl),
(12) '1' (Jl) f(nJl) +(n -1)/(0) f(Jl) n
zge g . 1; : = = 1 g'i; g ; . g, gpy'ia g ; g g og .
(9) :
(13) _!_ [tfl) + (t -l)f(O)] t
t (9) t = 1, (9) t = n t 1
; , l t i, u.ahe g
171
(14) = ~ 1( ~) + (1- ~)f(O),
'ige ~ g _!_ 1, n
f(x), ,.

.
(14) : g h g :
Jl =~
()+8l() =,
. 1
(14) ,. n~; n~=~,

h g f(x) 'i g g.
1 ~ = - ~ = 1 .
n , , g
(16)
(17)

(18)
ga ., f(x), g g gpy'ioj og '.
(10)
(19)
n s
(20) +8
~ ~ 1
~r(;;) f(s) +(n -l)f(O)
.'ih ga g g g gpy'ioj og '.
2. , , -
.
+ ,
• s

,
2 " + + ... +
+ 8,
"-1 11 -1 ."-1 x- A=na"x-i, = x-i +n-1-na"x-1 .
h . 2 2

,~, , h
/()+/(~)+ /()= +8,
=(~ B=/(1t)+2/(0)-3f(~} he , ,
+ + = + 8 ,

. 1 , 2 , ••. , ", he
( [ ) + ... + ( ~) = + 8,
,
he , h S - - 1 + ... +",

173
, he

r ,
-~ ne n
sk = st = (-~i, ;_1 ::;; ::;; 1. n
h Sk, h 1 ~- 22 •

;
log (!' + ... + xi,) = log s +
::;; ::;; (1-p)logn < 1 (1 -) log n ::;; ::;; > 1.
, ,
; = ;, = ,
al,a2•···•an t t = t = , t , 1. v t, (, ). he

v dt + eD v dt,
D
= 1og(ea1 + 2 + ... +"), D = (1-N)logn,
N (, ), 1.
, t (, ), 1, D he ( 1- ) log n, (, ).
n . ,.
(22) ~~+ ... +~ =8(1 + ... +11 ), }~~l.

i = l,2, ... ,n = l,2, ... ,n
he ,. -
.
1 ,2 , ••• ,11 n

176 ~
s ; , h
h. ; ; . he h :
s g
(23)
z Fn 1.
, ,
(22) ; dx; .
~ =
=, 7071 ... 1; - 1
.fj = 0,5774 ... 1 .
3.
,
- .
, , h

he . , ,
(24) s = f(x,y)

s f(x, ) . h s
[(- ) +(- )] [(- )- (- )],
h h
h , h , ,

(25) f(x,y)-8[x+ -(0 + )] = f(x,y)-8[x- -(0 - )] =,
177
8 0,7071 ... 1, (0 , 0 ) .

(26)
.
h <. () ~f(x), = 0 = 1 ,
(0 ,0 ), ( ) -; D - = <. (), .
0 ,
h 1 0
h 0

0
= <. ().
(27)
(28) ~~+= 1 <.(), 1-.:;, ..:;, ...fi.

178
(29) = 0 (, 0 ) + , 11 (, 0 ),

(30) ( ) = -(-) ' ' 1 (, 0 ) =- < () dx

, 1 , , , 1 , 1 -2.
g (29) gi i 1 ga i . ogioapajyhx 2 g .
(31) = (,0 )+ (,0 ) = 0(,0 )+-2(,0 ).
2 (27)
(32) 1 < 8 < 1 .Jl- 2-'

(30), gih ' 1 ; . ogioapajyhx 'i g
(34) = 0(,0 )- l(,0 )
= 0(,0 )-.2(,0 ).
~(0 ,-0 ), 0 -, ,
yha ,, 2 , 1 2 -.
1 2 ( , 2 )

i- f(x)= ,

(35) ( V) 2
179
D V V V . . .
- - . -
V V . - - h .

