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1 5 Paper I.R Paper HP Paper A 16 A A A A A A B R- 21 B B C R- 23 1

2 D 24 D D D D D.5 Hom

3 5 [ ] R R- Paper G (group) G 2 G G G (G1)[ ] a, b, c G (ab)c = a(bc) (G2)[ ] e G a G ae = ea = a (G3)[ ] a G G a 1 aa 1 = a 1 a = e G (G4)[ ] a, b G ab = ba G 5.2. R M R M R M M; (r, m) r m M R- (M1) r (m + m ) = r m + r m (M2) (r + r ) m = r m + r m (M3) (rr ) m = r (r m) (M4) 1 m = m R- M R M R, S M R- S- (r m) s = r (m s) M (R, S)- R R- M M R M; (m, r) m r r m M R- R- R- R- K K- K 3

4 5.2 R- 1. R R- R R 2. 1 [resp. ] M p- M [resp. ] A 3. 2 K L ϕ K- f(t) K[t] f(t) x = f(ϕ)(x) ( x L) L K[t]- (M1) f (x + y) = f(ϕ)(x + y) = f(ϕ)(x) + f(ϕ)(y) = f x + f y 2 ϕ (M2) (f + f ) x = ((f + f )(ϕ))(x) = (f(ϕ) + f (ϕ))(x) = f(ϕ)(x) + f (ϕ)(x) = f x + f x 2 (M3) (ff ) x = (ff )(ϕ)(x) = f(ϕ)f(ϕ)(x) = f (f x) 2 (M4) 1 m = m L K[t] R n f (support) 3 R[x 1,, x n ] = {f : n } supp f := {x R n : f(x) 0} D K = {f : R n C } S = {f : R n C } 1 2 L K[t]- 3 4

5 4 0 (fg)(x) := f(x)g(x) 3 3 / x 1,, / x n [ ] R,, x 1 x n / x i t i n 3 R[x 1,, x n ] D K S R C R n C 5 5. R 2 R- M, N M N {(m, n) m M, n N} (m 1, n 1 ) + (m 2, n 2 ) = (m 1 + m 2, n 1 + n 2 ) R r (m, n) = (r m, r n) R- M, N R- R- M N (direct sum) M N 6 1 R R- R- R R n n (free module) 7 R n R n m = (r!,, r n ) (r i R) e i = (0,, 0, 1, 0,, 0) m = r i e i 8 9 R σ = R ( σ Σ) R σ = {(r σ ) σinσ, r σ R σ, σ a σ = 0} σ Σ r σ e σ {e σ } M Z M M; (n, x) n x := x + x + + x(n ) 11 M Z- Z B 7 R C 8 C 9 10 B C 11 n > 0 x n n < 0 x n 5

6 ş ť ş ť 5.3 R M R- N M R N N N M R- (sub module) R N = {r n r R, n N} 13 R R- M, M f : M M R f(r m) = r f(m) ( r R, m M) f R- 14 f R- f R R n n r i 1 < <i r a i1,,i r dx i1 dx ir a i1,,i r R n A r n r A- A ( ) n r n C r i 1< <i r a i1,,i r dx i1 dx ir (a i1,,i r ) i1<i 2< <i r A 8. C[x 1,, x n ] M 1 C[x 1,, x n ] 4 N 2 M, N f N D Df = 0 15 n r Claim. R F f = a m X m + a m 1 X m 1 +, g = b n X n + b n 1 X n 1 + fg = a m b n X m+n + (a m b n 1 + a m 1 b n )X m+n 1 + R a m, b n 0 a m b n 0 fg 0 0 f M g gf = 0 16 M, N R N 14 R M R M 15 x i x i0 d 0 D d0+1 / x d 0+1 i 0 Df = 0 16 M 6

7 M, N A = C[x 1,, x n ] M N [ ] C[t 1,, t n ] = C,, x 1 x n M N A M = N C F α, β 0 a ( ) β xα F x a α, β (Z 0 ) n multi index α = (α 1,, α n), β = (β!,, β n) x α = x α 1 1 xα n n ( / x) β = β 1+ +β n / x β 1 1 x β n n R R- R I R- R R 5.1. A I A- A I I (0) I = (i) f : A I; a a i I I a i = 0 A a = 0 i = 0 i = 0 I = (0) f x I I x = yi y A f f A- A I A- f : A I A- A 1 f(1) = a a A f(a) = f(a 1) = a f(1) = ai f A- I = (i) d Z[ d] := {a + b d; a, b Z} Z[ d] Z[ d]- Z[ d] {m a } A M 20 n a i m ai i=1 M A- {m a } A- M A A {m a } {m a } M {m a } C 20 {m a } 7

