() n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (5) (6 ) n C + nc + 3 nc n nc n (7 ) n C + nc + 3 nc n nc n (
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1 3 n nc k+ k + 3 () n C r n C n r nc r C r + C r ( r n ) () n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (4) n C n n C + n C + n C + + n C n (5) k k n C k n C k (6) n C + nc + 3 nc n nc n n (7) n C + nc + 3 nc n nc n n(n + ) n (8) nc k k + n+ C k+ n + n (9) nc k k + n + (n+ ) () n ( ) k n C k k + n +
2 () n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (5) (6 ) n C + nc + 3 nc n nc n (7 ) n C + nc + 3 nc n nc n (8) n () () (3) n nc k k + n ( ) k nc k+ k + n C k k + (3) 3 (5) n k ( k n) n C k () m m C m () p k p p C k p (3) p n (n + ) p n p p 6
3 n () nc r C r + C r ( r n ) n () ( + b) n nc r n r b r () C r + C r (n )! (n r )!r! + (n )! (n r)!(r )! (n )!(n r) (n r)!r! + (n )! {(n r) + r} (n r)!r! n! (n r)!r! (n )!r (n r)!r! n C r n,,..., n r n C r n r C r n r C r n C r C r + C r ( r n ) ( + x) n ( + x)( + x) ( + x) + x( + x) x r r n x r r n n C r x r r n C r x r + x C r x r ( C r + C r ) x r 3
4 x r r n C r + C r n C r C r + C r () (i) n () ( + b) + b, () ( r n ) C r r b r C b + C b + b (ii) n k (k ) + b ( + b) k k kc r k r b r k ( + b) k+ ( + b) kc r k r b r k k kc r k+ r b r + kc r k r b r+ k k k C k+ + kc r k+ r b r + kc r k r b r+ + k C k b k+ r k C k+ + k C }{{} k+ C k+ + k kc r k+ r b r + r k r ( k C r + k C r ) }{{} k+ C r k kc s k+ s b s + k C k b k+ s k+ r b r + k k+ C k+ + k+c r k+ r b r + k+ C k+ b k+ r k+ k+c r k+ r b r n k + (i)(ii) k C }{{} k b k+ k+ C k+ 4
5 f n (x) n C + n C x + n C x + + n C n x n nc r C r + C r ( r n ) x r r,,..., n nc r x r C r x r + x r r n nc r x r ( nc x + n C n x n) C r x r C x C r x r r ( n ) + x C r x r C x r f n (x) x n f (x) + x ( f (x) x ) f n (x) ( + x)f (x) f (x) C + C x + x f n (x) ( + x) n n ( + x) n nc r x r \ x b ( + b ) n n nc r b r r n n ( + b) n nc r n r b r ( ) ( ) ( ) 5
6 [ ] n C r r > n () ( ) n r ( ) n n C r r ( ) n r ( ) n r ( ) n + r [ ] f (x) + x, f n (x) ( + x)f (x) ( ) n r (n ) f n (x) x \ f n (x) ( + x)f (x) ( + x) n f n (x) ( + x) n f (x) ( + x) (n ) f n(x) ( + x) n f (x) ( + x) f n (x) ( + x) n ( ) x f n (x) ( + x)f (x) x f n ( ) ( ) 6
7 n () nc + n C + n C + + n C n n () nc + n C + n C 4 + n C + n C 3 + n C 5 + (3) nc n n C + n C + n C + + n C n nc + n C x + n C x + + n C n x n ( + x) n () x nc + n C + n C + + n C n n n {,,..., n } N k n C k N n C + n C + n C + + n C n N n nc + n C + n C + + n C n n () x nc n C + n C n C ( ) n nc n nc + n C + n C 4 + n C + n C 3 + n C 5 + () nc + n C + n C 4 + n C + n C 3 + n C 5 + n 7
8 (3) n n n n n C n n n r r n C r n r n C n r n C r n C r nc r n C r r,,,..., n n n n C n nc n n C + n C + n C + + n C n ( + x) n ( + x) n ( + x) n x n x n n C n x n ( + x) n n C + n C x + n C x + + n C n x n nc nc n x n + n C x nc x + + n C n x n nc ( n C nc n + n C nc + + n C n nc ) x n ( n C nc + n C nc + + n C n nc n ) x n nc n n C + n C + n C + + n C n () n (i) n nc + n C + n C n C n C + n C 3 + n C n C n (ii) n nc + n C + n C n C n n C + n C 3 + n C n C 8
9 n () n nc k n () k k! n ( k n) k n k (3 ) () () () n k n k n n! n! n n! k!(n k)! n n nc k n! k!(n k)! 9 r C r ( C r C r+ ) (3 ) ( + x) 4 ( + x) ( + x) x x 4 C x ( + x) C + C x + C x + + C x C C x + C x C 9 x C x C ( C C + C C C C ) x ( C C + C C + + C C ) x C r x 9
10 C r 4 C ( + x) 4 ( + x) ( + x) x 8 x 8 4 C 8 x 8 ( + x) C + C x + C x + + C x C C 8 x 8 + C x C 7 x C 8 x 8 C ( C C 8 + C C C 8 C ) x 8 ( C C + C C C 8 C ) x 8 9 ( C r C r+ ) x 8 r 9 r ( C r C r+ ) 4 C 8 9 r C r ( C r C r+ ) 4 C 4C 8 4!!! 8!! 4! 9 3 9
11 3 n () nc + nc + 3 nc n nc n () nc + nc + 3 nc n nc n () n, k k n C k k n! k!(n k)! n (n )! (k )!{(n ) (k )}! n C k ( + x) nc + n C + + n nc n C r x r x C r n n k n C r k n n C k k n k n nc + n C + + n nc n n n C k C r C r nc + n C + + n nc n n
