F1
2

F1
1

F1
0

F0
2

F0
1

F0
0









" F0
2


F0
2


"





F1
0










F1
0

F1
0



F1
0




z1-c


1


"














m

























Cn C


Cn
1
ëÅ‚ öÅ‚2
n
íÅ‚
v = |vj|2Å‚Å‚ , v " Cn.
j=0




A Cn A
A := sup Ax .
x =1




"zv(z) = A(z)v(z).

v(z) " Cn


A(z) z0 z0





&! C
îÅ‚ Å‚Å‚
a11(z) . . . a1n(z)
ðÅ‚ ûÅ‚
&! z A(z) = . . .
an1(z) . . . ann(z)
îÅ‚ Å‚Å‚
w1


ðÅ‚ ûÅ‚
n × n w = . . . " Cn
wn
îÅ‚ Å‚Å‚
v1(z)


ðÅ‚ ûÅ‚
&! z v(z) = . . . " Cn
vn(z)


dv(z)
= A(z)v(z),


dz

v(z0) = w.



K(z0, r) K(z0, r) ‚" &!



z0 = 0


"
A(z) = Akzk
k=0

"
v(z) := vkzk,
k=0

v0 = w,
m
1
vm+1 := Am-kvk.
m+1 k=0







K(0, r)
Ak d" Cr-k.


p0 = w
m
1
pm+1 := Cr-m+kpk,
m+1 k=0





vm d" pm.


v0 = p0.


vk d" pk, k = 0, . . . , m.

m
1
vm+1 d" Am-kvk
k=0
m+1
m
1
d" Am-k vk
k=0
m+1
m
1
d" Crk-mpk = pm+1.
m+1 k=0




m
r(m + 1)pm+1 = Cr-m+k+1pk,
k=0
m-1
mpm = Cr-m+k+1pk,
k=0

r(m + 1)pm+1 = (Cr + m)pm.


pm+1
lim = r-1.
m"
pm



"
pkzk
k=0


K(0, r)
"
vkzk
k=0


K(0, r)


&!



&! v(z) &!






("z - 1)v(z) = 0, v(0) = 1.



"
v(z) = vnzn.
n=0


nvn = vn-1.


"
zn
v(z) = , z " C.
n!
n=0

v(z) = ez



µ " C z = -1

"z - µ(z + 1)-1 v(z), v(0) = 1.



"
v(z) = vnzn.
n=0


nvn = (µ - n + 1)vn-1.


"
µ . . . (µ - n + 1)zn
v(z) = , |z| < 1.
n!
n=0

v(z) = (1 + z)µ





2
"z + c(z)"z + d(z) u(z) = 0.



z0 c(z) d(z)


z0



c(z), d(z)

&!
2
"z + c(z)"z + d(z) u(z) = 0



u(z0) = w0, "zu(z0) = w1,


&!



u(z) w0
v(z) := , w :=
u (z) w1

0 1
A(z) :=
-d(z) -c(z).



dv(z)
= A(z)v(z),
dz
v(z0) = w.




"
u(z) := ukzk.
k=0


2
b(z)"z + c(z)"z + d(z) u(z) = 0,


b(0) = 0

u0 = w0, u1 = w1,
m m-1 m-2
k(k - 1)ukbm-k + kcm-k-1uk + dm-k-2uk = 0.
k=0 k=0 k=0




A(z) |z| > R "



w = z-1 0




"z = -w2"w
"wv(w-1) = -w-2A(w-1)v(w-1).


"
lim z2A(z).
z"



"



w " Cn

dv(z)
= A(z)v(z),


dz

limz" v(z) = w.






c(z) d(z) |z| > R "



w = z-1 0



2
"w + (2w-1 - w-2c(w-1))"w + w-4d(w-1) u(w-1) = 0.



"
lim (2z - z2c(z)), lim z4d(z).
z" z"



" w0 w1




2
"z + c(z)"z + d(z) u(z) = 0



limz" u(z) = w0, limz"(u(z) - w0)z = w1.









dv(z)

= Ã(z)v(z)
dz


z0 Ã(z) z0




(z - z0)"zv(z) = A(z)v(z),


A(z) z0 A(z0)


z0


0


&!


C 0
îÅ‚ Å‚Å‚
a11(z) . . . a1n(z)
ðÅ‚ ûÅ‚
&! z A(z) = . . .
an1(z) . . . ann(z)



n × n w " Cn  " C
(A(0) - )w = 0,





 + m A(0) m = 1, 2, . . . .



}(z) &! v(z) := z}(z)


Å„Å‚
dv(z)
òÅ‚
z = A(z)v(z),

dz

ół
lim z-v(z) = w.
z0



K(0, r) K(0, r) ‚" &!


"
A(z) = Akzk
k=0

"
v(z) := z vkzk,
k=0

v0 = w
vm := ( + m - A0)-1 m-1 Am-kvk.
k=0






K(0, r)
Ak d" Cr-k.


p0 = w
pm := ( + m - A0)-1 m-1 Cr-m+kpk,
k=0



vm d" pm.

m
r ( + m + 1 - A0)-1 -1 pm+1 = Cr-m+kpk,
k=0
m-1
( + m - A0)-1 -1 pm = Cr-m+kpk,
k=0

-1 -1
r ( + m + 1 - A0)-1 pm+1 = C + ( + m - A0)-1 ) pm.



lim m ( + m - A0)-1 = 1.
m"


pm+1
lim = r-1.
m"
pm



v(z) K(0, r)


}(z) &!



îÅ‚ Å‚Å‚
 . . .
ïÅ‚ śł
1  . . .
ïÅ‚ śł
ïÅ‚ śł
A = . . . .
ïÅ‚ śł
ðÅ‚ ûÅ‚
. . . 
. . . 1 



z"zv(z) = Av(z)

z"zv1 = v1,
v1 + z"zv2 = v2,
. . .
vn-1 + z"zvn = vn.



îÅ‚ Å‚Å‚
0
ïÅ‚ śł
. . .
ïÅ‚ śł
ïÅ‚ śł
0
ïÅ‚ śł
ïÅ‚
z śł , m = 1, . . . , n.
ïÅ‚ śł
ïÅ‚ śł
z log z
ïÅ‚ śł
ðÅ‚ ûÅ‚
. . .
z(log z)m-1



0
"zv(z) = (az-1 + b)v(z).


v(z) = zaebz






2
Ü
"z + b(z)"z + c(z) u(z) = 0
Ü


Ü
z0 b(z) z0 c(z)
Ü



z0



z0 = 0
2
z2"z + b(z)z"z + c(z) u(z) = 0.



b(z), c(z)



&! 0  " C
( - 1) + b(0) + c(0) = 0,
( + m)( + m - 1) + ( + m)b(0) + c(0) = 0, m = 1, 2, . . . .



i
(z) &! u(z) := zi
(z)


Å„Å‚
2
òÅ‚
z2"z + b(z)z"z + c(z) u(z) = 0,



ół lim z-u(z) = 1,
z0


u(z) 1
v(z) := , w :=
zu (z) 

0 1
A(z) := .
-c(z) 1 - b(z)


zu (z)
A(z)v(z) = ,
-c(z)u(z) - b(z)zu (z) + zu (z)
u(z) zu (z)
z"z = ,
zu (z) z2u (z) + zu (z)
i
(z)
z-v(z) = .
zi
(z) + i
(z)



Å„Å‚
dv(z)
òÅ‚
z = A(z)v(z),
dz
ół
lim z-v(z) = w.
z0




"
u(z) := ukz+k
k=0


2
a(z)z2"z + b(z)z"z + c(z) u(z) = 0,




a(0) = 0

Å„Å‚
ôÅ‚ u0 = 1,
ôÅ‚
ôÅ‚
òÅ‚
um = - (( + m)( + m - 1)a0 + ( + m)b0 + c0)-1
ôÅ‚
ôÅ‚
ôÅ‚
ół
m-1
× (( + k)( + k - 1)am-k + ( + k)bm-k + cm-k)uk.
k=0



2
a(z)z2"z + b(z)z"z + c(z) u(z) = 0,


a(0) = 0 1, 2

( - 1)a(0) + b(0) + c(0) = 0.


