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§ & ' ( ) * + , - . / * 0 1 2 0
3 4 5 6 7 8 9 : ; < = > ? @ A B
d2w dw
+ p(z) + q(z)w = 0, ()
dz2 dz
D E F G H I J K
p(z) C q(z)
• F G H L M N O PF G H I J Q R H K
• S T M UF G L H L V W M N O PF G I J H L V W Q R H K
c d e e m n op q e r s
X Y Z [ \ [ ] ^ _ ` a b [ f g hi j k l Y Z
K
U z0
e e
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` a tZ l [ v w x y k z Y Z [ l [ v w x y k
u U U
p(z), q(z) u z0 z0
e ~
z Y Z [ g
Y Z g Y Z
U U
Taylor Laurent K
L V U D E F G H K
• p(z), q(z) z0 z0
L V U D E F G H K
• p(z), q(z) z0 z0
F G (Hypergeometric equation)
d2w dw
z(1 − z) + γ−(1 + α+ β)z −αβw = 0
dz2 dz
H I J M
γ−(1 + α+ β)z αβ
p(z) = C q(z) = −.
z(1 − z) z(1 − z)
U C C K U C M F G
p(z) q(z) z = 0 z = 1 z = 0 z = 1
H U H   M F G H K
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Legendre F G
d2y dy
1 − x2 − 2x + l(l + 1)y = 0,
dx2 dx
H E K
x = ±1
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M M F G H U § ¨ © ª H K
z = ∞ () z = 1/t
dw dw d2w d2w dw
= −t2 , = t4 + 2t3 .
dz dt dz2 dt2 dt
«
® ¯
E
UF G ()
d2w 2 1 1 dw 1 1
+ − p + q w = 0. ()
dt2 t t2 t dt t4 t
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M F G H U D M F G H K
t = 0 () ( ) z = ∞() ( )
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§ ° ± ² °
§ 2
³ E F G H ´ µ M
t = 0 ( z = ∞)
1
p = 2t + a t2 + a t3 + · · · ,
t 2 3
1
q = b t4 + b t5 + · · · ,
t 4 5
³
2 a a
p(z) = + 2 + 3 + · · · ,
z z2 z3
b b
q(z) = 4 + 5 + · · · .
z4 z5
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M F G C
F G H K
Legendre
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3
§ < = 9 ¶ · ¸ ¹ > º
» ¼
U H R Å K
½ ¾ ¿À Á Â Ã Ä
Æ Ç
¯
C
È ÉÊ Ë L V U
È É Ì Í F G Î Ë Ï Ð
p(z) q(z) |z − z