12 ( + x) n n nc r x r x n n( + x) r n C r x r r x n r n C r n r nc + n C + + n nc n n () n n ( + x) n nc r x r x n nc r n C r m m mc r m ( ) n, k k n C k k n! k!(n k)! n (n )! (k )!{(n ) (k )}! n C k k n C k n C k (n, k ) ( )
13 n k nc k n n k k n C k k n k n C k n k n {(k ) + } C k k n (k ) C k + n k n r C r + n n, r ( ) n k C k C r n r r C r + n C r ( ) n r C r (n ) n C r r C r (n ) n C r r r n (n ) s n C s (n ) n ( ( ) ) C r n k nc k n r C r + n r C r n (n ) n + n n(n + ) n n k nc k C n n k nc k n(n + ) n 3
14 n, k n k(k ) n C k k(k ) n(n ) n(n ) n! k!(n k)! n(n ) n C k n k(k ) n C k k (n )! (k )!(n k)! (n )! (k )! {(n ) (k )}! n k(k ) n C k k n n(n ) n C k k n n(n ) n C k x () ( )( ) n ( + x) n n C k n n C k n n k(k ) n C k n(n ) n k n k n C k n k ( ) ( ) 4
15 n n n k nc k k(k ) n C k + k n C k k k k n(n ) n + n n(n + ) n n n ( + x) n n nc r x r x n n( + x) r n C r x r r x n n(n )( + x) n r(r ) n C r x r r x n r r n C r n, n r(r ) n C r n(n ) n r n n n n k nc k k nc k (k k) n C k + k n C k k k n k(k ) n C k + k k n k n C k k n(n ) n + n n(n + ) n n k nc k n(n + ) n n 5
16 3 n () nc + n C + n C + + n C n n () nc + nc + + n nc n n (3) n+c + n+ C + + n+ C n n 7 4 n n C + n C x + n C x + + n C n x n ( + x) n () nc n C + n C + ( ) n nc n () nc + nc + + n nc n n (3) nc + nc +3 nc + +(n+) nc n (n+) 4 5 () k n C k n C k (k,,..., n) n () () S k nc k 6 n 3 () k n k k(k ) n C k n(n ) n C k () (3) n k(k ) n C k k n k nc k k 8 6
17 4 n () () (3) n nc k k + n ( ) k nc k+ k + n C k k + (3) 3 () k n nc k k + k + n + n+ C k+ n + n! k!(n k)! (n + )! (k + )! {(n + ) (k + )}! n nc k k + n n+c k+ n + n + n+ k n+c k x n+ ( + x) n+ n+c k x k n+ n+c k n+ n+ n+ n+c k k n+c k n+ C n+ n nc k k + n + n+ k n+c k n + (n+ ) 7
18 ( + x) n n nc r x r x ( + x) n dx n nc r x r dx [ ( + x) n+ n + ] x n [ x r+ nc r r + ] x ( + x) n+ n + n nc r r + xr+ x n+ n + n nc r r + n nc k k + n + (n+ ) () k n nc k k + k + n + n+ C k+ n + n! k!(n k)! (n + )! (k + )! {(n + ) (k + )}! n ( ) k n C k k + n ( ) k n+ C k+ n + n + n+ ( ) k n+c k k n+ ( x) n+ n+c k ( x) k 8
19 x n+ ( ) k n+c k n+ n+ ( ) k n+c k ( ) k n+c k k n ( ) k k n C k k + n + ( n+ ) ( ) k n+c k n+ C n+ ( ) k n+c k n + ( x) n n ( ) r nc r x r (3) k n ( x) n dx [ ] ( x) n+ n + n + n ( ) k nc k+ k + k + k + n + n + k n ( ) r nc r x r dx n [ ( ) r x r+ nc r r + n ( ) r nc r r + n C k k + n + ] (n)! (k + )!{n (k + )}! (n)! (k + )!(n k )! (n + )(n)! (k + )(k + )!{n + (k + )}! (n + )! (k + )!{n + (k + )}! n + n+ C k+ 9
20 ( + x) n+ nc k+ k + n + n + n+ C k+ n l n+c l n+ C + n+ C x + n+ C x + + n+ C n x n + n+ C n+ x n+ x ± n n+ C + n+ C + n+ C + + n+ C n + n+ C n+ n+ C n+ C + n+ C + n+ C n n+ C n+ n+ ( n+ C + n+ C + + n+ C n+ ) n+c + n+ C + + n+ C n+ n ( + x) n n nc k+ k + n + n + n l n+c l ( n ) n+c l n+ C l n + (n ) nc r x r x ( + x) n dx n nc r x r dx
21 [ ( + x) n+ n + ] x n [ ] x x r+ nc r r + x ± ( + x) n+ n + n nc r r + xr+ n+ n + n n n + nc r r + nc r r + ( )r+ + n+ n + n r nc r r + nc k+ k + nc k+ k + n n + 7 n A n nc + 3 nc nc n (n ) nc n B n n C n C A n B + n C 3 + ( ) n ( n + nc n n + ) n+
22 8 () n n ( ) k nc k () k k n (n + ) n C k (k + ) n+ C k+ n ( ) k (3) k + n C k n + 9 () n ( ) m nc m n m m n! nc m 3 n ( ) m m + n C m n + m () f(x) f n+ (x) (n,,,... ) f n+ (x) n! (n )! C m f(t)(x t) n dt () d dx f n+(x) f n (x) (n,,... ) f(x) f n (x) (98 ) () [] 6
23 6 () (b) (c) () f(x) n (x t) f(t) dt f(x) x n (n )! (b) f (x) f(t) dt, f (x) (k ) () f (x) () f k (x) (k )! f (t) dt,, f k (x) f k (t) dt (x t)f(t) dt (x t) k f(t) dt (k ) 69 () (x t) n f(t) (c) F n (x) dt (n ) n! d dx F n(x) F (x) (n ) ()(b)() n k () n () n (x t) (n )! f(t) dt (x t)! f(t) dt f(t) dt f(x) x () n k, f(x) x k n k + F (x) (x t) k (k )! f(t) dt F (x) f(x), F () 3 f(t) dt f(x)
