Section 4: Applied Mathematics
KEY Section 1: Algebra 4.1 1.1 1.2 1.3 1.4 1.5 1.6
9 b,c The multiplicative a,c a − 2b − c = 0 1 1 0 0
group {−1, 1} 4.2 c= 3 0 1 2
0
4 2 0
4.3
1 3
π √ 7 y(x) =
sin 1 4 e
b,c a,b R\{−1} b
4.10
1
Z
3
0
|f (t)| dt
Section 5: Miscellaneous
0
5.1 ∞ X
(−1)
n−1
x (2n − 1)(2n) 5.2 5.3 5.4 5.5 5.6 5.7 5.8
(4n+1)π
2.9 e , n∈Z 2.10 {z = x + iy : y ≥ 0} Section 3: Topology 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10
m!(m + 1)! (m − n + 1)!
2n
n=1
x2 − x 4
4.5 L(y 00 )(s) = s2 L(y)(s) − sy(0) − y 0 (0) 4.6 x = 0, irregular singular point, x = 1, regular singular point 4.7 0 4.8 y(x) = x3 + C1 x2 + C2 4.9 Z b (b − a) f (x) dx ∼ [f (a) + 4f ((a + b)/2)) + f (b)] 6 a
Section 2: Analysis
2.8
−2 b−a
4.4
1.7 0 1.8 a,b,c 1.9 a √ 1.10 λ2 − 3 2λ + 4
2.1 2.2 2.3 2.4 2.5 2.6 2.7
2
λ = (2n + 1)2 π4 , u = C sin(2n + 1) π2 , n = 0, 1, 2, · · ·
b a,c B and D are homeomorphic a,b,c a,b a,b b a,b c b,c
2n−1 (3n + 2) a,b log N 0 3 2 x √ − 16x + 64x − 9 = 0 37
5.9 a,b 5.10 a,b
1
a + 3b 4
