+ + 3 + 4 +... π 6, ( ) 3 + 5 7 +... π 4, ( ). ( 3 + ( 5) + 7 + ) ( 9 ( ( + 3) 5 + ) ( 7 + 9 + + 3 ) +... log( + ), ) +... π. ) ( 3 + 5 e x dx π....................................................................... 3.3 Γ- B-................................... 6 9............................... 9................... 3 3 7 3.............................. 7 3. ( ).................. 9 3.3........................... [ ] [ ], [ ] 7 http://www.mth.kyoto-u.c.jp/ obuo/tch/biseki/7/otes.html [ ] 7 9 3. [URL] http://www.mth.kyoto-u.c.jp/ e obuo
... D f, f,,,... D D (uiform orm) 3 () lim f f D. f D sup f (.) D (f ) f D coerge uiformly () x D lim f (x) f(x) (f ) f (D ) coerge poitwise x D f (x) f(x) f f D (), () (.) D R d K D lim f f K (f ) f (D ) coerge loclly uiformly coerge uiformly i wider sese x D K {x} K D f f K f f D. (.3) {.. x [, ],f (x) x, f(x) def., x [, ) (i) f, x. [, ] f (ii) < δ < f [, δ] f f f f.. f.. D R d, f C(D),,,... f : D C f C(D). 3 sup D f (.)
x, x D, x x f(x ) f(x) m, f(x ) f(x) f(x ) f m (x ) + f }{{} m (x ) f m (x) + f }{{} m (x) f(x). }{{} () () (3) K def. {x } {x} f f ( ) ( ) ε >, m, f f m K < ε/3, ( ) m () + (3) < ε/3. ( ) m f m C(D),, () < ε/3. () + () + (3) < ε...... ( M- ) D f : D C, f D < s f D lim s f D. x D f (x) f D. f D f D. x D () f (x) f D f D. () s f D () s f D f + D + f D. ( )... Arcsi y ( )!!y + ()!!(+) y [, ] 3
M- (..) (..4) g(x) ( ) x, x [, ] x g(x) M- g() (..3) p /, q ( ) ( g() ).. p [, ), q C ( ) () p p +. (b) Q q +... + q (c) lim p. (c) Q (),(b) (c) (c) s s p m q m m s s m p m+ sup Q j Q. (.4) j m (c) Q C (c) Q lim Q. (),(b) m, Q C q j (Q j Q) (Q j Q) s s m jm+ jm+ p j q j jm+ p j (Q j Q) } {{ } () jm p j+ (Q j Q) (p j p j+ ) (Q j Q) + p (Q Q) p }{{} m+ (Q m Q). () s (),() ( ) () : jm+ (p j p j+ ) (Q j Q) sup Q j Q j m+ jm+ (p j p j+ ) p m+ sup Q j Q ( ). j m+ 4
() () : (c) () p sup Q j Q,. j (c) Q lim Q () p Q Q,. s s s m () + () + p m+ Q m Q p m+ sup j m+ Q j Q + () + p m+ Q m Q p m+ sup Q j Q + (). j m lim () (.4)..3 ( 4 ) D p : D [, ), q : D C, ( ) () x D, p (x) p + (x). (b) Q q +... + q sup Q D <. (c) lim p D. (c) p D < Q D (),(b) (c) (c) s p m q m m D s s(x) s m (x) p m+ (x) sup Q j Q (x) x D. (.5) j m (c) Q : D C (c) Q(x) lim Q (x). x D.. s (.5) (.5) s s m D p m+ D sup Q j Q D. j m (c) (c) lim m s s m D. 4 5