24 (x t) k k! (x t) k f(t) dt F (t) dt k! [ ] (x t) k x k(x t) k ( ) F (t) F (t) dt k! k! (x t) k (k )! F (t) dt. F (x) x k f(x) x k + n k + ()() n (b) () f (x) xf (x) t f (t) dt [ tf (t) ] x tf (t) dt tf(t) dt x f(t) dt () (i) k tf(t) dt (x t)f(t) dt. (ii) n k f (x) k f k+ (x) f k+ (x) (k )! (x t) k f (t) dt k! f k+ (x) (k )! (k )! (k )! (x t) k f (t) dt ( ) (x t)k f k (t) dt {[ (x t)k f k (t) ] x + k (x t) k f(t) dt ( f () ) n k + ()() n } (x t) k f (t) dt 4
25 (c) G n (x) (x t) n f(t) dt G n (x) d dx G n(x) { n } nc k x n k ( t) k f(t) dt n } {nc k x n k ( t) k f(t) dt n } {(n k) n C k x n k ( t) k f(t) dt + n C k x n k ( x) k f(x) (n k) n C k (n k) n! (n k)!k! n n nc k x n k ( x) k (x x) n (n )! (n k)!k! n C k, d dx G n(x) n n n n C k x k ( t) k f(t) dt { n } C k x k ( t) k f(t) dt (x t) f(t) dt ng (x). d dx G n(x) n G (x) n! d dx F n(x) F (x) 5
26 3 5 () n n ( x) n ( ) r nc r x r n ( ) r r + n C r ( x ) n dx () n ( ) r r + n C r n () ( x ) n n ( ) r nc r x r () I n ( x ) n dx ( x ) n dx n ( ) r nc r x r dx n [ ( ) r x r+ nc r r + n ( ) r r + n C r ] 6
27 I n ( x ) n dx (x) ( x ) n dx [ x( x ) n] x n( x ) ( x) dx n ( x )( x ) dx { n ( x )( x ) dx n (I n I ) } ( x ) dx I n n n + I I ( x ) dx [x 3 x3] 3 I n n n + I n n n n + n n + (n)!! (n + )!! (n ) n I n (n ) n 4 5 I (n ) n (n)!! n (n ) 4, (n )!! (n ) (n 3) 3 (n)!! (n + )!! {(n)!!} (n + )!!(n)!! {n n!} (n + )! 4n (n!) (n + )! 7
28 n ( ) r r + n C r 4n (n!) (n + )! n n (n + )! (n!), J n ( x ) n dx, K n n J n () n+ n n () K, K J n+ J n (3) J n+ K n+ x ( x ) n dx J n+ J n (x) ( x ) n+ dx J n+ J n K n (4) K n J n n (5) n ( ) k L n k + n C k n C 3 n C + 5 n C 7 n C ( )n n + n C n L n J n (4 ) J n x x 3 () ( x 3 ) n n ( ) r nc r x 3r () I n ( x 3 ) n dx ( x 3 ) n dx 8 n ( ) r nc r x 3r dx n [ ( ) r x 3r+ nc r 3r + n ( ) r 3r + n C r ]
29 I n ( x 3 ) n dx (x) ( x 3 ) n dx [ x( x 3 ) n] x n( x 3 ) ( 3x ) dx 3n ( x 3 )( x 3 ) dx { 3n ( x 3 )( x 3 ) dx 3n (I n I ) } ( x 3 ) dx I n 3n 3n + I I ( x 3 ) dx [x 4 x4] 3 4 I n 3n 3n + I 3n 3n n 3n + 3n 3n + (3n)!!! (3n + )!!! 3(n ) 3n I n 3(n ) 3n 6 7 I 3(n ) 3n (3n)!!! 3n 3(n ) 6 3, (3n )!!! (3n ) (3n 5) 4 (!!! ) n ( ) r 3r + n C r (3n)!!! (3n + )!!! 9
30 6 n, r r n () n+c r n C r + n C r () nc n C + n C + ( ) n nc n (3) nc n C + 3 n C 3 + ( ) n n C n n () (3) ( x) n n C n C x + n C x + ( ) n nc n x n nc x n C x + + ( ) nc n x n ( x) n x { nc n C x + + ( ) nc n x } ( x) n dx x dx { nc n C x + + ( ) nc n x } dx [ nc x n C x + x nc ( ) C x n n n C n C + 3 n C 3 + ( ) n n C n t x n n ] ( x) n x dx t n t t n t ( dt) dt ( + t + t + + t ) dt [ t + t + t tn n ] n 3
31 nc n C + 3 n C 3 +( ) n n C n n () (3) (i) n () C () (ii) n m (m ) () () mc m C + 3 m C 3 +( ) m m m C m m n m + m+c m+ C + 3 m+ C 3 + ( ) m m m+ C m + ( ) m m + m+ C m+ (mc m C + 3 m C 3 + ( ) m ) m m C m m+ C m C ( m+c m C ) + }{{} 3 ( m+c 3 m C 3 ) + }{{} m C m C + ( ) m m ( m m+c m m C m ) +( ) }{{} m + m+ C m+ m C m m C + 3 m C + + ( ) m m m C m + ( ) m m + m C m m C + 3 m C + +( ) m m m C m +( ) m m + m C m ( x) m m C m C x + m C x + ( ) m mc m x m m + n m + (i)(ii) n 3
32 n n ( ) r n nc r r r r r n C + n C + ( ) n n C n ( n ) n () () () nc r C r C r n, r n () ( ) r n C r r ( )n n r (3) n n n ( ) r nc r r r n n (6 ) 3
33 7 m, n I n, m x n ( x) m dx () m I n, m I n+, m () I n, m n!m! (n + m + )! (3) n!m! m C (n + m + )! n + m C n ( )m m C n n + m + () m I n, m x n ( x) m dx ( x n+ n + [ x n+ ( x)m n + m n + () m I n, m m n + I n+, m I n, m ) ( x) m dx ] x n+ ( x) m dx [ x n dx x n+ n + m n + I n+, m x n+ n + m( x)m ( ) dx ] m n + m n + I n+, m n + m n + m n + n + m I n+m, m n + m n + n + m m! (n + m + )(n + m) (n + ) n + m + m! n! (n + m + )(n + m) (n + ) n! n!m! (n + m + )! 33
34 m I n, m n!m! (n + m + )! (3) x n ( x) m m C m C x + + ( ) m m C m x m x n ( x) m m C x n m C x n+ + + ( ) m m C m x n+m x [ x n ( x) m dx m C x n+ n + m C x n+ n ( )m m C m x n+m+ n + m + ] () m C n + m C n ( )m m C n n + m + n!m! (n + m + )! m C n + m C n ( )m m C n n + m + m, n I n,m π () n I n, m I n, m+ () I n+, m+ x n ( x) m dx (3) n!m! m C (n + m + )! n + m C n ( )m! cos n θ sin m θ dθ m C n n + m + (4 ) 34