8 5.5 R- M R- N 21 M N x y x y N M [m] = m + N [m] + [m ] = [m + m ] 22 well-defined m 1 m 1, m 2 m 2 m 1 m 1 + m 2 m 2 N [m 1 + m 2 ] = [m 1 + m 2] M/N R M/N (r, [m]) r [m] := [r m] M/N R well-defined m m r m r m = r (m m ) r N N [r m] = [r m ] M/N R- R- R- M M/N : m [m] r [m] = [r m] R- R- f : M M Ker f = {x M f(x) = 0}, Im f = {f(x) x M} M, M R- R x Ker f f(r x) = r f(x) = r 0 = 0 r x Ker f y Im f x M r y = r f(x) = f(r x) r x M y Im f R- R- R- f : M M M/ Ker f = Im f (R- ) [m] f(m) f(m) = 0 m ker f [m] = [0] R- r [m] = [r m] = f(r m) = r f(m) well-defined [m] = [m ] m m ker f 0 = f(m m ) = f(m) f(m ) f(m) = f(m ) 4 f : R S Im f Ker f R 1 1 R- f : M M R- Im f R- Ker f M R- 1 1 Im f R- N f 1 (N ) Ker f M R- M R- N f(n) Im f R G N x y xy 1 N G G/N [x] [y] = [xy] well-defined G g G gng 1 = N G N 8

9 5.6 R- M N 2 R- 3 R- universarity R R M R-,N R- A ϕ : M N A R- R-bilinear map; R- ;R-balanced map m M, n N, r R ϕ(m 1 + m 2, n) = ϕ(m 1, n) + ϕ(m 2, n), ϕ(m, n 1 + n 2 ) = ϕ(m, n 1 ) + ϕ(m, n 2 ), ϕ(mr, n) = ϕ(m, rn) R R- M R- N M R N R- τ : M N M R N (M R N, τ) A ϕ : M N A R- f : M R N A f τ = ϕ (U) M R N R- M, N R M N Z- P = Z(m, n) (m,n) M N (m 1 + m 2, n) (m 1, n) (m 2, n) (m, n 1 + n 2 ) (m, n 1 ) (m, n 2 ) (ma, n) (m, an) Z- Q M R N := P/Q, τ(m, n) = (m, n) (U) τ R- Q ϕ : M N A R- ϕ : P A ϕ R- ϕ Q 0 ϕ ϕ = f τ f P M N M R N τ f τ = ϕ f f (U) T R- τ : M N T (U) (U) A = M R N, ϕ = τ f τ = τ f (U) A = T, ϕ = τ M R N (U) τ = f τ f f f : T T f f : M R N M R N (U) A = T, ϕ = τ τ = g τ g g = id T f f f f = g = id T f f = id M R N M R N = T 9

10 M N τ M R N T f f M R N ϕ f τ τ A fig.1 M N fig.2 τ(m, n) = m n τ R- (m 1 + m 2 ) n = m 1 n + m 2 n m (n 1 + n 2 ) = m n 1 + m n 2 (mr) n = m (rn) M R N τ(m N) M R N n m i n i (m i M, n i N) i=1 3 R R, S, T M (R, S)- N (S, T )- M S N (R, T )- R R- M, N M R N R- M, N (rm)s = r(ms), (sn)t = s(nt) θ (r,t) : M N (m, n) (rm) (nt) M S N (r(ms)) (nt) = ((rm)s) (nt) = (rm) (s(nt)) = (rm) ((sn)t) θ (r,t) (ms, n) = θ (r,s) (m, sn) θ (r,s) S- M R N (U) M R N f f τ = θ (r,t) f(m n) = θ (r,s) (m, n) = (rm) (nt) (R, T ) r (m n) t = f(m n) = (rm) (nt) M S N (R, T )- R R- M R- (r m) r = r (r m) = (r r) m = (rr ) m = r (r m) = r (m r ) (R, R)- M, N K 10