1 -2 " Z


z1 z2 1 - 2 " Z



z1 1 - 2 e" 0





Ã(z) |z| > R "



w = z-1


0



"
lim zÃ(z).
z"



z"zv(z) = A(z)v(z),

A(z) " -A(") "



&! C

"
îÅ‚ Å‚Å‚
a11(z) . . . a1n(z)
ðÅ‚ ûÅ‚
&! z A(z) = . . .
an1(z) . . . ann(z)



n × n w " Cn  " C
(A(") + )w = 0,





 + m A(") m = 1, 2, . . . .


}(z) &! v(z) := z-}(z)


Å„Å‚
dv(z)
òÅ‚
z = A(z)v(z),


dz

ół
lim zv(z) = w.
z"




C



z1 z2 "


"zv(z) = a1(z - z1)-1 + a2(z - z2)-1 v(z)



z1 : a1, z2 : a2, " : -a1 - a2,


(z - z1)a1(z - z2)a2



b(z), c(z)



&! ‚" C "  " C
( + 1) - b(") + c(") = 0,
( + m)( + m + 1) - ( + m)b(") + c(") = 0, m = 1, 2, . . . .



i
(z) &! u(z) := z-i
(z)

Å„Å‚
2
òÅ‚
z2"z + b(z)z"z + c(z) u(z) = 0,



ół
lim zu(z) = 1.
z"



C


0 "


2
(z2"z + bz"c + c)u(z) = 0.





0 : ( - 1) + b + c = 0,
" : ( + 1) - b + c = 0.



Á, Á 0 -Á, - zÁ
Ü Á "
Ü




Ü
zÁ Á = Á zÁ zÁ log z Á = Á
Ü Ü

(z2"z + (1 - Á - Á)z"z + ÁÁ)u(z) = 0.
Ü Ü



C


z1 z2


2
"z + g1(z - z1)-1 + g2(z - z2)-1 "z + h(z - z1)-2(z - z2)-2 u(z) = 0,


g1 + g2 = 2
z1 : ( - 1) + g1 + h(z1 - z2)-2 = 0,
z2 : ( - 1) + g2 + h(z1 - z2)-2 = 0.



Á, Á z1 -Á, - Á z2 (z-z1)Á(z-
Ü Ü

Ü Ü
z2)-Á (z -z1)Á(z -z2)-Á Á = Á (z -z1)Á(z -z2)-Á (z-z1)Á(z-z2)-Á log(z -z1)(z-z2)-1
Ü


Á = Á
Ü




2
"z + (1 - Á - Á)(z - z1)-1 + (1 + Á + Á)(z - z2)-1 "z
Ü Ü
+ÁÁ(z1 - z2)2(z - z1)-2(z - z2)-2 u(z) = 0.
Ü



C


z1 z2 "
2
"z + g1(z - z1)-1 + g2(z - z2)-1 "z



+h1(z - z1)-2 + h2(z - z2)-2 + h(z - z1)-1(z - z2)-1 u(z) = 0.





z1 : ( - 1) + g1 + h1 = 0,
z2 : ( - 1) + g2 + h2 = 0,
" : ( + 1) - (g1 + g2) + h1 + h2 + h = 0.

Á1 Á1 z1 Á2 Á2 z2 Á3 Á3 "
Ü Ü Ü


Á1 + Á1 + Á2 + Á2 + Á3 + Á3 = 1.
Ü Ü Ü



2
"z + (1 - Á1 - Á1)(z - z1)-1 + (1 - Á2 - Á2)(z - z2)-1 "z
Ü Ü


+Á1Á1(z1 - z2)(z - z1)-2(z - z2)-1 + Á2Á2(z2 - z1)(z - z2)-2(z - z1)-1
Ü Ü

+Á3Á3(z - z1)-1(z - z2)-1 u(z) = 0.
Ü





îÅ‚ Å‚Å‚
z1, z2, "
ðÅ‚
P Á1, Á2, Á3 ûÅ‚ .
Á1, Á2, Á3
Ü Ü Ü



u(z1(1 - t) + z2t) = tÁ1(1 - t)Á2
îÅ‚ Å‚Å‚
0, 1, "


ðÅ‚
P 0, 0, Á3 - Á1 - Á2 ûÅ‚ w(t) = 0.
Á1 - Á1, Á2 - Á2, Á3 - Á1 - Á2
Ü Ü Ü



a, b, c

îÅ‚ Å‚Å‚
0, 1, "


ðÅ‚ ûÅ‚
P 0, 0, a w(t) = 0.
1 - c, c - a - b, b



t(1 - t)






u1(z) u2(z)
2
("z + b(z)"z + c(z))u(z) = 0.


W (u1, u2)(z) = W (z) := u1(z)u 2(z) - u 1(z)u2(z).


("z + b(z))W (z) = 0.

i
1(z) = a11u1(z) + a12u2(z), i
2(z) = a21u1(z) + a22u2(z)



W (i
1, i
2) = (a11a22 - a12a21)W (u1, u2).





a " C
(a)n := a(a + 1) · · · (a + n - 1).


a1, . . . , ak " C c1, . . . , cm " C\{0, -1, -2, . . .}

Fm
k
"
(a1)j · · · (ak)jzj

Fm(a1, . . . , ak; c1, . . . , cm; z) := .
k
(c1)j · · · (cm)jj!
j=0




m + 1 > k z " C


m + 1 = k |z| < 1


m + 1 < k


Fm
k




fj j
fj+1 (a1 + j) · · · (ak + j)
= .
fj (c1 + j) · · · (cm + j)


Fm(a1, . . . , ak; c1, . . . , cm; z)
k
Åšm(a1, . . . , ak; c1, . . . , cm; z) :=
k
“(c1) · · · “(cm)
"
(a1)j · · · (ak)jzj

= .
“(c1 + j) · · · “(cm + j)j!
j=0


c1, . . . , cm " C ci " {0, -1, -2 . . .}


Åš





F1
2
" (a)n(b)n
F (a, b; c; z) = zn.
n=0
n!(c)n


|z| < 1 C\{0, 1}


1
2
z(1 - z)"z + (c - (a + b + 1)z)"z - ab u(z) = 0.


F1
1
" (a)n
F (a; c; z) = zn.
n=0
n!(c)n


z " C


2
(z"z + (c - z)"z - a)u(z) = 0,


F1
0
"
1
F (-; c; z) = F (c; z) = zn.
n=0
n!(c)n


1


2
(z"z + c"z - 1)u(z) = 0.



