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4 elemet, set A, A A, A A, A A, b, c, {, b, c, }, x P x, P x x {x P x}, A x, P x {x A P x} 3 { {,, 3,, } } N, {, 3,,, 0,,, 3,, } Z, p p, q Z, q 0 Q, R, {x + yi x, y R} C q 4 A, B x A x B x B x A, A B A B, A B, A B B A A B x A x B 3, 5 A, B uio, cup A B, itersectio, cp A B A B {x x A x B}, A B {x x A x B} A, B A B A B {x x A x B} 3 A, A, A 3,, A,, A {x x A }, A, A {x x A } A 6, b R b,, b,, b], [, b, [, b],,, [,,, b,, b], b {x R < x < b},, b] {x R < x b}, [, b {x R x < b}, [, b] {x R x b},, {x R x > }, [, {x R x },, b {x R x < b},, b] {x R x b},, b,,,, b, [, b], [,,, b] 7 X, Y, X Y, X Y, f : X Y X f Y x X f : X Y Y fx, x f f : X Y g : Z W, X Z Y W, x X fx gx, f g 8 X, Y, Z, f : X Y, g : Y Z x X, gfx Z X Z f g g f X x x X, id X X

5 3 f : X Y, g f id X, f g id Y g : Y X f, g f 4 f : X Y, Y c, fx c x X, f 5 f : X Y, fx fy x, y X x y, f 6 9 f : X Y, f f f : X Y g, y Y fgy f gy id Y y y f x, z X fx fz, x id X x g fx gfx gfz g fz id X z z f f, y Y fx y x X y Y fx y x X g : X Y y Y f gy fgy y id Y y f g id Y x Y fx y, gy x g fx gfx gy x id X x, g f id X g f g, h : Y X f f h id Y, g f id X, y Y gy gid Y y gf hy gfhy g fhy id X hy hy h g, f f : X Y, f : Y X { },, C, α C ε > 0 N, > N α < ε, { },, α, lim α z C, z < lim z 0 z 0 z 0 z > z > 0, k k + z z + +, ε > 0 k > ε z < ε 3 { },,, {b },,, c C lim + b lim + lim b, lim c c lim lim b lim lim b, lim b 0 lim b lim lim b lim α, lim b β, ε N, N, > N α < ε, > N b β < ε N, N N 3 > N 3 α < ε b β < ε + b α + β α + β β α + β β < ε + ε ε, lim + b α + β, N 4 > N 4 α < ε + c c cα c cα ε c + c < ε lim c cα

6 ε N 5, N 6, N 7, > N 5 α < + β, > N ε 6 b β < + α, > N 7 α < N 5, N 6, N 7 N 8 > N 8 α+α α + α + α, b αβ b β+ αβ b β + α β + α ε + α + ε + β β < ε + ε ε, lim b αβ, β 0 N 9, N 0, > N 9 b β < ε β, > N 0 b β < β N 9, N 0 N > N β β b + b β b + b < β + b b > β, β b β b β < b β β < ε lim b β lim b lim b b lim lim 4 { },,, {b },, b α β α β N b lim lim b {c },,, N c b, lim {c },, lim c lim lim b lim α, lim b β α > β N, N, > N α < α β, > N b β < α β N, N N 3 > N 3 α β < α b β < α β b < α+β <, N b α β ε, N 4, N 5, > N 4 α < ε, > N 5 b α < ε N 4, N 5 N 6 α α c α b α b α, > N 6 c α < ε {c },, α 5 ε > 0 N, m, > N m < ε, { },, 6 { },, lim α, ε N, > N α < ε m, > N m m α + α m α + α < ε + ε ε { },, 7 S C, r, x S x r S { },,, {,,,, }, { },, S R, M, x S x M S, M S, L, x S x L S 3 { },,, {,,,, }, { },,, {,,,, }, { },, 8 { },,, < < < k < k+ < { k } k,,, { k } k,, { },, 9 { },, α α { k } k,, { },, ε > 0, N N α < ε < < < k < k k k, k N k α < ε { k } k,, α 3