..4 ε, D ε {z C ; z, + z > ε} ε > ( ) z g(z) D ε..3 D D ε, p /, q (z) ( ) z Q (z) z ( z) m z ( z) + z m Q (z) + z ε..3 (),(b),(c)..5 ( 5 ) ( ) s (x) m x m, m x [, ] s(x) [, ] D [, ], q (x), p (x) x..3 (), (b), (c) s(x)...3 Γ- B- Γ- B-.3. u > Γ(u) x u e x dx (.6) u Γ(u) ((, ) (, )) Γ- Γ(u + ) uγ(u), (.7) Γ(u + ) (u + )(u + ) (u + )uγ(u), (.8) Γ( + )!. (.9) (.6) f(x) x u e x (x > ) e x!x () f(x)!x u, () f () > u f f (.7): Γ(u + ) 5 iels Herik Abel (8 89) x u e x dx [ x u e x ] +u x u e x dx. }{{} }{{} Γ(u) 6
(.8): (.7) (.9): Γ() (.8) e x dx..3. u, (, ) Γ(u) u y u e y dy (.) u ( log s ) u s ds, (.) u r u e r dr (.).3. u, > x u ( x) dx π/ cos u θ si θ dθ (.3) B(u, ) B- B(u, ) B(u, ) B(, u), (.4) B(, u) B(u, ) /u, (.5) B(, ) π. (.6) (.3) f(x) x u ( x), M u (/) u x (, /] f(x) M x u x [/, ) f(x) M u ( x). / f, / f f x cos θ (.4): y x (.5): B(, u) (.4) B(u, ) x u dx /u. (.6): (.3) u /.3. t, u, > (i) B(u, ) u B(u, ). (i) y dy. (ii) (+y) u+ x u dx (+x t ).3.3 u, > xu ( x t ) dx t B( u t, ). (iii) B(u, u) t B( u t, u t ), (t > u). B(u +, ) u B(u, + ), B(u, + ) B(u +, ), (.7) u u B(u +, ) B(u, ), B(u, + ) B(u, ). (.8) u + u + 7
k, l B(k + u, l + ) (k + u) k(l + ) l (k + l + u + ) k+l B(u, ), (.9) k B(k +, l + ) ( ) k k+ }{{} k x k ( x) l dx k!l! (k + l + )! (.) (.).3.4 < < b <, u, > (x ) u (b x) dx (b ) u+ B(u, ) (.) (x )m (b x) dx (b )m++ m!! (m++)! Γ B.3.3 u, > 3.3.3.3.4 Γ( ) π. > Γ( ) π. (m, ) B(u, ) Γ(u)Γ() Γ(u + ). (.) e r dr π/. π (.6) B(, ) (.) Γ( )Γ( ) Γ( Γ() ) e r dr (.) Γ( ) π/..3.5 m, ( ) { + ( )!! π,, Γ / ( )!!,. ( )/ ( m + B, + ) π/ cos m θ si θ dθ { π (m )!!( )!! (m+)!!, {m, }, (m )!!( )!! (m+)!!, {m, }. 8
.. lim f lim f, ( ).. ( ) R d f R() ( { }) () (b) (c) () lim f f. (b) sup f < f f ( ) (c) lim f f. lim f f... () (c) () (b) (b) (c).. (b) (c) ( [, 5,.4.] ) sup f < f f ( ) f f ( ) (c) (, ], f (x) ( x ) (x, ) f f /. (c).... x [, ] Arct x ( ) x +. x + 3 + 5 7 +... π 4, ( ) x (, ) : f (y) ( ) y () x (, ) lim f (y) + y. () x (, ) x f ( ) x +. + 9
x (, ) () [ x, x] Arct x () x dy x + y lim ( ) x + + f ( ) y +. + y ± : f(y) f(y) [, ].. y ± f(y) [, ] f() y....5 f(y) [, ].. x [, ] ( ) x + (+) 6..3 π 6. f (x) π/ f si ( )!! x + ()!! +.3.5 π/ () lim f si ( )!! ()!! ( )!! ()!! ( + ). x + Arct y y dy x π/ si + ()!! + ( + )!! ( + )... lim f Arcsi ([, ] ). lim f si θ θ (θ [, π/] ). (..) π/ () lim f si (),() s def. (+) π/ π 8. θ dθ π 8. ( + ) } {{ } π /8 + (), s π }{{} s/4..3 735 k, k 4, 6, 8,, ([,, 35 ] k k B k π k, k,,.. (k)! [,, 337 ] B, B,.. 6 Eugée Chrles Ctl (84 894). 6.