F0
2




C\{0}


" (a)n(b)n
F (a, b; -; z) <" zn
n=0 n!



2
z2"z + (-1 + (a + b + 1)z)"z + ab u(z) = 0.


F0
1
" (a)n
F (a; -; z) = (1 - z)-a = zn
n=0
n!


|z| < 1 C\{1}


((z - 1)"z - a)u(z) = 0.


F0
0
"
1
F (-; -; z) = ez = zn.
n=0
n!


("z - 1)u(z) = 0.







2
(z"z + (c - z)"z - a)u(z) = 0.


0 1 - c


Å‚
z-ł"zzł = "z +
z
2
z-ł(z"z + (c - z)"z - a)zł


2
= z"z + (2ł + c - z)"z - ł - a + (-ł + ł2 + cł)z-1.



2
zc-1(z"z + (c - z)"z - a)z1-c


2
= z"z + (2 - c - z)"z - 1 + c - a,



1 + a - c 2 - c


fn(n + )(n +  - 1 + c) = (n +  - 1 + a)fn-1.


"
(a)n
F (a; c; z) = zn.
n!(c)n
n=0



z1-c
"
(a - c + 1)n
z1-cF (a - c + 1; 2 - c; z) = z1-c+n.
n!(2 - c)n
n=0




e-z"zez = "z + 1
2
e-z(z"z + (c - z)"z - a)ez


2
= z"z + (c + z)"z + c - a.


z = -w -1
2
w"w + (c - w)"w - c + a.


c - a c ezF (c - a; c; -z)


1


F (a; c; z) = ezF (c - a; c; -z).




[0, 1] t Å‚(t) " &! f &!
f := f(Å‚(1)) - f(Å‚(0)).
Å‚


Å‚


ezssa(1 - s)c-a = 0.
Å‚




ezssa-1(1 - s)c-a-1ds
Å‚





2
(z"z + (c - z)"z - a)ezssa-1(1 - s)c-a-1
= zezssa+1(1 - s)c-a-1 + (c - z)ezssa(1 - s)c-a-1 - aezssa-1(1 - s)c-a-1
= -zezssa(1 - s)c-a - aezssa-1(1 - s)c-a + (c - a)ezssa(1 - s)c-a-1
= -"sezssa(1 - s)c-a.


Rea > 0 Re(c - a) > 0
1
“(a)“(c - a)

ezssa-1(1 - s)c-a-1ds = F (a; c; z).
“(c)
0






1
“(a)“(c - a)
sa-1(1 - s)c-a-1ds = .
“(c)
0


n = -a "
{0, 1, 2, . . .} F (-n; c; z) n

Å‚


0
(1 + Ä…)n
LÄ…(z) := F (-n; 1 + Ä…; z)
n
n!
1
= e-tzt-n-1(1 - t)Ä…+ndt.
2Ä„i
[0+]


" F0
2


Å‚ = -a
2
za+1(z"z + (c - z)"z - a)z-a


2
= z2"z + z(-2a + c - z)"z + a(1 + a - c).


2
= z2"z + z(1 - a - b - z)"z + ab,


b := 1 + a - c w = -z-1



z = -w-1 "z = w2"w
2
w2"w + (-1 + (1 + a + b)w)"w + ab.


F0 0
2



"


2
(w2"w + (-1 + (1 + a + b)w)"w + ab)g(w) = 0,





2
(z"z + (c - z)"z - a)z-ag(-z-1) = 0.



2
(z"z + (c - z)"z - a)f(z) = 0,

2
(w2"w + (-1 + (1 + a + b)w)"w + ab)w-af(-w-1) = 0.


F0
2



"
g(w) = gnwn.
n=0


"
n(n - 1)gnwn - ngnwn-1 + (1 + a + b)ngnwn + abgnwn = 0
n=0


(n - 1 + a)(n - 1 + b)gn-1 = ngn.


(a)n(b)n
gn = g0
n!




Å‚


e-tta(1 - wt)1-b = 0
Å‚


e-tta-1(1 - wt)-bdt
Å‚




ezssa-1(1 - s)c-a-1ds
Å‚


-1
w-a e-sw sa-1(1 - s)c-a-1ds,
Å‚


s
b = 1 + a - c t =
w




w " C\[0, "[ Rea > 0
"
1

F (a, b; -; w) := e-tta-1(1 - wt)-bdt.
“(a)
0


a


"
(a)n(b)n
F (a, b; -; w) <" wn,
n!
n=0


n |argw| e" > 0
ëÅ‚ öÅ‚
n
(a)j(b)j
lim w-n íÅ‚F (a, b; -; w) - wjÅ‚Å‚ = 0.
w0
j!
j=0



n-1
1
f(j)(0)zj f(n)(sz)n(1 - s)n-1
f(z) = + zn ds,
j! n!
0
j=0


n-1
(b)jzj (b)nzn 1
(1 - z)-b = + n(1 - s)n-1(1 - zs)-b-nds.
j! n!
0
j=0


F (a, b; -; w)
"
1
= e-tta-1(1 - wt)-bdt
“(a)
0
n-1
"
1 (b)jwjtj
= e-tta-1 dt
“(a) j!
0
j=0
"
1 (b)nwntn 1
+ e-tta-1 (1 - wts)-b-nn(1 - s)n-1ds
“(a) n!
0 0
n-1
(b)j“(a + j)wj
=
“(a)j!
j=0
"
wn(b)n 1
+ n(1 - s)n-1ds e-tta-1+n(1 - wts)-b-ndt
“(a)n!
0 0
n-1
(b)j(a)jwj
=
j!
j=0
wn(b)n(a)n 1
+ n(1 - s)n-1dsF (a + n, b + n; -; ws).
n!
0




"


s ezssa-1(1 - s)c-a-1,


Rea > 0 Re(c - a) > 0


1
“(a)“(c - a)
F (z) = ezssa-1(1 - s)c-a-1ds = F (a; c; z)
“(c)
0


Imz > 0


eiĆ"
F0(z) = ezssa-1(1 - s)c-a-1ds,
0
eiĆ"
F1(z) = ezssa-1(1 - s)c-a-1ds,
1


3Ä„
Ć "]Ą - argz, - argz[ ezs
2 2





F (z) + F1(z) - F0(z) = 0.


s = -z-1t t " [0, "[ Rea > 0
"
F0(z) = e-t(-z-1)a-1(1 + z-1t)c-a-1(-z-1)dt
0
= (-z)-a“(a)F (a, a + 1 - c; -, -z-1).


s = 1 - z-1t t " [0, "[ Re(c - a) > 0
"
F1(z) = -ez e-t(1 - z-1t)a-1z-c+atc-a-1dt
0
= -ezz-c+a“(c - a)F (c - a, 1 - a; -, z-1).