11 5.5. K M, N K K dim K (M K N) = dim K M dim K N M, N {x i }, {y j } M K N {x i y j } 5.6. {M i } R- {N i } R- ( ) M i R ( ) N j = (M i R N j ) ; m i n j (m i n j ) j J i,j j J i,j i I i I {M i } M {x i } K- x i {N j } N {y j } K- y j 23 R- M K N M, N X, Y X, Y x 11 Y x 12 Y x 1n Y x 21 Y x 22 Y x 2n Y x n1 Y x n2 Y x nn Y X n Y m n m {x i y j } {z ij } z 11, z 12,, z 1m, z 21,, z n1, z nm 5.7 R- R S S R- 24 r s = f(r)s R- M R- S S R M R K R K M R K R- M K- K V K L V K L L K = R, L = C 23 x y 24 Z Q Q Z- 11

12 5.7. R, S R S (1) S R R n = S n (2) S R R[X] = S[X] (3) S R M n (R) = M n (S) (1) Q Z Z n = Q n Z/nZ Z Q = 0 m r r = n (r/n) m r = m n (r/n) Z Z 1 m r = m n (r/n) = n m (r/n) 1 Z/nZ 0 m r = 0 (r/n) = 0 25 Z/nZ Z Q 0 Z/nZ Z Q = 0 Z/nZ Z Z/mZ = Z/dZ ( d m n ) 1 Z/nZ Z Z/mZ x ȳ 0 ȳ 0 = ȳ = 0 0 = 0 x ȳ x = 1 x x 2 1 x ȳ = 1 xy Z/nZ Z Z/mZ 0 1 ȳ m, n d ms + nt = d s, t Z y d y = pd + q = p(ms + nt) + q (d > q) 1 ȳ = 1 p(ms + nt) + q = 1 pms + 1 pnt + 1 q = m ps q = 1 q Z/nZ Z Z/mZ 0 1 q (1 q d 1) f : Z/nZ Z/mZ Z/dZ; (ā, b) ab Z- (U) τ :Z/nZ Z/mZ Z/nZ Z Z/mZ f : Z/nZ Z Z/mZ Z/dZ f = f τ f f f = (r n) = 0 0 (r/n) = 0 0 = 0 12

13 5.8. I R N R- IN = ax a I, x N N R- R I (R/I) R N = N/IN f R g R/I 0 I R N f Id N R R N g Id N (R/I) R N 0 R R N = N Im f Id N = Ker g Id N = IN N N/IN = (R R N)/ Im f Id N = (R R N)/ Ker g Id N = (R/I) R N R- Z/mZ Z Z/nZ = (Z/nZ)/(mZ)(Z/nZ) = Z/(mZ + nz) = Z/dZ 3 G = Z/NZ n 1 n 2 = N n 1, n 2 G = Z/n 1 Z Z/n 2 Z n = p e, m = q f (p, q ) Z/p e Z Z Z/q f Z = { C[X] C C[Y ] = C[X, Y ] 0 (p q) Z/p min{e,f} Z (p = q) C[X, Y ] C[X] C[X, W ] = C[X, Y, W ] M n (R) R M m (R) = M mn (R) S 1 R R M = S 1 M Z/pZ Z Z (q) = { 0 (p q) Z/pZ (p = q) R S S 1 R R S 1 M R- M S 1 R- Z (q) Z qz 26 N/IN 13

14 M 1, M 2 S 1, S 2 M 1 M 2 S 1 S 2 S 1, S R- M r M r r 27 T r (M) M 28 T (M) = 11. R- M R M R- T r (M) p=0 I = x y y x x, y M I M R M M/I M S 2 M S r M T r M I = x 1 x i x i+1 x r x 1 x i+1 x i x r 1 = 1,, r 1, x i M R n K- K M S r M r M r x 1 x r T r (M) S r M T r M T r (M) S r M x 1 x 2 = x 2 x 1 M {x 1,, x k } S r M {x i1 x ir 1 i 1 i r k} M V T (V ) 14

15 12. M r M r x 1 x r i, j x i = x j T r (M) r (M) 29 r 1 M r (M) M r r (M) M r x 1 x n r (M) x i = x j x 1 x r = 0 x 1 x i x i+1 x r = x 1 x i+1 x i x r M k {x 1,, x k } r (M) {x i1 x ir 1 i 1 < i r k} r > k r (M) = 0 r k M dim ( r (M)) = rc k M (M) = n r (M) r= R- M R R- R- M R M M R (f + g)(m) = f(m) + g(m) (f, g M, m M) (r f)(m) = r (f(m)) (f M, m M, r R) M M M M q T q (M ) q p q (p, q) p q 32 15