35 8 x, y () n, b n + y(y + ) (y + n)(y + n + ) y(y + ) (y + n) + b (y + )(y + ) (y + n + ) () n n! x(x + ) (x + n) n ( ) r n C r x + r 6 () () n + y(y + ) (y + n)(y + n + ) y(y + ) (y + n) + y(y + ) (y + n)(y + n + ) b (y + )(y + ) (y + n + ) n + (y + n + ) + by n + ( + b)y + (n + ) y + b, (n + ) n +, b () n! x(x + ) (x + n) n ( ) r n C r x + r (i) n! x(x + ) x(x + ) ( ) r C r x + r ( ) C x + ( ) C x + x x + x(x + ) 35
36 (ii) n n! y(y + ) (y + n) n + (n + )! x(x + ) (x + n)(x + n + ) n ( ) r n C r y + r n! n + x(x + ) (x + n)(x + n + ) { } n! x(x + ) (x + n) (x + )(x + ) (x + n + ) n! x(x + ) (x + n) n! (x + )(x + ) (x + n + ) n n ( ) r n C r x + r ( ) r n C r (x + ) + r ( ) n C x + ( ) n C x + + ( ) n C x ( )n n C n x + n { } ( ) n C x + + ( ) n C x ( ) n C x + n + ( )n n C n x + n + x + ( ) n C + n C x + + ( ) n C + n C x ( ) n n C n + n C x + n + ( ) n+ n C n x + n + x + ( ) n+ C x + + ( ) n+ C x ( )n n+ C n x + n }{{} + ( ) n+ n+ C n+ x + n + n+ ( ) r n+ C r x + r n ( ) r n+ C r x+r r n + (i)(ii) n [] () [4] 36
37 8 () n! x(x + ) (x + n) n ( ) r n C r x + r rising fctoril () n rising fctoril () n () n () n { ( + ) ( + n ) ( n ) ( n ) n ( ) r n C r x + r n! x(x + ) (x + n) n! (x) n+ ( ) ( ) x () n ( ) r ( ) x n C r + r n! (n + ) n + n ( ) r n C r r + ( n! ) n+ ( ) n+ 3 n (n + ) n+ 3 5 (n + ) 4 6 (n) n+ n n! (n + )! n! n+ 37
38 n ( ) r ( ) n C r r + n! n+ 4n (n!) (n + )! 5 (3) p ( ) x p n ( ) r ( ) p p n C r pr + n! n+ ( ) x m + ( m ) n ( ) r 7 (3) n C r m + + r + n! (m + )(m + ) (m + n + ) m!n! m!(m + )(m + ) (m + n + ) m!n! (m + n + )! ( ) x ( ) n ( ) r r n C r x + r n ( ) r n C r x + r x n! x(x + ) (x + n) x { } n! x (x + ) (x + n) f(x) n! (x + ) (x + n) f() 38
39 { f (x) n! (x + ) (x + ) (x + n) + (x + ) (x + ) (x + n) lim x r n ( ) r + + (x + ) (x + ) (x + n) } n C r x + r lim x x { lim x f(x) x lim x f(x) f() x } n! (x + ) (x + n) f () ( ) n! n + n + + n ( ) n n ( ) r n C r r r ( ) n 6 (3) 39
40 x n ( x) m dx 3 p, q B(p, q) () B(p, q) B(q, p) x p ( x) q dx () B(p, q + ) q B(p +, q), B(p +, q) + B(p, q + ) B(p, q) p (3) B(p +, q) p p + q (4) B(5, 4) B(p, q), B(p, q + ) q B(p, q) p + q (4 ) B(p, q) (p )!(q )! (p + q )! b (x ) p (b x) q dx (p )!(q )! (p + q )! (b ) p+q 4 p, q B(p, q) () q > B(p, q) q B(p +, q ) p (p )!(q )! () B(p, q) (p + q )! (3) b x p ( x) q dx () (x ) p (b x) q dx (b ) p+q B(p, q) () [] 4 4
41 4 m, n α, β I(m, n) β α () n I(m, n) () β α () I(m, n) (x α) m (β x) n dx (x α) m (β x) n dx β α { (x α) m+ n I(m +, n ) m + m + m!n! (β α)m+n+ (m + n + )! } (β x) n dx [ m + (x α)m+ (x β) n] β β + (x α) m+ n(β x) dx m + α n I(m +, n ). m + () I(m, n) n I(m +, n ) n!m! m + α I(m, n) m!n! I(m +, n ) (m + )!(n )!. J(m, n) I(m, n) m!n! J(m, n) J(m +, n ). J(m, n) J(m +, n ) J(m +, n ) J(m + n, ) β α I(m, n) β α (x α) m+n dx [ m + n + (x α)m+n+] β α m + n + (β α)m+n+. (x α) m (β x) n dx m!n! (m + n + )! (β α)m+n+ m!n! (m + n + )! (β α)m+n+. 4
42 5 f(x), g(x) (f g)(x) (f g)(x) f(t)g(x t) dt () f(x) x m, g(x) x n (m, n ) (f g)(x) m!n! (m + n + )! xm+n+ () m, n β α (x α) m (β x) n dx m!n! (β α)m+n+ (m + n + )! m n () (i) n m (f g)(x) t m dt xm+ m +, m!n! (m + n + )! xm+n+ m!! (m + + )! xm++ xm+ m + (f g)(x) m!n! (m + n + )! xm+n+ (ii) n k( ) m f m (x) x m, g k (x) x k (f m g k )(x) m m!k! (m + k + )! xm+k+ (f m+ g k )(x) (m + )!k! (m + + k + )! xm++k+ 4
43 n k + g k+ (x) x k+ (f m g k+ )(x) t m (x t) k+ dt ( ) t m+ (x t) k+ dt m + [ ] x t m+ x (x t)k+ m + k + m + t m+ (x t) k dt k + m + (f m+ g k )(x) dt k + m + t m+ m + (k + )(x t)k ( ) dt (m + )!k! (m + + k + )! xm++k+ ( ) m!(k + )! {m + (k + ) + }! xm+(k+)+. n k + (i)(ii) m, n () () t m (x t) n dt m!n! (m + n + )! xm+n+ β α (x α) m (β x) n dx β α t m (β α t) n dt ( x α t ) m!n! (β α)m+n+ (m + n + )! 4 43
44 5 m, n f m, n (x) f m, n (x) (x ) m (x b) n () m d dx () I m, n b I m, n { } n + f m, n+(x) f m, n (x) + m n + f m, n+(x) f m, n (x) dx m () m n + I m, n+ (3) () I m, n I m, n ( ) n m!n! (m + n + )! (b )m+n