7 0 S S U, ε > 0 M ε, S M U M ε, S M U, ε 0 > 0, M U M ε 0, S S, M ε 0 ] M ε 0 U M U S, M ε 0 ], S, M ε 0 ] M ε 0 U, S, K ε 0 ] K U, K M ε 0 S, M + ε 0 ], S, M ε 0 ], S, ε > 0 M ε, S M U { },,, ρ > 0, k k mk ρ k > m k > k {s l },, s m, s s l+ m sl, s l+ sl l,,, s > s, s l+ > s l+ > s l, sl sl > ρ l,, sp p sl sl + p sl+ sl + s > pρ + s l, { },, { sp } p,, { },,, i ii iii iv v vi i ii; { },, { },, i { },,, 3 { },, lim ii iii; S S U U, { },,,,, U > > > i i, S i,,, 0 M U M, S U, + < + < M + U +, M, +, S S U { },,, ii α lim, x S x 4 α x α U, U, β < α β U, β] S β, S, α α,, α, S α β > k k β, α k, β, S, β, S U iii iv; S S {x R x S} S c c S, S d d S S l 4

8 , S {M R x S x M} L L {M R M L} {M R x S x M} {M R x S x M} {M R y S y M} L S iii L µ µ L iv S, S {M R x S x M} U U {M R M U} {M R x S x M} {M R x S x M} {M R y S y M} U S iv U µ µ U iii iii i; { },, S {,,,, } S α, ε > 0, α α ε S N > α ε N { },, > N α ε < α, { },, α i v; { },, {α },, {β },,,,, β α + β α i [α, β ] i, [α, β ] α, β α, α,, α k, β, β,, β k,,, k [α k, α k+β k ], [ α k+β k, β k ] i i i [α k, α k+β k ] i α k+ α k, β k+ α k+β k α k+, β k+, α k+ α k+β k, β k+ β k α k+, β k+ α k α k+, β k β k+ β k+ α k+ β k α k k β α, i [α k+, β k+ ] i {α },, [α, β ], {α },, lim α α β α + + β α lim β α, < < < k, j,,, k j [α j, β j ], i [α k+, β k+ ] i, k+ [α k+, β k+ ] k+ k { },, { k } k,,, k α k k β k 4 α v vi; { },, N, m, > N m <, m > N N+ < m < N+ +,,,, N, N+ A,,,, N, N+ + B, A B { },, { },, { k } k,, lim k k α, ε > 0 N, N k > N k α < ε, p, q > N p q < ε k > N k > N k N + N +, p > N p α p k + k α < ε + ε ε { },, α vi i;,,, iii, vi,, 3 S, S S, S, S S 5

9 4 {c },,, c + ib, b R,,, {c },,, { },,, {b },, {c },,, { },,, {b },,, b c + b, m, b b m c c m leqq m + b b m 5 {c },,, c + ib, b R,,,, { },, {b },, v { },, { k } k,, {b },, {b k } k,, v {b k } k,, {b ki } i,, 9 { k } k,, { ki } i,, 3 {c },, {c ki } i,, 6 {c },,, c + ib, b R,,,, { },, {b },, 4, vi { },, {b },, 3 {c },, 7 { },, C, α C S k lim S α, α k 3, 8, b, + b + b, c c c C S k k, T k k b S k + T k k + b, cs k k c 3 +b lim S k+t k lim S k+ lim T k + b, k k k c lim cs k c lim S k c k k 9 { } 0,, 0, S k k { } 0,, S k+ S k + k k+ S k, S k+ S k k+ k+ S k {S k+ } k0,,, {S k } k0,, S k+ S k k+ S k, k, l S k+ S l k l S k+ S l+ S l, k l S k+ S k S l {S k+ } k0,, {S k } k0,, i, ii, lim S k α S k+ S k k+ lim k+ 0 lim S k+ α k k k lim S k α k 0, ε > 0 N N, m > N m k < ε k 6

10 S k k m S m S m k, {S k } k,, k, ε > 0 N N, m > N m k < ε 6, { k } k,, { },,, k k k 3 { },, { k } k,, k k k k,, 0 ε > 0 N N, m > N m k < ε m k m k < ε 0 k k k s l l k {s l } l,,, s l l k k k l m m m m, i {s l } l,, 3 t l l k, s l l k l,,, l ml k k {,,, l } {,,, ml} t l s ml k {t l } l,, i k k k p,, k p k, k kp, l k, k,, kl µl k, k,, kl µl l N µl {,,, l} { k, k,, kl } {,,, µl } {,,, N } {,,, N } {,,, l} {l +, l +,, mn} S l l k, T l l mn k S l T N k k k k k s mn s l k N, k >l k N, k >l kl+ k s N µl S l T N s s l k {s l } l,, s, ε > 0 N l > N s s l < ε {T l } l,,, ε > 0 N l, m > N T l T m < ε N N N 3, l > N 3 S l T l S l T µl + T µl T l < s s l + ε < ε lim l S l lim l T l { },, {c },,,, c, N N r <, N + c r 3 N N r <, N r S k k k,, S k k k c c {S k } k,, i 7