16 A 33 A.1 A.1. M {U α } ϕ α : U α R n (M, {U α, ϕ α }) n (M1) ϕ α U α R n V α (M, {U α, ϕ α }) n C r (M2) U α U β φ α, β R n R n C r (0 r ω) (U α, ϕ α ) M 34 ϕ β ϕ 1 α : ϕ α (U α U β ) ϕ β (U α U β ) R n n S n n T n n P n (K) C A.2 f : M R M, N f : M N C M A.2. M m C r N n C r f : M N M 1 x (U, ϕ) f(x) N (V, ψ) ψ f ϕ 1 : ϕ(u f 1 (V )) ψ(v ) R m R n C r f x M C r f M C r f M C r 35 M C r C r (M) f + g, fg, λf

17 C r (M) R 36 A.3 A.3. M p C f v(f) v : C (M) R v p M (1) v(λf + µg) = λv(f) + µv(g) (2) v(fg) = v(f)g(p) + f(p)v(g) p T p M p M T p M p (x 1,, x n ) ( ) (v + v )(f) = v(f) + v (f), (λv)(f) = λv(f) x i f := p f ϕ 1 x i (ϕ(p)) (1 i n) ( / x i ) p 2 {( / x i ) p 1 i n} T p M A.1. n M M p p M n p { ( ) } ; 1 i n x i p 37 A.4 m C M n C C f f A.4. f : M N p f (= df) : T p M v f (v) T f(p) N f (v) : C (N) g v(g f) R 36 K V R- K K- V V V V a(vv ) = (av)v = v(av ) (a K, v, v V ) 37 17

18 f (v) T f(p) N f (v)(ag + bg ) = v((ag + bg ) f) = v((ag) f) + v((bg ) f) = av(g f) + bv(g f) = af (v)(g) + bf (v)(g ) f (v)(gg ) = v((gg ) f) = v((g f)(g f)) = v(g f)g (f(p)) + g(f(p))v(g f) = f (v)(g)g (f(p)) + g(f(p))f (v)(g ) (2) f (v) T f(p) N f : v f (v) f (av + bv )(g) = (av + bv )(g f) = av(g f) + bv (g f) = af (v)(g) + bf (v )(g) A.5 A.5. M p p X : M T p M; p X p p M M (x 1,, x n ) M U ( / x i ) p T p M U p X p X p = n i=1 ( ) ξ i (p) x i p ξ i (1 i n) X (x 1,, x n ) ξ U U U C r X U C r M C r X M C M C X(M) M C C (M) X, Y X(M), f C (M) p f(p)x p, p X p + Y p M C fx, X + Y f(gx) = (fg)x, f(x + Y ) = fx + fy X(M) C (M)- X(M) Lie X X(M), f C (M) (Xf)(p) := X p f (p M) X p T p M p (x 1,, x n ) (Xf)(p) = n i=1 18 ξ i (p) f x i (p)

19 Xf M C Xf C (M) Xf X f X(λf) = λ(xf), X(f + g) = Xf + xg, X(fg) = (Xf)g + f(xg) A.6. K A D : A A A D(aa ) = D(a)a + ad(a ) X X(M), f C (M) D X f := Xf D X C (M) A.2. D C (M) D = D X M C X C (M) D X, D Y [D X, D Y ] := D X D Y D y D X C (M) 38 [D X, D Y ] = D Z M Z Z [X, Y ] [X, Y ]f = X(Y f) Y (Xf) [X + Y, Z] = [X, Z] + [Y, Z], [X, Y + Z] = [X, Y ] + [X, Z], [X, Y ] = [Y, X], [fx, gy ] = fg[x, Y ] + f(xg)y g(y f)x, [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] = 0 Lie Lie 39 A.7. K g Lie (Lie algebra) [, ] : g g g (L1) [, ] (L2) [x, y] = [y, x] ( x, y g) (L3)[Jacobi ] [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 ( x, y, z g) Remark A.3. (L2) [x, x] = 0 ( x g) M C [, ] Lie A Der A Lie [D, D ] A A.4. M C X(M) Lie Lie C (M) Lie Der (C (M)) X(M) Der(C (M)); X D X 38 D X D Y, D Y D X 39 Lie Lie 19

