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f ∈ C ∞ C 1
−1 < k < 0 Γ x = k (k 2 + y 2 )(y 2 +
k(k + 1)) = 4ky 2 y 4 + (2k 2 − 3k)y 2 + k 4 + k 3 = 0 t = y 2
pk(t) := t 2 + (2k 2 − 3k)t + (k 4 + k 3 ) = 0
y t
limt→±∞ pk(t) = +∞ pk(0) = k 4 + k 3 < 0
t − < 0 t + > 0
t− y± = ± √ t +
t+ = −(2k2 − 3k) + k2 (9 − 16k)
.
2
Γ
y = ϕ + (x) y = ϕ− (x)
C1 y = y(m) y ′ (m) = 0
0 = (9m 2 − 1)(1 + m 2 ) 2 − 4m 2 (3m 2 − 1)(1 + m 2 )
= (1 + m 2 )((9m 2 − 1)(1 + m 2 ) − 4m 2 (3m 2 − 1))
= (1 + m 2 )(−1 + 12m 2 − 3m 4 ),
m m = ± (6 ± √ 33)/3
y(m) m = ± (6 ± √ 33)/3
Si i = 1, . . . , 4 −1 < x < 0
y2 x = −1 x = 0
y = 0 ( 3
32 (1+√ 33) 2 −1 < x < 0
y = ϕ + (x)
(−1, 0) y P2
y = ϕ + (−) ±2π/3
x = x(m) x ′ (m) = 0
0 = 6m(1 + m 2 ) 2 − 4m(3m 2 − 1)(1 + m 2 )
= 2m(1 + m 2 )(3(1 + m 2 ) − 2(3m 2 − 1)) = 2m(1 + m 2 )(5 − 3m 2 ),
m = 0 m = ± 5/3 Qi
i = 1, 2, 3
Si Qj i = 1, ..., 4 j = 1, 2, 3 Γ
P1, P2, P3 Γ
r, R > 0 R > r
D R r = {(x, y) ∈ R 2 : x ≥ 0, y ≥ 0, r 2 ≤ x 2 + y 2 ≤ R 2 },
lim
r→0 +
DR r
√
ye x2 +y2 x2 dxdy.
+ y2