11 N N N+ N+ N N+ N r N N c N r N N {c },,,, c, c N + N r 3 N r N c {c },,,, r N c, c N + r r, N S S {,,, k, } i j i j,, 3, k k, S, S {m, m,, m k, } i j m i m j mk k k k S S S 3 α S k k,, N S k N, k l S k S l s k S S k, k s k α S k { k,i i,, } i < j k,i < k,j, t k k,i, t k,l i k i l k,i, m, l N m t k,l, l m t k k { m t k } m,, t k s k t k k N k s k k k ε > 0 0, N N m > N m i < ε i α < ε > N i ε {,,, N} S S S M i m M N m > M s k α m s k N+ N+ i + i α N+ i + i α < ε k k i i in+ i s k α k i k i 4 m, b c k m0 0 c k c k m b m0 0 k0 k0 m+k m b k m0 m b k m, p i i l, q i i b l, r i i c l, α, β, i p i α, q i β l0 l0 l0 r i c k k0 i l0 k0 l k b l k p i q i αβ {r i } i0,, 8

12 , d l l k0 k b l k, e i i d l, {e i } i0,, l0 Cuchy S l, T b l, S i i l, T i i b l, R i i c l, S i T i R i u,v i u+v>i l0 u b v i li+ l0 d l e i e i l0 l0 l0 u,v i u+v>i u b v, ST R i S S i T + S i T T i + S i T i R i, ST R i S S i T + S i T T i + e i e i i, S S i, T T i 0, S i S, {e i } i0,, Cuchy e i e i 0 lim i R i ST 5 z C, z! { } 0,,, lim 0 z! exp z expz + w exp zexp w, exp 0, exp z exp z z!, b w! { } 0,,,, {b } 0,,,, 4 c k m+k exp zexp w c k k0 m b, m+k z m m! k0 lim z + 0 z!, exp : C C m, b m0 0 c k c k m b m0 0 w! m+k k! k m k0 z m w k! z + wk c k k! z + wk expz + w, exp 0 exp zexp z expz + z exp 0 exp z exp z 7 x m > x x x k k! < xm m! km x m m! x! x x x x m m x m+ < x m m! < x x x m x x! < xm m! x x xm m! 8 x C, + x { },, m+ lim, m+ km x k k! < x,, < exp x lim exp x + k k k+ k! <, < + x k + x + k k x k k! k0 x! 0, x m m! k m+ km k x k k! < exp x i + > i + + k k x k k! + + k + x k k! + + k + + x+ +! k > 0 x k k! k k 9

13 k x k k! x K K! < ε 4 kk K ε 4+ x K K, ε > 0, 7 K > x 3 K N, K 0 < ε < 4, K >, > K K K < ε, k K k k 4+ x K K 7 K k! + K k! <, K K k3 N 0 < ε <, > N x k k! x k k! k k0 < k K k K k K k x k k! x k k! x k k! k k + k + x k k! kk kk K ε + ε 4+ x K 4 < ε 4 k! k K x k x k! k k! k x + x K ε 4+ x K K K + x K K! k K + ε 4 < ε 4 k! + ε 4 < ε 4 + ε 4 ε k x k k! < ε, x k k! exp x k, ε > 0 N N > N < ε k0 k0, exp x x k k! + x k k! exp x < ε + ε ε k0 k0 3 3 X, Y C, p, q C, f : X Y ε > 0 δ > 0 0 < x p < δ x X fx q < ε, f p q, lim fx q 3 p C, r > 0 Bp ; r {x C x p < r}, lim fx q, ε > 0 δ > 0 x Bp ; δ X x p fx Bq ; ε 33 X, Y C, p, q C, f : X Y, lim fx q, p { },,, N, p X lim f q lim fx q { },, N, p X, p ε > 0 δ > 0 0 < x p < δ x X fx q < ε, N > N p < δ > N 0 < p < δ, X f q < ε, lim f q lim fx q, ρ > 0,, 0 < p < f q ρ X { },, p, N, p X, lim f q p { },,, N, p X lim f q lim f q 0