20 A.6 V T (V ) I = x y + y x x, y V A ω : V V R V k A k (V ) A (V ) = A k (V ) k=0 k > dim(v ) A k (V ) = 0 V (V ) i : (V ) A (V ) k i k : k (V ) A k (V ) : i k (ω)(x 1,, X k ) = 1 k! det(α i(x j )) (ω = α 1 α k k (V ), α i V ) A.5. i : (V ) A (V ) A (V ) (V ) ω k (V ), η l (V ) ω η k+l (V ) ω η(x 1,, X k+l ) = 1 (k + l)! σ S k+l sgn(σ)ω(x σ(1),, X σ(k) )η(x σ(k+1),, X σ(k+l) ) (X i V ) M p T p M T p M (T p M) A.9. M C ω M k p M ω p k (T p M) ω p p C r 1 M r (M) U M (x 1,, x n ) p U ( ) ( ), x 1 p x n p T p M dx i,, dx n ( ) (dx i ) = δ ij x j 20

21 Tp M ω p = f i1,,i k (p)dx i1 dx ik i 1 < <i k ω p ω p C f i1,,i k (p) p C ω M k p ω p : T p M T p M R p ω : X(M) X(M) C (M) A.6. M C M k A k (M) X(M) k C (M) C - C (M) 1 A.7. M C M p k ω p k (Tp (M)) M k ω p k M C (M) B R- B.1 R {M i } i I R- M i M = {(m i ) i I m i M i } {M i } i I R- (m i ) + (m i) = (m i + m i), r (m i ) = (r m i ) R- R- M i R- (direct product) i I p i : M M i ; (m i ) i I m i 21

22 R R- (universarity) N R- i I R- f i : N M i R- f : N M f i = p i f ( i) ( ) N f i f M i M = M i p i fig.b-1 f(n) = (p 1 i (f i (n))) i I M = M i f i = f p i f i, p i R- f R- ( ) B.2 {M i } i I { M i := (m i ) } M i ; i m i = 0 M i i I i I i I M i R- {M i } i I i I i I i m i M i 0 j i : M i i I M i R- N R- i I R- g i : M i N R- g : M i N g i = g j i i ( ) M i j i M i g g i N fig.b-2 22

23 g((m i ) i I ) := g i (j 1 i (m i )) ( ) I R- Z C Z- Z Z C R- M R- B B R- n r i b i (r i R, b i B) i=1 M R- R (M1),(M3) R- B B R- R- M B M = B M M B B {b i } n i=1 n r i b i = 0 (r i R) = r i = 0 ( i) i=1 B R M R M R- M R M B M R M m m = n r i b i (r i R, b i B) B M R M m i=1 m = n m r i b i = r jb j i=1 2 0 B 0 {b i }, {b j } 0 2 B M m = 0 r i = 0 B R R- M R M R- B = {b i } R- M j=1 R r rb i Rb i M R Rb i (M2) R (M3) R- M Rb i = R M = i Rb i Rb i = R 42 D D

24 {b i } M Rb i = R R- M M (rank) R 44 C.1. R M I M R- M = R = R I i I 45 C.2. R m mm = am a m, m M R- M/mM = (R/m) I C.3. R m R/m R/m R/m M/mM I D D.1 D.1. M, M R- f : M M R- N, N R- g : N N R- f g : M R N M R N (f g)(m n) = f(m) g(n) f, g h : M N M R N h(m, n) = f(m) g(n) R- M R N (U) f g 44 R 45 Zorn 24