F (a; c; z) F (a, a + 1 - c; -; -z-1) ezF (c - a, 1 - a; -; z-1)
= (-z)-a + z-c+a
“(c) “(c - a) “(a)






2
e-z/2 z"z + (c - z)"z - a ez/2
c z
2
= z"z + c"z + - a - ;
2 4



c z

2
z-(1-c)/2 z"z + c"z + - a - z(1-c)/2
2 4
z c (1 - c)2

2
= z"z + "z - + - a - .
4 2 4z


z z = 2w
2
1 1 1 - c 1

2
"w + "w - 1 + (c - 2a) - ,
w w 2 w2


w2
2
1 - c

2
w2"w + w"w - w2 + (c - 2a)w - .
2










f eww(1-c)/2f(2w)






F1
0




c = 2a m := (1 - c)/2


1 m2

2
"w + "w - 1 - ,
w w2


f
1
2
z"z + (2m + 1 - z)"z - (m + ) f(z) = 0,
2


wme-wf(2w) wmewf(-2w)





wm

2m“(m+1)



m
1 w 1
I(w) := e-wF (m + ; 2m + 1; 2w)
“(m + 1) 2 2
m
1 w 1

:= ewF (m + ; 2m + 1; -2w).
“(m + 1) 2 2








w = iz
1 m2

2
"z + "z + 1 - .
z z2


wm

2m“(m+1)
m
1 z 1
J(z) := e-izF (m + ; 2m + 1; 2iz)
“(m + 1) 2 2
m
1 z 1
:= eizF (m + ; 2m + 1; -2iz)
“(m + 1) 2 2
Ä„

= e-i 2 mI(iz).




(" + º)F = 0.


d = 2


F = f(r)g(Ć) "Ćg(Ć) =



Cg(Ć) g(Ć) = eimĆ m2 = -C
1 m2
"rr"r - + º f(r) = 0.
r r2


º > 0
"
F (r, Ć) = eimĆg( ºr),


g º < 0
"
F (r, Ć) = eimĆg( -ºr),


g


º = 0
1 m2
"rr"r - f(r) = 0
r r2


rm r-m rmeimĆ r-meimĆ


(x + iy)m (x - iy)m





l(l + d - 2)
r-d+1"rrd-1"r - + º f(r) = 0
r2



d-2 l(l + d - 2) d-2
2
r rd-1"rrd-1"r - + º r- 2
r2
2
d - 2
= r-1"rr"r - l + ,
2




F1
0


F1
0


2
(w"w + c"w - 1)f(w) = 0.
Ü




c = 2a
z

2
z"z + c"z - .
4
1
1

2
z2 w

2
w = z = 4w "z = "w
16 2
1
2
3 w c 1 1

2
2 2 2
w "w + "w + w "w - w .
2 2

1
1+c
2
w c =
Ü
2


F1
0




2
(z"z + c"z - 1)f(z) = 0.

0 0, 1-c "



2 2
zc-1("z + c"z - 1)z1-c = "z + (2 - c)"z - 1.


1
"
zn
F (c; z) := .
(c)nn!
n=0



z1-c

z1-cF (2 - c; z)


F (c; z)
"
F (c; z) zn
Åš(c; z) := = .
“(c) “(c + j)j!
n=0




Åš1
0
m
w
Im(w) = Åš(1 + m; w2/4);
2
m
w
Jm(w) = Åš(1 + m; -w2/4);
2


F1
0


"
"
2c - 1
F (c; z) = e-2 zF ; 2c - 1; 4 z
2
"
"
2c - 1
= e2 zF ; 2c - 1; -4 z .
2






1




Å‚
etez/tt-c = 0.
Å‚


et+z/tt-cdt
Å‚




2
(z"z + c"z - 1)et+z/tt-c = "tet+z/tt-c.


1

Åš(c; z) = et+z/tt-cdt.
2Ä„i
]-",0+,"[







1 1
ett-cdt = .
2Ä„i “(c)
]-",0+,"[






Å‚
"
1
(t2 - 1)c- 2 e2t z = 0.
Å‚


"
3
(t2 - 1)c- 2 e2 ztdt
Å‚




"
1
1
"
3 “(c - ) Ä„


2
(1 - t2)c- 2 e2t zdt = F (c; z) .
“(c)
-1






"
1
1
3 “(c - ) Ä„
2
(1 - t2)c- 2 dt = .
“(c)
-1


F1
0


k " Z
Åš(1 + k; z) = z-kÅš(1 - k; z).

" "
zn-k zm
z-kÅš(2 - c; z) = = = Åš(1 + k; z).
(-k + n)!n! m!(k + m)!
n=k m=0


et+z/t = tnÅš(n + 1, z)
n"Z

1

Åš(n + 1; z) = et+z/tt-n-1dt.
2Ä„i
[0+]











2
z(1 - z)"z + (c - (a + b + 1)z)"z - ab f(z) = 0.




0 0 1 - c


1 0 c - a - b


" a b



1
"
(a)j(b)j zj
F (a, b; c; z) = ,
(c)j j!
j=0


F (a, b; c; z) c = 0, -1, -2, . . .



"
F (a, b, c, z) (a)j(b)j zj
Åš(a, b; c; z) := =
“(c) “(c + j) j!
j=0


a, b, c


z1-c


2
zc-1 z(1 - z)"z + (c - (a + b + 1)z)"z - ab z1-c
2
= z(1 - z)"z + (2 - c - (a + b + -2c + 3)z)"z - (b - c + 1)(a - c + 1)


z1-c


z1-cF (b + 1 - c, a + 1 - c; 2 - c; z)


1



0 1 w = 1 - z
2
z(1 - z)"z + (c - (a + b + 1)z)"z - ab
2
= w(1 - w)"w + (c - a - b + 1 - (a + b + 1)w)"w - ab.



1
F (a, b; a + b + 1 - c; 1 - z).



"

a, b


0 " w = z-1


2
(-z)1+a z(1 - z)"z + (c - (a + b + 1)z)"z - ab (-z)-a


2
= w(1 - w)"w + (a - b + 1 - (2a - c + 2)w)"w - a(a - c + 1) .


z-a
z-aF (a, a - c + 1; a - b + 1; z-1),


a b
z-bF (b - c + 1, b; b - a + 1; z-1).




z
0 1 " z w =
z-1
2
-(1 - z)1+a z(1 - z)"z + (c - (a + b + 1)z)"z - ab (1 - z)-a


2
= z(1 - z)"z + (c - (c + 1)z)"z - a(c - b) ,


a b
F (a, b; c; z)
= (1 - z)c-a-bF (c - a, c - b; c; z)
z
= (1 - z)-aF a, c - b; c;
z-1
z
= (1 - z)-bF c - a, b; c; .
z-1




Å‚
ta-c+1(1 - t)c-b(t - z)-a-1 = 0.
Å‚




ta-c(1 - t)c-b-1(t - z)-adt
Å‚




2
z(1 - z)"z + (c - (a + b + 1)z)"z - ab ta-c(1 - t)c-b-1(t - z)-adt
= -a"tta-c+1(1 - t)c-b(t - z)-a-1.




"
ta-c(t - 1)c-b-1(t - z)-adt


1

= “(b)“(c - b)Åš(a, b; c; z), Re(c - b) > 0, Reb > 0.






"
“(b)“(c - b)
ta-c(t - 1)c-b-1(t - z)-adt = .
“(c)
1












2
(z2"z + z"z + z2 - m2)v(z) = 0.


d
2
(z2"z + (d - 1)z"z + z2 - l(l + d - 2))u(z) = 0.


d
u(z) = z1- 2 v(z)


2 d
(z2"z + z"z + z2 - (l + - 1)2)v(z) = 0.
2


v(z) = zm}(z)




2
(z"z + (1 + 2m)"z + z)}(z) = 0.