.. (i) ( x log x)! x (, ] (ii) x x dx. (.9),(.)...3 (i) x (ii) x log x dx x x log x dx..4 θ (, π), r [, ] ϕ(r, θ) { e iθ r e iθ y log y x e x r cos(θ) dy (x [, ]). dx π 6 + i...3. r si(θ) log ( r cos θ + r ) + iϕ(r, θ), (.) Arct ( ) r cos θ si θ, θ π,, θ π. cos(θ) + i si(θ) r ( log si θ ) + i π θ. (.) r < θ (, π) r θ (, π) () (.) Log ( re iθ ) () (.) si(θ) π θ θ, π. θ θ π (..) r [, ) θ (, π) ρ e i(+)θ eiθ ρe cos θ ρ + i si θ, (ρ [, r] ). iθ ρ cos θ + ρ ρ [, r].. (.) r θ (, π)..4 r e iθ ( ) ( re iθ ) r [, ] (.) r [, ] r < (.) r e iθ cos(θ) + i si(θ) log ( cos θ) + iϕ(, θ). cos θ si θ, si θ cos θ si θ (.)
..4 θ ( π, π)\{} e( )iθ θ π 4 cos(( )θ) + i ( si(( )θ) log ( 3 + ( 5) + 7 + ) ( 9 ( ( + 3) 5 + ) ( 7 + 9 + + 3 ) + ( ) ( 3 + 5 5 + 7 t θ ) + i θ π θ 4. )... log(t π 8 ) log( + ), ) +... π. {..5 π π log( r cos θ + r, r <, )dθ log r, r >. (.) r < r > r <..6 (i) π log si π log si π log cos π log. (ii) Arcsi x x dx π ϕ t ϕ dϕ π ( π θ) t θ dθ π log. (iii) ( )!! π ()!!(+) log. (i) (.) θ [ε, π ε] (ε > ) ε..7 (.) f (θ) (i) f (θ) θ..8 f (θ) si (+ )ϕ si ϕ dϕ θ e iθ, g (θ) si(θ) (θ def. (, π)). (ii) sup f <. b e iθ (, b C, θ def. (, π)) f(θ), g(θ) lim f f, lim g g ( ) sup( f + g ) < 7 b fg π ( ) (.),..7 π 6..9 : (i) θ [, π] cos(θ) (θ π) 4 π (ii) π4 4 9. (i):..7 (.)..7 θ [ε, π ε] (ε > ) ε (ii): (..8) :..9 k (k 4, 6,...).. () (b) (b) (c) () (c) f f f f f f ( ). 7 799 Mrc-A. Prsel (755 836)
(b) (c) M sup { } f ε > J J < ε 4M+ ε lim f f J f f ε/. f f M, f f f f + M( J ) ε + M ε 4M + ε. J J ε > (c)...3.... (, b) R, J f t R loc () (t J) t J f t lim u u> sup t J u f t lim b <b sup t J f t t J f t (.3)....() M-..(b)..3.. ( ).. f t () g : [, ) (x, t) J f t (x) g(x), g (b) f t fg t, f, g t : C f, g t C () (t J), sup t J g t <, sup t J g t <. 3
(): t J f t (.3) lim b <b (b): f sup t J F (x) x lim f t f t f t g g. ( f, M sup g t + t J sup b x [,b) ( b). F (x) F (b). ) g t (F (x) F (b))g t(x) dx sup F (x) F () M <. x [,b) f t (x) (F (x) F (b)) g t (x) f t f t b lim b <b (F () F (b))g t () }{{} () (F (x) F (b))g t(x) dx } {{ } () () M F (b) F (), () sup F (x) F () M. x [,b) sup t J sup t J f t u f t (u ) (.3)....3 ( ) (, b) R, f R loc () ( { }) f t () lim f f, (b) f { } lim f f, lim f f. 4
..4 x ( + ) b Γ(b) e s s b dt xe s Γ(b) t ( log t xt ) b (i) >, b >, x. (ii), b >, x [, ). f (s) e (+)s s b g(s) def. e s s b xe s x f (s), f Γ(b) ( + ) b. g (s) def. e s s b (xe s ) + xe s dt x f (s) () lim g g ((, ) ) (b) g..3 g lim g x x f Γ(b) ( + ) b. () ε > [ε, ε ] s xe s e ε >. s e s s b M xe s g g (s) e s s b xe s + (b) xe s M xe s + Me ε(+),. g (s) e s s b xe s (i) (ii) s (, )..() (b)....4 ( ) +b x (, ]) ( ) 3+, ( ) 3+... y R y y + x b/ x / si xy e x dx. u b +u, (, b >, ( ) + ( ),..3.. < u < < b f f sup f f + { } u f + } {{ } () 5 f f + u u f f f f + sup } {{ } () { } u f } {{ } (3)