45 9 F (x) x ()(b) () F () (b) F (x)f (y) F (x + y) (i) F (x + ) xf (x) π (ii) n F (n) (n )! (iii) F (x) > ( ) (iv) F π (v) F (x) x π F (x)f (x + F (x) Γ (x) π Γ (x) (cos θ) x (sin θ) y dθ ) e t t x dt (x > ) (cos θ) x (sin θ) y dθ B(x, y) p >, q > B(p, q) x p ( x) q dx ( ) ( ) x cos θ B(p, q) x p ( x) q dx (b) [] P.3 π Γ (x)γ (y) Γ (x + y)b(x, y) (cos θ) p (sin θ) q dθ 45
46 (v) Γ (x)γ [] P.97 () (i) (b) y ( x + ) π x Γ (x) F (x)f () F (x + ) F (x + ) F (x + ) x π [ x () F (x + ) xf (x)f () xf (x) (ii) () (i) F (n) (n )F (n ) (cos θ) x sin θ dθ (cos θ)x] π (n )(n )F (n ) (n )(n ) F () (n )! (iii) s F (s) > ( (b) y x ) {F (x)} F (x) π (cos θ sin θ) x dθ F (x) x π (sin θ) x dθ {F (x)}, π (sin θ) x dθ > F (x) s F (s) F (s) \ s n s F (s)f (n s) F (n) n (b) x s, y n s π (n )! (cos θ) s (sin θ) (n s) dθ π 46 (cos θ) s (sin θ) (n s) dθ >
47 F (s) \ s < 3 s + (i) F (s) F (s + ) s > x F (x) > (iv) x ( ) F > { ( )} F π F () dθ π ( ) F π (v) (b) y ( ) F (x)f F ( x + ) π (cos θ) x dθ π (sin θ) x dθ π π π x (sin t) dt x (sin t) dt + x (sin t) dt + [ t θ ] π π π x (sin t) dt x (sin(π u)) du [ u π t ] π π π π x (sin t) dt + (sin t) x dt π (sin u) x du [ ( )] sin π x [ v ( dv) v π ] t (cos v) x dv 3 47
48 F (x) x F π ( x + (sin θ) x dθ ) π (cos θ) x dθ {F (x)} ( ) F (x)f 3 x F F (x) ( x + ) F ( (x) ) F ( ) F π F (x) (x x π F (x)f + ) 6 F (x) x ()(b) () F () (b) F (x)f (y) F (x + y) π (cos θ) x (sin θ) y dθ (i) F (x + ) F (x) (ii) F (x) > ( ) (iii) F π (iv) F (x) (x x π F (x)f + ) (978 ) 48
49 4 k p C k p p C k n k ( k n) nc k () m m C m () p k p p C k p (3) p n (n + ) p n p p 6 () () m m(m ) mc m m(m ) mq q m q q m q + q m 3 () p k < p pc k p! (p k)!k! p k (p )! {(p ) (k )}!(k )! p k p C k p p k (k,,..., p ) p C k p 49
50 (3) p (n + ) p n p p C n p + p C n p + p C n p + + p C p n + p C p n p p C n p + p C n p + + p C p n ( ) () p C k p (k,,..., p ) p ( ) 7 p () k k < p p C k p () n p (n + )p p np p (3 ) p r () x, x,..., x r (x + x + + x r ) p p x p p x x r r p, p,..., p r p + p + + p r p () x, x,..., x r (x + x + + x r ) p (x p + x p + + x p r ) p (3) r p r p p () () (x + x + + x r ) p p x p p x x r r (x + x + + x r ) p (x + x + + x r )(x + x + + x r ) (x + x + + x r ) 5
51 p x p x p x r p r x p x p x r p r p! p!p! p r! () (x + x + + x r ) p x p, x p,, x r p (x + x + + x r ) p (x p + x p + + x r p ) p! p!p! p r! x p x p x r p r p + p + + p r p, p i p (i,,..., r) ( ) p! p!p! p r! p(p )! p!p! p r! p ( ) p!p! p r! p p! p p!p! p r! (3) () x x x r r p r r(r p ) p r p r p p 5
52 () n, r n, r n r nc r n C r () p r r p p C r p (3) p 3 (x + ) p p pc r x r p p (4) p 5 3 p p () ()() (3) (x + ) p p pc r x r x p p C + p C + p C + + p C p + p C p + p C + p C + + p C p ( ) () p C r (r,,..., p ) p p 3 ( ) p p (4) (x + ) p p pc r x r x 3 p p C + p C + pc + + p pc p + p pc p p + + p C + pc + + p pc p ( ) () p C r (r,,..., p ) p ( ) 3 p p p + 3 (3) p p p 5 3 p p () p r p r r p C r p p C r p C r p () p p p 998 5
53 3 p () r p r p C r p p C r p r () s q (q, s) q C s q (3) m, n (m + n) p (m p + n p ) p (4) n n p n p n 4 () () () 4 C 6 4 (3) m, n (m + n) p (m p + n p ) p C m p + p C m p n + p C m p n + + p C p mn p + p C p n p (m p + n p ) p C m p n + p C m p n + + p C p mn p ( ) () p C r (r,,..., p ) p ( ) p m, n (m + n) p (m p + n p ) p (4) n p n p (i) n p p (ii) n k (k ) k p k p (3) (k + ) p (k p + ) p (k + ) p (k + ) (k + ) p (k p + ) + k p k p n k + (i)(ii) n 53
54 4 m m mc, m C,..., m C m d m d m () m d m m () k k m k d m k (3) m d m 9 () () m m C m d m d m m k m mc k m! (m k)!k! m k (m )! {(m ) (k )}!(k )! m k m C k m m k (k,,..., m ) m C k m m C, m C,..., m C m, m d m m () k m k d m (i) k m d m (ii) k l (l ) l m l d m (l + ) m (l + ) l m + m C l m + m C l m + + m C m l + (l + ) (l m l) + m C l m + m C l m + + m C m l l m l d m,d m m C, m C,..., m C m m C l m + m C l m + + m C m l d m k l + 54