2 2
v(z) = (z )mu(-z ), c = 1 + m, t = -z
2 4 4


2
(t"t + c"t - 1)u(t) = 0.


"


zv(z) = w(z)


2
"z + (1 - m2)z1 + 1 w(z) = 0.
2
4
"


tv(t´) = w(t)


2
"t + (´t´-1)2 + (1 - m2´2)t1 + 1 w(t) = 0.
2
4


1
v 1
Á+1
Á+1
"
Á
2
u(t) = tv 1 t1+ 2
Á+1
Á + 2


2
("t + tÁ)u = 0.







Å‚


t t-m
Å‚(1)
z z 1

(t + t-1) + m exp (t - t-1) = 0,
2 2 tm Å‚(0)

C
z dt

C exp (t - t-1)
2 tm+1
Å‚




z
2 z dt
(z2"z + z"z + z2 - m2) exp (t - t-1)
Å‚ 2 tm+1



2
z z z dt
= t - t-1 2 + t - t-1 + z2 - m2 exp (t - t-1) .
Å‚ 2 2 2 tm+1




Å‚(1)
z z 1
0 = (t + t-1) + m exp (t - t-1)
2 2 tm
Å‚(0)


z 1
= "t z (t + t-1) + m exp (t - t-1) dt

Å‚ 2 2 tm
2
z z z dt
= t + t-1 2 + t - t-1 - m2 exp (t - t-1) .
Å‚ 2 2 2 tm+1










Å‚(1)
1
(1 - t2)m+ 2 eizt = 0.
Å‚(0)


1
v(z) = zm (1 - t2)m- 2 eiztdt
Å‚




2
(z2"z + z"z + z2 - m2)v(z)
1 1
= m(m - 1)v(z) + 2mizm+1 Å‚(1 - t2)m- 2 - zm+2 Å‚(1 - t2)m- 2
eizttdt eiztt2dt
1
+mv(z) + izm+1 Å‚(1 - t2)m- 2
eizttdt + (z2 - m2)v(z)
1 1
1
= 2i(m + )zm+1 Å‚(1 - t2)m- 2
eizttdt + zm+2 Å‚(1 - t2)m+ 2 eiztdt
2
1
2
= -zm+1i "t(1 - t2)m+ eizt dt = 0
Å‚




0
( - 1) +  - m2 = 0.


0 Ä… = Ä…m



"
v(z) = vkzk+m,
k=0

+ - - = m - (-m) = 2m = -1, -2, . . .



vk (m + k)(m + k - 1) + (m + k) - m2 + vk-2 = 0.


vk-2
vk = - .
k(2m + k)


m = -1, -2, . . .

(-1)nv0
v2n+1 = 0, v2n = .
22nn!(m+1)...(m+n)



m = -1, -3, . . .
2 2

1
v0 :=
2m“(m+1)
(-1)n
v2n = .
22n+mn!“(m + n + 1)



vk m




Jm(z)
"
(-1)n z 2n+m
2
Jm(z) = .
n!“(m + n + 1)
n=0


Jm Ä…m


1

= 0 m = -1, -2, . . . 2m = -1, -2, . . . Jm
“(m+1)


m
z 1
Jm(z) <" , z <" 0,
2 “(m + 1)


f(z) <" g(z) z <" 0


f(z)
1
g(z)


m " Z J-m(z) Jm(z)




m

Rez > 0
1 z dt
Jm(z) = exp (t - t-1)
2Ä„i ]-",0+,-"[ 2 tm+1



m
1 z z2 ds
= exp s - .
2Ä„i 2 ]-",0+,-"[ 4s sm+1


z z 1
lim (t + t-1) + m exp (t - t-1) = 0,
Ret-" 2 2 tm


] - ", 0+, -"[
z dt
v(z) = C exp (t - t-1)
2 tm+1
]-",0+,-"[



zt
s =
2
m
z z2 ds
v(z) = C exp s - .
2 4s sm+1
]-",0+,-"[


-m
z ds 2Ä„i
lim v(z) = C es = C .
z0
2 sm+1 “(m + 1)
]-",0+,-"[



1
C = m = -1, -2, . . .

2Ä„i
v(z) = Jm(z).


m = -1, -2, . . .


0 < argz < Ä„ ]i", 0+, i"[




Ä„ "
1 1
Jm(z) = cos(z sin Ć - mĆ)dĆ - sin(mÄ„) e-z(sh²+m²)d², Rez > 0.
Ä„ Ä„
0 0


z
1
( )m 1
2
"
Jm(z) = (1 - t2)m- 2
eiztdt, m > -1.
1
Ä„“(m+ ) -1 2
2



1 1 z 1 1
Jm(z) = " - m)( )m (t - 1)m- 2 (t + 1)m- 2 ,
“(
2Ä„i Ä„ 2 2
[1,-1-,1+]
1 1 z 1 1
J-m(z) = e-iÄ„m " “( - m)( )m (t - 1)m- 2 (t + 1)m- 2 ,
2Ä„i Ä„ 2 2
[i",-1+,1+,i"]


F1 F1
0 1
2
1
Jm(z) = (z )m0F1(-; 1 + m; -z )
“(m+1) 2 4
1 1
= (z )me-iz1F1(m + ; 2m + 1; 2iz).
“(m+1) 2 2


m



m " Z

Jm(z) = (-1)mJ-m(z), m " Z.



m = 0, 1, . . .
(-1)n z
( )2n+m
"
2
Jm(z) =
n=0
n!(n+m)!
(-1)n+m
(z )2(n+m)-m
2
= (-1)m " (n+m)!(n+m-m)!
n=0
(-1)n
(z )2n-m
2
= (-1)m " n!(n-m)!
n=m
(-1)n 2
(z )2n-m
= (-1)m " n!“(n-m+1) = J-m(z).
n=0



m " Z


0 0



C


Jm(z)


m " Z
1 z dt
Jm(z) = exp (t - t-1)
2Ä„i [0+] 2 tm+1
m
1 z z2 ds
= exp s - .
2Ä„i 2 [0+] 4s sm+1




z dt
v(z) = C exp (t - t-1)
2 tm+1
[0+]


zt
s =
2
m
z z2 ds
v(z) = C exp s - .
2 4s sm+1
[0+]


-m
z ds 2Ä„i
lim v(z) = C es = C .
z0 2 sm+1 m!
[0+]


1
C = m = 0, 1, 2, . . .
2Ä„i
v(z) = Jm(z).


w = -1 t - t-1 = w - w-1 dt = -dw [0+]
t t w


[0-]
z dt z dw
exp (t - t-1) = (-1)-m+1 [0-] exp (w - w-1)
2 tm+1 2 w-m+1


[0+]

z dw
= (-1)-m [0+] exp (w - w-1)
2 w-m+1



m = 0, -1, -2, . . . (-1)-mJ-m(z)


Jm(z)


m = -1, -2, . . .