ε > (b) u, () ε/6, (3) ε/6 [u, ] () f f ([u, ] )... () ε/3. f f ε. 6
3 3. 3.. ( ) R d J R k, (x, t) f t (x) ( J C) () ( ) F (t) f t (x) dx t J (b) ( ) J R (x, t) J l t f t (x) (l,..., m) (x, t) J F C m (J) F (l) (t) t l f t (x)dx, l,..., m. 3.. t [, ) b (, ) tx si x e x dx π Arct t. f t (x) def. tx si x e x, tf t (x) e tx si x (x, t) [, b] [, ) 3.. F b (t) f t(x) dx t [, ) C F b (t) ( ) e tx si x dx m e (t i)x dx e tb (cos b t si b) + t. F b (t) t t e sb (cos b s si b) + s + C, (C ) ds + s r b (t) + C, }{{} π Arct t r b (t) t e sb (cos b s si b) + s. F b (t) f t e tx dx t (t ) cos b s si b C. +s r b (t) t e sb ds b (b ). lim F b (t) π Arct t. b 3.. q(θ) cos θ + b si θ (, b > ) (i) π π cos q π q π 4, π b si q π b q π 4b. (iii) π b q π q π 4 b. (ii) b ( + ) b. 7
( ) t 3.. t exp( x ) dx + exp( x ) dx π/ 3..3 q(θ) cos θ + b si θ (, b > ) π (i) log q π ( +. (ii) π b) log q π log + b. 3..4 (ii),(iii) 3.. exp( (+x )t ) +x dx π 4 (i) x (, ) cos x x x {. cos x (ii) e tx log + (/t) dx, t >, x t. { cos x π (iii) x e tx dx Arct t t log + (/t), t >, π t. cos x si x (iii) ( si x ) x e tx dx. 3.. (): t, t J, t t K {t } {t} f t (x) (x, t) K 3..3 f t (x) f t (x) (x ).. f t (x) dx F f t (x) dx. (b): A m, P m A m P m m m J K J J J ( ) f t+h (x) f t (x) lim t f t (x) (x ) h h h t f t (x) J. 3..3 ε > δ > t, t J, t t δ sup t f t (x) t f t (x) ε. x h δ f t+h (x) f t (x) t f t (x) h ( t f t+θh (x) t f t (x)) dθ t f t+θh (x) t f t (x) dθ ε. x ( ) h F (t + h) F (t) h f t+h (x) f t (x) dx. h 8
h ( ).. F (t) t f t (x)dx 3.. F C(J). P m 3..5 3..(b) 3.. 3..3 A i R d i (i, ), f C u (A A ) x, x A, x x lim sup y A f(x, y) f(x, y). ε > δ > x, x A, x x δ sup y A f(x, y) f(x, y) ε. ε > δ > (x, y), (x, y ) A A, (x, y) (x, y ) δ f(x, y) f(x, y ) ε. y y x, x A, y A, x x δ f(x, y) f(x, y) ε. y A 3. ( ) ( 3..) 3.. ( ) (, b) R, J R k, (x, t) f t (x) ( J C) F (t) f t (x)dx t J K J t K () ( ) F C(J). (b) ( ) J R l,..., m (x, t) J l t f t (x) J F C m (J) t J F (l) (t) l t f t (x)dx t J l t f t (x)dx, l,..., m. 9
3.. f : C((, )), f () F (t) e tx f(x) dx t [, ) lim lim F (t). t (b) F C ((, )), F (l) (t) ( x) l e tx f(x) dx. t F (t) (): g t (x) e tx, f t fg t..(b) f t t [, ) lim f t f, t lim t f t. (, ) K (, ) K [ε, /ε] ε > def. [ε, /ε] f t f ( e t/ε ) f (t ). f t e tε f (t )...3 (b): t l f t (x) ( x) l f t (x) (x, t) (, ) t l f t t (, ) 3.. () t J def. [ε, /ε] t J l t f t (x) (x) l e tx f(x) ε l e εx f. t J x (, )..() t l f t t J 3.. Γ C ((, )) q Γ (q) (t) x t (log x) q e x dx 3..3 π π e (x m) e (x m) +itx t imt dx e, m R, >, t R. m > : 3..3 8 3..3 m m e x e x +itx dx e x cos(tx) dx. > f t (x) e x cos(tx) f t (x), t f t (x) xe x si(tx) (x, t) R R f t (x) e x, t f t (x) x e x 8 m, 5 f,