( V) 2
( V) 2
(37) V V
+ = 8<(,)
1 .fi.
(38)
1 1 8<
=-+1
V = 8<,
1 . (40) h (39), (40)
(41) V ( ) - = 8< ,-+1 .
< , he
(42) v = 8' < (,~ + 1 ) dx + 2 ,
180
2 , 8' 1 -fi.
, (39) ( 42)

he
' (t) , , t. , h,
( V) 2
+ ... +(V) 2
= f(xl,x2, ... ,x") 1 11
V n 1, ••• , 11 V
V - ; V
;

V='+8
' 1 , ••• , 11 8 1 .;;.
*
1
() = 1+~+ 2 2 + + ... 1! 2! !
(1) !() = 1- 2 2 + 4 4- ... 2! 4!
l 2 (x) =~;-~~ 3 + ~~ xs - ... ,
11
(2) h
" = udt,
r t , t (, ).

* , , . , , . 55, 1926, . 1-17. 1. 1926. .
1 Su des tanscendantes entiees gemimlisant les fonctions exponentielles et tigonome tiques (Compt. rend. de I'Acad. des Sciences de Paiis, t. 156. NQ du 21. Avril 1913). Series hypetigonometl"iques (Compt. J"end. de I'Acad. des Sc. de Paris, t. 156. NQ du 16. Juin 1913). 4 'i4} .h g ogpebeux l'i ( XCI . . ).- Fonctions entiees se attachant aux nombres premies (Compt. rend. de 1 'Acad. des Sciences de Paris, t. 168. NQ du 17. Mars 1919).
182

2 () = ~ u(t)smrxdt.
(1)
(4) J 1(x)+il2(x)=J(xi),
= const.
(5) () = Jl(x) = COSI"X 2() = sinrx.
t, ( 4) , ,
.
(1) h , , ,

, . , ,

(1) (3). h 1 ()
2 () . . , ,
. he, , 1 () 2 () , h
.
*
1 () 2 () " (1)
(5) 2t
183
u(t) : f(z) z,
= 1 z = , ,
.f(z) = h0 + h1z + l12 z2 + ...
L.

()
L
(7)
z2.f(z) z = re1 ;
=
(5)
, (1) (3), 1 () 2 () h () V(x), , V(x) h (1). , ,
(9)
(10)
21t
(12) (x)+iV(x) = W(xi),
(13) W(x)'= _!_ u(t)ex1dt.
V: , (geme = 1),
g 'i ,
g g og z .
, < (t) t , 27t, (0, 2n) t, t -
m=oo
(14) < (t) = L ( cos mx + bm sin mx), m=2
, bm hm_2 •

m=oo
(15) u(t) = L,.L,.A11 (a 111 cosmnt+b111 sinmnt) m=2n=2
,
(16)
(17)

(18)
(19)
21t
Ck(x) = cos kt · cos xt · dt = 2 2: k 2 sin 2nx
21t
f . k d 2k . 2 S k () = Slll t · COS Xt · t = 2 2 Slll 7tX k -

185
(20) 1 =~=~
Q(x) = I :L,s). m=2n=2
Sk(x) ; Ck(x) = k, 2n. , Q(x) , ( ) .
() ( ), ( ).
, (5) , (t) , 2 " ()
(21) M(27t)2n+l
(2 + 1)! '
h u(t) . (z) . , S :
,
. ,
,
, .
S , () , . , .

V(x), , 111 , 111 . . l1m
(22)
i () V(x)
,
1 () 2 () , :
186
,
, ;
, = ±.
l. , .
, - +. ,
.
, ()

(23) 1, 3, 5, 7, 11, 13, 17, ...
2 . h : U(x) . g (23) .z
., ga . g . . .
, (16), (18), (19), (20), , h ,
(24)
(25)

(27) () = R(x)- ().
(. )
( - ) ( + )
( ) ,
2 U(x), V,

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