"
z
exp (t - t-1) = tmJm(z).
2
m=-"



z
t exp (t - t-1)
2


C\{0}


Ä„
1
Jm(z) = cos(z sin Ć - mĆ)dĆ, m " Z.
Ä„
0












m " Z


Rez > 0
1 z dt
(1)
Hm (z) = - exp (t - t-1) ,
Ä„i 2 tm+1
]-",(0+1·0)-[
1 z dt
(2)
Hm (z) = exp (t - t-1)
Ä„i 2 tm+1
]-",(0+1·0)+[



] - ", (0 + 1 · 0)-[ -" 0

] - ", (0 + 1 ·



0)+[ -" 0





z z 1
lim (t + t-1) + m exp (t - t-1) = 0,
t0+1·0 2 2 tm

t 0+1·0 t


t 0+ ] - ", (0 + 1 · 0)+[ ] - ", (0 + 1 · 0)-[





(1)
0 < argz < Ä„ Hm [i", 0] -Ä„ <


(2)
arg < 0 Hm [-i", 0]




(1) (1)
H-m(z) = emĄiHm (z),
(2) (2)
H-m(z) = e-mĄiHm (z),
(1) (2)
1
Jm(z) = Hm (z) + Hm (z) ,
2
(1) (2)
1
J-m(z) = emĄiHm (z) + e-mĄiHm (z) ,
2
(1) ie-mĄiJm(z)-iJ-m
(z)
Hm (z) = ,
sin mĄ
(2) -iemĄiJm(z)+iJ-m
(z)
Hm (z) = .
sin mĄ



t = -1
s


]-", (0+1·0)+]

Ä„ 1
] - ", -1] *" {-ieiĆ, Ć " [-Ą , ]} *" [1, 0] w = -
2 2 t



t - t-1 = w - w-1 t-1dt = -w-1dw t-1 =



(-1)-mw-m = eiĄmwm





] - ", (0 + 1 · 0)+[ *" ](0 + 1 · 0)+, -"[


] - ", 0+, -"[





0
Jm(eÄ…i2Ä„z) = eÄ…im2Ä„Jm(z),
(1) (2)
Hm (eiĄz) = -e-im2ĄHm (z),
(2) (1)
Hm (eiĄz) = -eim2ĄHm (z).


ie-miĄJm(eiĄz) - iJ-m(eiĄz) iJm(z) - ie-imĄJ-m(z)
(1) (2)
Hm (eiĄz) = = = -e-imĄHm (z).
sin mĄ sin mĄ


w = -t



-Ä„+´ < argz < 2Ä„-´
´ > 0
(1)
Hm (z)
lim = 1,
1
imĄ iĄ
z"
2 -
2
eize- 2 4
Ä„z
(2)
Hm (z)
lim = 1.
1
imĄ iĄ
z"
2 +
2
2 4
e-ize
Ä„z


1 dt
(1)
Hm (z) = - eĆ(t) ,
Ä„i tm+1
]-",(0+1·0)-[

z
Ć(t) = (t - t-1),
2
z
Ć (t) = (1 + t-2),
2
Ć (t) = -zt-3.



Ć(t) tą = ąi
Ć(ąi) = ązi, Ć (ąi) = "zi.



H(1) t+ = i t = t+


ReĆ(t) - Ć(t+) = Rez (t-i)2 < 0
2 t
(1)
1 eĆ(i) " 1 Ć (i)(t-i)2
2
Hm (z) <" -Ä„i im+1 -" e dt
1
imĄ iĄ
1 eiz 2 -
2
= -Ąi eiĄ(m+1)/2 2Ą = eize- 2 4
.
iz Ä„z


0 < argz < Ą [0, "[ -Ą < argz < 0 [0, -i]*"{eiĆ, :

3Ä„
Ć " [-Ą , -5Ą ]} *" [-i, -i"[ Ą < argz < 2Ą [0, -i] *" {eiĆ, : Ć " [-Ą , ]} *" [-i, -i"[
2 2 2 2


imĄ
-
(1)
Hm (z) = -ie 2 " eizht-mtdt, 0 < argz < Ä„,
Ä„ -"
imĄ
(2) " Ä„
2
2ie
Hm (z) = eizhth(mt - imĄ)dt - i e-iz cos t cos mtdt , 0 < argz < Ą,
Ä„ 0 0
-imĄ
(1) Ä„
Hm (z) = -2ie 2 " e-izhth(mt + imĄ)dt + i eiz cos t cos mtdt , -Ą < argz < 0,
Ä„ 0 0
imĄ
(2) "
2
ie
Hm (z) = e-izht-mtdt, -Ä„ < argz < 0
Ä„ -"


“(1 - m) 1 1
z
(1) 2
Hm (z) = " ( )m eizt(t - 1)m- 2 (t + 1)m- 2 dt,
Ä„i Ä„ 2
]i",1+,i"[
“(1 - m) 1 1
z
(2) 2
Hm (z) = " ( )m eizt(t - 1)m- 2 (t + 1)m- 2 dt,
Ä„i Ä„ 2
]i",-1-,i"[


m e" -1
2
2 z 1
(1)
Hm (z) = -" ( )m eizt(1 - t2)m- 2 dt,
1
2
Ä„“(m + )
]1,i"[
2
2 z 1
(2)
Hm (z) = " ( )m eizt(1 - t2)m- 2 dt,
1
2
Ä„“(m + )
]-1,i"[
2


m Ä„
- ) 1 1
(1)
2 it
Hm (z) = zm ei(z-Ä„ 2 4 " e-zttm- 2
(1 + )m- 2
dt,
1
Ä„ 0 2
“(m+ )
2
m Ä„
- ) 1 1
(2)
2 it
Hm (z) = zm e-i(z-Ä„ 2 4 " e-zttm- 2 - )m- 2
(1 dt.
1
Ä„ 0 2
“(m+ )
2






"
1 + u2

lim = 1,
u"
u
"


u 1 + u2 C\[-i, i]


1

C\{0} t u(t) = (t - t-1).
2

C\{0}
&!+ := {t " C : |t| > 1},
&!0 := {t " C : |t| = 1},
&!- := {t " C : |t| < 1}.


&!+ &!- C\[-i, i] &!0 [-i, i]


u(-1) = u(1) = 0 u(-i) = -i u(i) = i

u t(u)


"
C\[-i, i] u t+(u) = u + 1 + u2 " &!+,
"
C\[-i, i] u t-(u) = u - 1 + u2 " &!-,







t u(t)


Rez > 0
1 ezudu
" "
Jm(z) =
2Ä„i [-",-i+,i+,-"]
1+u2(u+ 1+u2)m


t u(t)



Ä„ Ä„
] - ", -i] *" {eiĆ : Ć " [- , ]} *" [i, "[,
2 2


] - ", 0+, -"[
] - ", -i+, i+, -"[.