55 (i)(ii) k (3) m n (n ) ( + x) n + n C x + n C x + + n C x + x n x n C + n C + n C + nc n C + n C n + n C mc m C + m C m + m C m d m () d m \ d m k d m k k m k (d m ) m (d m ) m C d m m m C d m m + + m C r d m r m ( ) r + + m C m d m ( ) m + m C m ( ) m (d m ) d m (mc d m m m C d m m + + m C m ) + () k m k d m d m d m d m d m 55
56 5 nc k n p, q n pq n ( k n ) 997 () n C n pq n n C k ( k n ) p q n n C k ( k n ) p q n n C k ( k n ) nc p pq(pq )(pq ) (pq p + ) p(p )(p ) ( ) pq(pq )(pq ) (pq p + ) p p pq ( ) p n C q q n n C k ( k n ) 56
57 6, b > b p, d p > p b p d d p () 995 d p b p ( b)( p + p b + + b p ) d p >, > b ( p + p b + + b p ) > b d p + p b + + b p d p b p (b + ) p b p b p + p C b p + p C b p + + p C p b + b p + p C b p + p C b p + + p C p b pc b p + p C b p + + p C p b ( d ) ( ) p (i) k p pc k p! (p k)!k! p k (p )! {(p ) (k )}!(k )! p k p C k p p k (k,,..., p ) p C k p p C, p C,..., p C p p p C b p + p C b p + + pc p b p (ii) b pc b p + p C b p + + p C p b b ( pc b p + p C b p 3 ) + + p C p b d ( ) p C b p + p C b p + + p C p b d p > p (i)(ii) ( ) p p 57
58 9 p n n p n p n 977 () n p n p (i) n p p (ii) n k (k ) n k + (k + ) p (k + ) k p + p C k p + p C k p + + p C p k + (k + ) (k p k) + p C k p + p C k p + + p C p k ( ) k p pc k p! (p k)!k! p k (p )! {(p ) (k )}!(k )! p k p C k p p k (k,,..., p ) p C k p p C, p C,..., p C p p p C k p + p C k p + + pc p k p k p k p ( ) p n k + (i)(ii) k 58
59 5 7 n () < i < n i n C i C i + C i () m m n m nc i (3) i n i C i i ( ) j nc j ( ) i C i j 3 () ()() (3) i (i) i j ( ) j nc j ( ) nc ( ) C i (ii) i ( i n ) i+ i ( ) j nc j ( ) j nc j + ( ) i+ nc i+ j j ( ) i C i + ( ) i+ nc i+ ( ) i+ ( n C i+ C i ) ( ) i+ C i+ i + (i)(ii) i n i i ( ) j nc j ( ) i C i j 59
60 8 n s(n) n 6 s(n) k m s(n) m k n S(k, m) s(n) 3 n,, 3 S(, 3) 3 () k (k ) S(k, m) m S(k, i) (m,,......, 9) () n r n C r C n C r n r+ n ic n i r+ n, r (n r ) i (3) k (k ) S(k, m) k+m C m (m,,......, 9) i () () k n s(n) m n j ( j m ) k m j j ( j m ) S(k, m j) () n r+ S(k, m) i m j S(k, m j) S(k, m) + S(k, m ) + + S(k, ) + S(k, ) S(k, ) + S(k, ) + + S(k, m ) + S(k, m) m S(k, i) i n ic n i r+ n r+ i n ic r n r+ i n ic r n C r 6
61 n r+ i n ic r n r i n ic r }{{} n i+ C r n i C r + r C r n r ( n i+ C r n i C r ) + i ( n C r C r ) + ( C r n C r ) + + ( r+ C r r C r ) + n C r r C r + n C r (3) k S(k, m) k+m C m ( ) (i) k () m S(, m) S(, i) i +m C m m C m m m m i ( ) (ii) k l (l ) ( ) () m S(l +, m) S(l, i) m i i l+i C i l++m C m l+m C l m i l+m ic m i l+m C m + l+m 3 C m + + l C + l C l C + l C + + l+m 3 C m + l+m C m m j l+j C j S(l +, m) l++m C m k l + ( ) (i)(ii) k (k ) ( ) 6
62 () nc k + n C k+ n+ C k+ (k,,,..., n ) () () n nc + n C + + n C n n () () () n (i) n () C + C + () (ii) n k k+c + k+ C + + k+ C k + k+ C k+ k C + ( k C + k C ) + + ( k C k + k C k ) + k C k ( k C + k C + + k C k ) k k+ (i)(ii) n nc + n C + + n C n n () ( + b) n n nc r n r b r () n nc r () () + (n)! (n!) 98 6
63 n C r () ( x ) 6 x 4 () ( + x) 6 ( y) 6 xy 3 x y (3) 6C 6 C 4 6 C 6 C C 6 C 6 C 3 6 C + 6 C 4 6 C (4) n n 4 f(n) n C n C 4 n C n C 3 + n C n C n C 3 n C + n C 4 n C f(n) n 7 () () ( x ) 6 6 C r 6 r ( x ) r ( ) r 6C r x r x 4 r 4 r x 4 ( ) 6C 5 () ( + x) 6 6 C p 6 p x p 6 C p x p ( y) 6 6 C q 6 q ( y) q ( ) q 6C q y q ( + x) 6 ( y) 6 ( ) q 6C p 6C q x p y q xy 3 p, q 3 ( ) 3 6C 6C 3 6 x y p, q ( ) 6C 6C (3) 6 C 6 C 4 6 C 6 C C 6 C 6 C 3 6 C + 6 C 4 6 C (+x) 6 ( y) 6 y 4, xy 3, x y, x 3 y, x 4 y x ( + x) 6 ( x) 6 ( x ) 6 x 4 () 5 (4) f(n) ( + x) n ( y) n y 4, xy 3, x y, x 3 y, x 4 y x ( + x) n ( x) n ( x ) n x 4 ( x ) n n C r n r ( x ) r ( ) r nc r x r x 4 r 4 r x 4 ( ) nc n(n ) n 63
64 () (4) n 4 f(n) n C n C 4 n C n C 3 + n C n C n C 3 n C + n C 4 n C ( n C n C 4 n C n C 3 ) + ( n C ) { } { n(n )(n )(n 3) n(n )(n ) n(n ) n n(n ) ((n )(n 3) 4n(n ) + 3n(n )) n(n ) ( n 5n + 6 4n + 8n + 3n 3n ) n(n ) 6 n(n ) n } 9 ω + ω + ω n () ω 3 () ( + ω) 3n ( ) n (3) 3nC 3k+ n 3nC 3k 3nC 3k+ 3nC 3k+ ( ) n 7 () () + ω + ω ω 3 (ω )(ω + ω + ) ω 3 () + ω + ω + ω ω (+ω) 3n ( ω ) 3n ( ) 3n (ω ) 3n { ( ) 3} n (ω 3 ) n ( ) n n ( ) n 64