F (t) def. f t, tf t t R 3.. F C (R) F (t) xe x si(tx) dx [ e x si(tx) ] }{{} t xe x cos(tx) dx. } {{ } F (t) F () π (.3.4). 3.. m m e (x m) +itx dx e x +it(x+m) dx e itm e x +itx dx. m e itm m ( 3.. F C (R), h C(R) F t hf F (t) F () exp ) ( h ) t G(t) exp h (F/G) 3..3 3..3 (x m) π xe dx m, π (x m) e (x m) dx. 3..4 3.. 3.. e tx si x x dx π Arct t (t [, )) 3..5 3..4 3.. 3..6 ( ) t, u (i) ( exp ( x t ) ) x dx t exp (ii) ( ) exp x t dx πe t. 4x cos(tx)e u(+x ) π ( ( ) ) y t dy y π y. u exp ( x t 4x ) (iii) dx dx, +x (i), ( + )/ (iii) u 3.. cos(tx) dx π +x e t. 3..4 ( ) R f C m () (m, ) () f f (b) f (k) ( k m) f C m () x, k m f (k) (x) lim f (k) (x). A m, P m A m P m m m g(x) lim f (x) ( ).. g C()., x, x J J
() f (x) f () + x f. f g (J ). () (..) f(x) f() + f C (), f g. P m 3..7 3..4 x g. 3.. ():t, t J, t t 3.. lim f t (x) f t (x), (x ) f t..3 F lim f t (x) dx f t (x) dx. (b): 3..4 ( 3..) u, {, b} F u, (t) u f t F F,c + F c,b F F,c, F c,b F c,b ( 3..) F c, C m (J) F c, (l) (t) t l f t, l,.., m. c t J l,.., m lim F c,(t) F c,b (t) b lim F c, (l) (t) b c l t f t (t J ). 3..4 F c,b C m (J), F (l) c,b (t) t l f t c 3.3 3.3. : (, b), (c, d) R f C((, b) (c, d)) dx d c f(x, y) dy d c dy f(x, y) dx.
(, b) x d c f(x, y)dy, (c, d) x f(x, y)dy dx d c f(x, y)dy d c dy f(x, y)dx. 3.3. ( [, ] 3.3. e x def e x 3.3. dx π (.3.4. dx π/ e x dy dx e y dy e (+y )x xdx e x dx dy e x y xdy + y π/. 3.3.3 u, > B(u, ) Γ(u)Γ() Γ(u + ) (.3.3 ). B(u, ) Γ(u)Γ() Γ(u + ) (.7),(.8) (.7) (u + )(u + + ) B(u +, + ), u (u + )(u + + ) Γ(u + )Γ( + ). u Γ(u + + ) u, u +, + u, > u, > x, x < g u (x) x u e x, g u (x) g u C(R). g f(x, y) g u (x y)g (y) f C(R ) f(x, y)dx g (y) y g u (x y)dx g (y) g u (x)dx g (y)γ(u). () () dy f(x, y)dx Γ(u)Γ(). x f(x, y)dy e x (x y) u y dy x u+ e x B(u, ). dx f(x, y)dy Γ(u + )B(u, ). 3
3.3. dy 3.3.4 t f(x, y)dx dx f(x, y)dy tx si x e x dx π Arct t. ( 3.3.4 ) t > 3.. t ( ) x > e xt x t e xy dy. tx si x e x dx dx t e xy si x dy. () e xy si x dx () t dy e xy si x dx t dy + y π Arct t. +y () Fubii f(x, y) e xy si x f(x, y) e xy x. dx t f(x, y) dy ( ) x dx t e xy dy e xt dx /t <. Fubii 3.3. t cos(tx) dx π +x e t 3..6. +x e (+x )y dy 3.3. ( 9 cos x si x ) dx dx x x t e xy dy π x π. def. t π e ix tx x dx dx e (y +t i)x dy. t > f(x, y) e (y +t i)x 3.3. 9 Augusti Je Fresel (788 87) 4