25 M N τ M R N (f, g) h f g M N τ M R N fig.3 D.2 3 D.2. M, N, L R (1) R R M = M (2) M R N = N R M (3) (M R N) R L = M R (N R L) M n M n (1) M R R M; r 1 x R M M (r, x) r x R- (U) R R M M; r x rx 1 (rx) = r x 2 (2) M N N R M; (x, y) y x R- f : M R N N R M; x y y x g : N R M M R N; y x x y (3) z L φ z : M N M (N L); (x, y) x (y z) R- f z : M N M (N L) ψ : (M N) L M (N L) : (w, z) f z (w) R- f z f z (w 1 + w 2 ) = f z (w 1 ) + f z (w 2 ) f z1 +z 2 (w) w = x i y i f z1+z 2 (w) = i = i = i = i f z1+z 2 (x i y i ) x i (y i (z 1 + z 2 )) x i ((y i z 1 ) + (y i z 2 )) [(x i (y i z 1 ) + (x i (y i z 2 ))] = i (x i (y i z 1 )) + i (x i (y i z 2 )) = f z1 (w) + f z2 (w) 2 f rz (w) = i x i (y i (rz)) = i x i ((ry i ) z) = i f z (x i (ry i )) = f z (rw) 3 ψ R- (U) f : (M N) L M (N L) g : M (N L) (M N) L 25

26 D.3 D.3. {M i } R- {N i } R- ( ) M i R ( ) N j = (M i R N j ) ; m i n j (m i n j ) j J j J i,j i I i,j i I ( M i N j mi, ) n j (m i n j ) (M i R N j ) i,j R- (U) j i : M i R N j (M i R N j )g i : M i R N j ( M i ) R ( N j ) ( ) g : (M i R N j ) ( M i ) R ( N j ) f g D.4. R- M {x i } R- N {y j } M R N {x i y j } M, N M = x 0, N = y 0 M N M; (x, ry 0 ) xr R- (U) g : M R N M h : M M R N : x x y 0 M = M R N M R N x y 0 M R N = N M R N {x 0 y 0 } Remark D D.1. D.2. R M, N R- R- p : M N R- f : P N p g = f g : P M P R p.23 26

27 D.4 D.6. R R- 47 R- N M 1 f M 2 g M 3 0 M 1 R N f Id N M 2 R N g Id N M 3 R N 0 R- g Id N M 3 N m i n (m I M 3, n N) g Im f Id N = Ker g Id N M 1 M 2 M 3 0 Ker g = Im f M 3 = M2 / Ker g = M 2 /f(m 1 ) φ : M 3 N (M 2 N)/(f(M 1 ) N); (g(x), y) x y mod(f(m 1 ) N) (x M 2, y N) well-defined g(x) = g(x ) g(x x ) = 0 x x Ker g = Im f x x f(m 2 ) mod(f(m 1 ) N) x y x y R- (U) h : M 3 N (M 2 N)/(f(M 2 ) N) h h τ = φ τ M 3 N M 3 N x i y i Ker g Id N (x i M 2, y I N) g(x i ) y i = 0 0 = h(g(x i ) y i ) = φ(g(x i ), y i ) x i y i mod(f(m 1 ) N) i x i y i Im(f Id N ) 0 M 1 M 2 M M 1 N M 2 N M 3 N 0 (flatness) D.3. R- 0 M 1 M 2 M M 1 N M 2 N M 3 N 0 R- N R R- f g M N L Im f = Ker g 48 1 p.30 p.36 27

28 R 50 R K[x, y] I = (X, Y ) K- Z- Z/2Z D.5 Hom R- Hom R- D.4. R- M, N M N R- Hom R (M, N) Hom R (M, N) (r f)(m) = f(r m) R- R- R- f : M M Hom R (N, M) Hom R (N, M ); g f g Hom R (M, N) Hom R (M, N); g g f R- Hom R (N, ) R- Hom R (, N) R- R- Hom D.7. (1) R- 0 M 1 M 2 M 3 0 Hom R (N, M 1 ) Hom R (N, M 2 ) Hom R (N, M 3 ) R- (2) R- M 1 M 2 M 3 0 Hom R (M 1, N) Hom R (M 2, N) Hom R (M 3, N) 0 R- D.1,D.2 D.5. R- 0 M 1 M 2 M 3 0 R- N R- N Hom R (N, M 1 ) Hom R (N, M 2 ) Hom R (N, M 3 ) 0 0 Hom R (M 1, N) Hom R (M 2, N) Hom R (M 3, N) 0 28

29 Ext 29

数学Ⅱ演習（足助・09夏）

数学Ⅱ演習（足助・09夏） II I 9/4/4 9/4/2 z C z z z z, z 2 z, w C zw z w 3 z, w C z + w z + w 4 t R t C t t t t t z z z 2 z C re z z + z z z, im z 2 2 3 z C e z + z + 2 z2 + 3! z3 + z!, I 4 x R e x cos x + sin x 2 z, w C e z+w