65 (3) ( + ω) 3n 3n 3nC r ω r n 3nC 3k ω 3k + 3nC 3k+ ω 3k+ + n n ( n 3nC 3k + 3nC 3k+ ω + 3nC 3k + 3nC 3k+ ω 3nC 3k+ ω 3k+ 3nC 3k+ ω + 3nC 3k+ ( ω ) 3nC 3k ) 3nC 3k+ + () ( + ω) 3n ( ) n n ( ) n ( n 3nC 3k 3nC 3k ) 3nC 3k+ + 3nC 3k+, 3nC 3k+ ( 3nC 3k+ ( 3nC 3k+ 3nC 3k+ ) ω 3nC 3k+ ) ω 3nC 3k+ ω n 3nC 3k 3nC 3k+ n 3nC 3k 3nC 3k+ ( ) n 3nC 3k+ 3nC 3k+ ( ) n 65
66 ( + x + x ) n x r r () n r n+r () n 3 3 (3) () () ( + x + x ) n ( + x + x ) n + x + x + + n x n ( + x + x ) n x n ( + x + x ) n ( x n + x + ) n n x s x s x n+r n+r x r x x s ( + x + x ) n 3 ( x n + x + x ) n n s s x s ( + ) n n x + x s x n s s 3 ( + x + x ) n x r n s r (s n r) n r n r n+r () x n 3 3 (3) 3 ω ω 3 ω 3 (ω )(ω + ω + ) ω ω + ω + x ω 66
67 ( + ω + ω ) n n s [ n 3 ] [ n 3 ] [ n 3 ] s ω s 3k ω 3k + 3k + 3k + [ n 3 ] 3k + ω + ω [ n 3 ] 3k [ n 3 ] [ 3 ] [ 3 ] [ 3 ] [ n 3 ] 3k+ 3k+ ω 3k+ + 3k+ ω + 3k+ ω + 3k+ [ n 3 ] [ n 3 ] + + [ 3 ] [ n 3 ] 3k+ ω 3k+ ω 3k+ 3k+ ( ω ) [ 3 ] 3k+ 3k+ [ n 3 ] [ n 3 ] 3k+ 3k+ ω ω [ n 3 ] 3k [ n 3 ] 3k+, [ 3 ] 3k+ [ n 3 ] 3k+ ω [ n 3 ] 3k [ n 3 ] 3k+, [ 3 ] 3k+ [ n 3 ] 3k+ [ n 3 ] 3k [ 3 ] 3k+ [ n 3 ] 3k
68 n () n nc k n C + n C + + n C + n C n n () z z z + z z 4n (3) n ( ) k 4n C k 4n C 4n C + 4n C 4n + 4n C 4n ( 4) n () () ( + x) n x n nc k x k n nc k n () z z + z ± i z ± i [ ( cos ± π ) + sin 4 ( ± π 4 )] z 4n ( [ ( ) 4n cos ± π ) ( + sin ± π )] 4n 4 4 n [ cos ( ± nπ ) + sin (±nπ) ] 4 n ( ) n ( 4) n (3) ( + x) 4n 4n 4nC k x k x + i ( + i) 4n 4n 4nC k i k n n n 4nC 4r i 4r + 4nC 4r + 4nC 4r + 4nC 4r+ i 4r+ + 4nC 4r+ i + 4nC 4r+ i 68 4nC 4r+ i 4r+ + 4nC 4r+ i + 4nC 4r+ 4nC 4r+3 i 3 4nC 4r+3 i 4nC 4r+3 i 4r+3
69 () ( + i) 4n ( 4) n n 4nC 4r + 4nC 4r+ i n 4nC 4r 4nC 4r+ 4nC 4r+ ( 4) n 4nC 4r+3 i ( 4) n n ( ) k 4nC k 4n C 4n C + 4n C 4n + 4n C 4n ( 4) n [] () z 4n ( ± i) 4n { ( ± i) } n (±i) n { (±i) } n ( 4) n p, q (p q ) p C q! ( () n, k n+k+ C k+ n+kc k p! q!(p q)! n+k+c k ) n () m C 3 4C 3 5C 3 mc 3 3 () () n, k 69
70 ( n+k+c k+ (n + k + )! (k + )!n! n + k + k + k k + n () () n+k+c k+ n+kc k n + k + ( ( k!n! (n + k)! n+kc k n+k+c k ) k!(n + )! (n + k + )! n+k+c k ) ) k k + k n+3c 3 ( n+c n+3c ) 3 n+3c 3 3 ( n+c n+3c ) 3C 3 + 4C 3 + 5C mc m 3 m 3 k+3c 3 3 ( ( k+c C mc ( ) m(m ) 3 m m m(m ) 3(m + )(m ) m(m ) k+3c ) ) 7
71 [] : [] : [3] : [4] : () 7
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数論 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/008142 このサンプルページの内容は, 第 2 版 1 刷発行当時のものです. Daniel DUVERNEY: THÉORIE DES NOMBRES c Dunod, Paris, 1998, This book is published
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2000年度『数学展望 I』講義録
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x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 1 1977 x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y) ( x 2 y + xy 2 x 2 2xy y 2) = 15 (x y) (x + y) (xy
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3 3.1 3.1.1 A f(a + h) f(a) f(x) lim f(x) x = a h 0 h f(x) x = a f 0 (a) f 0 (a) = lim h!0 f(a + h) f(a) h = lim x!a f(x) f(a) x a a + h = x h = x a h 0 x a 3.1 f(x) = x x = 3 f 0 (3) f (3) = lim h 0 (
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chap1.dvi
1 1 007 1 e iθ = cos θ + isin θ 1) θ = π e iπ + 1 = 0 1 ) 3 11 f 0 r 1 1 ) k f k = 1 + r) k f 0 f k k = 01) f k+1 = 1 + r)f k ) f k+1 f k = rf k 3) 1 ) ) ) 1+r/)f 0 1 1 + r/) f 0 = 1 + r + r /4)f 0 1 f
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II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )
II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11
ORIGINAL TEXT I II A B 1 4 13 21 27 44 54 64 84 98 113 126 138 146 165 175 181 188 198 213 225 234 244 261 268 273 2 281 I II A B 292 3 I II A B c 1 1 (1) x 2 + 4xy + 4y 2 x 2y 2 (2) 8x 2 + 16xy + 6y 2
微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.