Ä…i




m " Z
1 ezudu
" "
Jm(z) =
2Ä„i [-i,i+,-i+]
1+u2(u+ 1+u2)m
1 ezudu
" "
= .
2Ä„i [-i,i+,-i+]
1+u2(u- 1+u2)m


r > 0
1 z dt
Jm(z) = exp (t - t-1)
2Ä„i 2 tm+1
"K(0,r)


r > 1 "K(0, r) ‚" &!+ t u(t) "K(0, r)


[-i, i+, -i+] t " &!+
"
du u + 1 + u2
= " .
dt
1 + u2




1 > r > 0 "K(0, r) ‚" &!- t u(t)


"K(0, r) [-i, i-, -i-] t " &!-
"
du u - 1 + u2
= " .
dt
- 1 + u2


[-i, i-, -i-] [-i, i+, -i+]





(1)
1 ezudu
" "
Hm =
Ä„i [-",i+,-"]
1+u2(u+ 1+u2)m
1 ezudu
" "
=
Ä„i [-",i-,-"]
1+u2(u+ 1+u2)m
1 ezudu 1 ezudu
" " " "
= + ,
Ä„i [-",i]
1+u2(u+ 1+u2)m Ä„i [-",i] 1+u2(u- 1+u2)m
(2)
1 ezudu
Hm = -Ä„i [-",-i+,-"] "1+u2(u+"1+u2)m
1 ezudu
= -Ä„i [-",-i-,-"] "1+u2(u+"1+u2)m
1 ezudu 1 ezudu
" "
= -Ä„i [-",-i] "1+u2(u+"1+u2)m -
Ä„i [-",-i]
1+u2(u- 1+u2)m


t u(t).





Ä…i


Ä…i


" "


1 + u2 1 + u2


(1) (2)
Hm (z) Hm (z)



(1) (1)
H-m(z) = emĄiHm (z),
(2) (2)
H-m(z) = e-mĄiHm (z).


"
(1) ezu(u+ 1+u2)mdu
1
"
H-m(z) =
Ä„i [-",i+,-"]

1+u2

m
1 -1 ezudu
" "
= .
Ä„i [-",i+,-"]
u- 1+u2 1+u2
"


(-1)m = eiĄm - 1 + u2
"



1 + u2 ] - ", i+, -"[
"


1 + u2 ] - ", i-, -"[


eiĄm ezudu
(1)
" " = eimĄHm (z).
Ä„i
1 + u2(u + 1 + u2)m
[-",i-,-"]





(1)
Hm (z) Ä„
lim = 1, |argz| < - ´,
1
imĄ iĄ
z"
2 - 2
2
eize- 2 4
Ä„z
(2)
Hm (z) Ä„
lim = 1 |argz| < - ´.
1
imĄ Ą
z"
2 + 2
2
2 4
e-ize
Ä„z


w2
u = i -
2


] - ", i+, -"[


] - ", "[
"
1 w2
(1)
Hm (z) = f(w)ez(i- 2 )dw,
Ä„i
-"

1
f(w) := .
m
w2 w2 w2
-i + i - + w -i +
4 2 4

q
w2
-i+

1
limw" w 4 =
2
1 iĄ imĄ
-
4 2
"
f(0) = = e .
iim


(1)
Hm (z)
1 1
"
2 2
1 zw2 1 iĄ imĄ 2Ą 2 imĄ iĄ
-
4 2
f(0)eiz e- 2 2dw = e eiz = eize- 2 - 4 .
Ä„i Ä„i z Ä„z
-"




(1) (2)
1
Ym(z) = (Hm (z) - Hm (z))
2i
cos Ä„mJm(z)-J-m
(z)
= .
sin Ä„m


(1) (2)
Hm (z) = Jm(z) + iYm(z), Hm (z) = Jm(z) - iYm(z).


m " Z
2
Ym(z) = (log(z ) + Å‚)Jm(z)
Ä„ 2
m-1 "
(-1)k
1 1
-Ä„ (m-k-1)!(z )2k-m - (z )m+2k(h(k) + h(m + k)),
k! 2 Ä„ k!(m+k)! 2
k=0 k=0

k
1
h(k) :=
j=1
k


d 1 1
Ć(z) := = - "z log “(z).
dz “(z) “(z)

Ć(-n) = (-1)nn!, n = 0, 1, 2, . . . ,
Å‚-h(n)
Ć(n + 1) = , n = 0, 1, 2, . . .
n!

" (-1)kĆ(m+k+1)
"mJm(z) = log(z )Jm(z) + (z )m+2k.
k=0
2 k! 2


n = 0, 1, 2, . . .
" (-1)kh(n+k)
z
"mJm(z) = (log + Å‚)Jn(z) - (z )n+2k,
k=0
2 (n+k)!k! 2
m=n
z
"mJm(z) = (log + Å‚)J-n(z) - (-1)n n-1 (n-k-1)!(z )2k-n
2 k=0 k! 2
m=-n
" (-1)kh(-n+k)
- (z )-n+2k.
k=n
(-n+k)!k! 2



"
(-1)kh(k) z
-(-1)n ( )n+2k.
k!(k + n)! 2
k=n


"m( cos Ä„mJm(z)-J-m
(z))
Yn(z) =
"m sin Ä„m
m=n
cos Ä„m"mJm(z)+"(-m)J-m
(z)
1
= = "mJm(z) + (-1)m"mJm(z) .
Ä„ cos Ä„m Ä„
m=n m=n m=-n




z = iz
Ü
2
(z2"z + z"z - z2 - m2)u(z).



v(z) v(iz)


Im(z) = i-mJm(iz)
"
1
= (z )2n+m
n=0
n!“(n+m+1) 2
1
= exp(z (t + t-1))t-m-1dt
2Ä„i ]-",0+,-"[ 2
1 z2
= (z )mF (m + 1; )
“(m+1) 2 4
1 1
= (z )me-zF (m + ; 2m + 1; 2z).
“(m+1) 2 2


Ä„
Km(z) = K-m(z) = (I-m(z) - Im(z))
2 sin mĄ
"
1
= exp(-z (t + t-1))t-m-1dt
2 0 2
(1) (2)
= im+1 Ä„ Hm (iz) = i-m-1 Ä„ Hm (-iz).
2 2


(1)
Hm (z) = -2iKm(-iz),
Ä„
(2)
2i
Hm (z) = Km(iz).
Ä„


n = 0, 1, 2, . . .
In(z) = I-n(z),
z
Kn(z) = (-1)n+1(log + Å‚)In(z)
2
(-1)n h(k)+h(n+k)
"
+1 n-1 (-1)m(z )2m-n (n-m-1)! + (z )2m+n.
2 m=0 2 m! 2 k=0
k!(n+k)! 2









2"zJm(z) = Jm-1(z) - Jm+1(z),
2mJm(z) = zJm-1(z) + zJm+1(z).