微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)
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85 4 86 Copright c 005 Kumanekosha 4.1 ( ) ( t ) t, t 4.1.1 t Step! (Step 1) (, 0) (Step ) ±V t (, t) I Check! P P V t π 54 t = 0 + V (, t) π θ : = θ : π ) θ = π ± sin ± cos t = 0 (, 0) = sin π V + t +V
r 1 m A r/m i) t ii) m i) t B(t; m) ( B(t; m) = A 1 + r ) mt m ii) B(t; m) ( B(t; m) = A 1 + r ) mt m { ( = A 1 + r ) m } rt r m n = m r m n B
1 1.1 1 r 1 m A r/m i) t ii) m i) t Bt; m) Bt; m) = A 1 + r ) mt m ii) Bt; m) Bt; m) = A 1 + r ) mt m { = A 1 + r ) m } rt r m n = m r m n Bt; m) Aert e lim 1 + 1 n 1.1) n!1 n) e a 1, a 2, a 3,... {a n
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D xy D (x, y) z = f(x, y) f D (2 ) (x, y, z) f R z = 1 x 2 y 2 {(x, y); x 2 +y 2 1} x 2 +y 2 +z 2 = 1 1 z (x, y) R 2 z = x 2 y
5 5. 2 D xy D (x, y z = f(x, y f D (2 (x, y, z f R 2 5.. z = x 2 y 2 {(x, y; x 2 +y 2 } x 2 +y 2 +z 2 = z 5.2. (x, y R 2 z = x 2 y + 3 (2,,, (, 3,, 3 (,, 5.3 (. (3 ( (a, b, c A : (x, y, z P : (x, y, x
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December 28, 2018
e-mail : kigami@i.kyoto-u.ac.jp December 28, 28 Contents 2............................. 3.2......................... 7.3..................... 9.4................ 4.5............. 2.6.... 22 2 36 2..........................
( ) 2.1. C. (1) x 4 dx = 1 5 x5 + C 1 (2) x dx = x 2 dx = x 1 + C = 1 2 x + C xdx (3) = x dx = 3 x C (4) (x + 1) 3 dx = (x 3 + 3x 2 + 3x +
(.. C. ( d 5 5 + C ( d d + C + C d ( d + C ( ( + d ( + + + d + + + + C (5 9 + d + d tan + C cos (sin (6 sin d d log sin + C sin + (7 + + d ( + + + + d log( + + + C ( (8 d 7 6 d + 6 + C ( (9 ( d 6 + 8 d
(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y
[ ] 7 0.1 2 2 + y = t sin t IC ( 9) ( s090101) 0.2 y = d2 y 2, y = x 3 y + y 2 = 0 (2) y + 2y 3y = e 2x 0.3 1 ( y ) = f x C u = y x ( 15) ( s150102) [ ] y/x du x = Cexp f(u) u (2) x y = xey/x ( 16) ( s160101)
ma22-9 u ( v w) = u v w sin θê = v w sin θ u cos φ = = 2.3 ( a b) ( c d) = ( a c)( b d) ( a d)( b c) ( a b) ( c d) = (a 2 b 3 a 3 b 2 )(c 2 d 3 c 3 d
A 2. x F (t) =f sin ωt x(0) = ẋ(0) = 0 ω θ sin θ θ 3! θ3 v = f mω cos ωt x = f mω (t sin ωt) ω t 0 = f ( cos ωt) mω x ma2-2 t ω x f (t mω ω (ωt ) 6 (ωt)3 = f 6m ωt3 2.2 u ( v w) = v ( w u) = w ( u v) ma22-9
1 8, : 8.1 1, 2 z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = n i=1 a ii x 2 i + i<j 2a ij x i x j = ( x, A x), f =
1 8, : 8.1 1, z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = a ii x i + i
4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5.
A 1. Boltzmann Planck u(ν, T )dν = 8πh ν 3 c 3 kt 1 dν h 6.63 10 34 J s Planck k 1.38 10 23 J K 1 Boltzmann u(ν, T ) T ν e hν c = 3 10 8 m s 1 2. Planck λ = c/ν Rayleigh-Jeans u(ν, T )dν = 8πν2 kt dν c
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II (1 4 ) 1. p.13 1 (x, y) (a, b) ε(x, y; a, b) f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a x a A = f x (a, b) y x 3 3y 3 (x, y) (, ) f (x, y) = x + y (x, y) = (, )
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数学Ⅱ演習（足助・09夏）
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(1) 3 A B E e AE = e AB OE = OA + e AB = (1 35 e ) e OE z 1 1 e E xy e = 0 e = 5 OE = ( 2 0 0) E ( 2 0 0) (2) 3 E P Q k EQ = k EP E y 0
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Chap11.dvi
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Chap9.dvi
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