(1) (2)

Hm (z) Hm (z) Ym(z)





Å‚


z dt
2"z Å‚ exp (t - t-1)
2 tm+1
z dt z dt
= exp (t - t-1) - exp (t - t-1) .
Å‚ 2 tm Å‚ 2 tm-2





z 1
t exp (t - t-1)
2 tm

Å‚
z 1
0 = 2 "t exp (t - t-1) dt
Å‚ 2 tm
z dt z dt z dt
= -2m exp (t - t-1) + z exp (t - t-1) + z exp (t - t-1) .
Å‚ 2 tm+1 Å‚ 2 tm Å‚ 2 tm+2




1 m
"z (zmJm(z)) = zm-1Jm-1(z), "z + Jm(z) = Jm-1(z),
z z
1 m
- "z z-mJm(z) = z-m-1Jm+1(z), -"z + Jm(z) = Jm+1(z).
z z


n
1
"z zmJm(z) = zm-nJm-n(z),
z
n
1
- "z z-mJm(z) = z-m-nJm+n(z).
z




"

m = -1 v(z) = z}(z)
2


}
2
("z + 1)} = 0,


1
eiz e-iz m =
2
1 1


z- 2 2
eiz z- e-iz
1

z 1
2
1
2
“(1+ )
2
1
1
2
z 1 sin z 2
2
J1 (z) = = sin z.
1
2
2 z Ä„z
“(1 + )
2


1
1
- 2
z 1 2
2
J- 1 (z) = cos z = cos z.
1
2
2 Ä„z
“(1 - )
2
1
Ä„
2 )
2
H(1,2)(z) = eÄ…i(z- 2
,
1
Ä„z
2
1
(1,2)
2
2
H- 1 (z) = eÄ…iz
Ä„z
2


m



1
2
2 1
(1)
Hn+ 1 (z) = eizpn ,
Ä„z iz
2

pn




1
"z + W (z) = 0.
z


1
W (z)
z
1 1
JÄ…m(z) <" (z )Ä…m, JÄ…m(z) <" (z )Ä…m-1,
“(Ä…m+1) 2 “(Ä…m) 2


Jm(z) J-m(z)
2
W (Jm, J-m) = -Ä„z sin Ä„m,
(1) (2)
4i
W (Hm , Hm ) = -Ä„z ,
2
W (Jm, Ym) = .
Ä„z




2 2
" = "x + "y


x = r cos Ć, y = r sin Ć


1 1
2 2
"r + "r + "Ć.
r r2


(" + 1)f = 0


È


fÈ(x, y) := ei(x cos È+y sin È)


fÈ(r, Ć) = eir cos(Ć-È).


L := x"x - y"x,


L = "Ć.


L "
(" + 1)f = 0, Lf = imf.



fm(r, Ć) = Jm(r)eimĆ.





"
Ä„

fÈ(r, Ć) = fm(r, Ć)e-im(È- 2 ).
m=-"





2Ä„
1 Ä„

fm(r, Ć) = fÈ(r, Ć)(-i)meim(È- 2 )dÈ.
2Ä„
0




"
eir sin Ć = eimĆJm(r),
m=-"



2Ä„
1
Jm(r) = eir sin È-imÈdÈ.
2Ä„
0









R r Á Åš Ć È

reiĆ + ÁeiÈ
R = (reiĆ + ÁeiÈ)(re-iĆ + Áe-iÈ), eiÅš = .
re-iĆ + Áe-iÈ


"
Jm(R)eimÅš = Jm-n(r)ei(m-n)ĆJn(Á)einÈ.
n=-"



m " Z m




Á < r



(i)

Ä„
|Ś - Ć| < Jm(R) Jm-n(r) Hm Ym

2


Ü Ü
È = È - Ć Åš = Åš - Ć Ć = 0
"
Jm-n(r)Jn(Á)einÈ
n=-"
"
1 r
= exp (2 (t - t-1))t-m-1Jn(Á)(teiÈ)n
n=-"
2Ä„i Å‚
Á
1 r
= exp (t - t-1)) + (teiÈ - (teiÈ)-1) t-m-1dt
2Ä„i Å‚ 2 2
1
= exp (R(s - s-1))s-m-1dseimÅš = eimÅšJm(R),
2Ä„i Å‚ 2



s = teiÅš r + ÁeiÈ = ReiÅš



x1 = r cos Ć, y = r sin Ć,
x2 = Á cos È, y2 = Á sin È,
x = R cos Åš, y = R sin Åš,


(x1, y1) + (x2, y2) = (x + y)


m m-n n
x x
2 2
"x+iy = n"Z Jm-n( x2 + y1) "1+iy1 Jn( x2 + y2) "2+iy2 .
Jm( x2 + y2)
1 2 2 2
x2+y2 x2+y1 x2+y2
1 2


L2(Z)
m-n
"x+iy
Um,n(x, y) := Jm-n( x2 + y2)
x2+y2



U(x, y)-1 = U(-x, -y) U(x, y)
Un,m(x, y) = Um,n(-x, -y),



R2 (x, y) U(x, y)
"
Uk,n(x2 + x1, y2 + y1) = Uk,m(x2, y2)Um,n(x1, y1).
m=-"


R2


R2 × S1
(x2, y2, ¸2)(x1, y1, ¸1) = (x2 cos ¸1 + y2 sin ¸1 + x1, -x2 sin ¸1 + y2 sin ¸1 + y1, ¸2 + ¸1).


R2 × S1 (x, y, ¸) U(x, y, ¸),
m-n
"x+iy .
Um,n,¸(x, y) := eim¸Jm-n( x2 + y2)
x2+y2



U(x, y, ¸)
"
Uk,n (x2, y2, ¸2)(x1, y1, ¸1) = Uk,m(x2, y2, ¸2)Um,n(x1, y1, ¸1).
m=-"




2
("z - z)u(z) = 0.


2Ä„i
u(z) u(eÄ… 3 z)





Å‚
Å‚(1)
i
e3 t3+itz = 0.
Å‚(0)


i
t3+itz
3
C e dt
Å‚




i
2 t3+itz
3
("z - z) e dt
Å‚
i i
t3+itz t3+itz
3 3
= (-t2 - z)e dt = i"te dt = 0.
Å‚ Å‚
1

t3-tz
3
t = is e


i 1
1 1 t3+tz
Ai(z) := e3 t3+itzdt = e- 3
dt
2Ą ]-","[ 2iĄ ]-i",i"[
1
= cos(-1t3 + tz)dt
Ä„ ]-","[ 3
iĄ
s3 iĄ -iĄ s3 -iĄ
3 " 3 3 " 3
e -sze e -sze
= e- 3
ds - e- 3
ds
2Ä„i 0 2Ä„i 0


"
0 u(z) = umzm
m=0


n(n - 1)un = un-3.


"
m
1
Ai(0)(z) := z3m,

3j(3j
j=1 - 1)
m=0
"
m
1
Ai(1)(z) := z3m+1,

3j(3j + 1)
j=1
m=0

Ai(0)(0) = 1, Ai(0) (0) = 0;
Ai(1)(0) = 0, Ai(1) (0) = 1.

2 1
3- 3
3- 3
Ai(z) = Ai(0)(z) + Ai(1)(z).
“(2) “(1)
3 3



Ä„ 2
s3
sin
"
3- 3
3
Ai(0) = e- 3
ds = ,
2
Ä„ 0
“( )
3
2Ä„ 1
s3
sin
"
3- 3
3
Ai (0) = - e- 3
sds = .
1
Ä„ 0
“( )
3


3 1 1
2 2
Ai(0)(z) = I- 1 (2z )z (2 )3 “(2)
3 3 3
3
3 1 1
2 2
= J- 1 (2(-z) )(-z) (1 )3 “(2),
3 3 3
3
3 1 1
2 2
Ai(1)(z) = I1 (2z )z (2 )- 3
“(4)
3 3 3
3
3 1 1
2 2
= -J1 (2(-z) )(-z) 3- 3
“(4),
3 3
3
1 3 1 3 3
1
3 2 2 2 2
Ai(z) = Ä„-1(z ) K1 (2 z ) = z I- 1 (2z ) - I1 (2z )
3 3 3 3 3
3 3 3
1 3 3
1
2 2 2
= (-z) J- 1 (2(-z) ) + J1 (2(-z) )
3 3 3
3 3