14 34 f, g : X Y, lim fx, lim gx, 4 lim gx 0 lim fx + gx lim fx + lim gx 3 lim fxgx lim fx lim gx lim cfx c lim fx 4 lim fx gx lim fx lim gx ε, lim fx, lim gx b δ, δ > 0 0 < x p < δ x X fx < ε, 0 < x p < δ x X gx b < ε δ, δ δ, 0 < x p < δ x X fx + gx + b fx + gx b < ε + ε ε lim fx + gx + b δ > 0 0 < x p < δ x X fx < ε + c, 0 < x p < δ x X cfx c c fx < ε c + c < ε, lim cfx c 3 δ > 0 0 < x p < δ x X fx <, δ, δ 3 > 0 0 < x p < δ x X fx < + b, 0 < x p < δ 3 x X ε gx b < + δ, δ, δ 3 δ, 0 < x p < δ x X fx fx + fx + < + fxgx b fxgx b + bfx fx gx b + b fx < b + b < ε + ε ε 4 b 0 δ > 0 0 < x p < δ x X gx b < b δ > 0 0 < x p < δ x X gx b < ε b δ, δ δ, 0 < x p < δ x X b b gx + gx < b b + gx gx > gx b gx b b gx < ε b b ε lim gx b, 3 35 f : X Y, p X, lim fx fp, f p p X f, f 34, 36 f, g : X Y p X, f + g, cf, fg : X Y p X, x X gx 0, f g : X Y p X,,,, 0 37 f : X Y, g : Y Z lim fx, g lim fx, lim gfx g lim fx, f p X, g fp, g f p, lim fx q ε > 0, g q, δ > 0 y q < δ y Y gy gq < ε lim fx q ρ > 0 0 < x p < ρ x X fx q < δ 0 < x p < ρ x X gfx gq < ε lim gfx q 38 f : X R [, b] X, f fb, f fb d, fc d c, b p, q [, b] y fp fq, y fp f p+q ε ε ε p+q f fq, up, q up, q 3p+q 4, up, q p+3q 4

15 , up, q p+q q p 4, y f up, q q p 4 f up, q + q p 4 x 0, x,, x, x 0, x b, x +b, x 0, x,, x 3 i 0,,, x i+ x i b i i,, y f x i b i f xi + b i x u x b, x + b x, x x b, y f x b f x + b, x+k x k x +i+ x +i k b < b +i b +i vi, lim x c i0 i0 i0, x 0, x,, [, b] 4 c [, b] fc y y f x b y f x + b 0, x b, x + b c 33 4 y fc 0, fc y 39 f : X R M, x X fx M fp M p X, M f, m, x X fx M fp M p X, M f 30 [, b] f : [, b] R S {y R fx y x [, b] }, f > [, b] v, [, b] { },, { k } k,, lim p k p, f lim f p k fp f k > k k, {f k } k,,, S, iii S M, l fb l > M l b l [, b] v, [, b] {b l } l,, {b lk } k,, lim b l p k q, f lim fb l p k fq M k M l k < fb lk M, {fb lk } k,, M M fq q X, M f f 3 X f : X R x < y fx fy, f, x < y fx < fy, f, f x < y fx fy, f, x < y fx > fy, f 3 [, b] f : [, b] R f,, f [f, fb], [fb, f] b f f fb f < fb f, x, b f < fx < fb, fx < f fx < p < f p, [, x] fc p c, x, fx < p < f < fb, [x, b] fd p d x, b fc fd p, c < x < d, f fx > fb,, x, b f < fx < fb

16 x < y b, f < fy, [, y] fx < fy f f < fb f f, f, f 33 I I, b, [, b],, b], [, b,, b,, b],,, [,, R, f : I R f, c, d I, c < d 3, f [c, d], f,, x, y I, x < y, c x p, d y q, f [p, q], [c, d] [p, q] f [p, q], x, y [p, q] fx < fy, f 34 I f : I J f : J I f 33 f, f,, p J, q f p [q r, q] I r > 0, f, [q r, q] [fq r, p], 0 < ε < r, f fq ε, p] q ε, q], δ p fq ε p δ < x p q ε < f x q [q, q + r] I r > 0, δ fq + ε p p x < p + δ q f x < q + ε, f p 35 X C, X D, D X p X, D p 36 X, Y C, f, g : X Y D X, x D fx gx f g p X, D { },, p,, f g f, g fp lim f lim g gp, f g 4 5 exp : C C exp z 4 x R exp x x, y R, x < y exp x < exp y 3 exp : C C 0 4 y, exp x y x x R exp x 0 z!, 6, x! R 6 exp x exp x + x exp x 0 exp x exp x expx + x exp 0 exp x 0 exp x x < y, 6 exp y exp x exp yexp x exp yexp x expy x 0 + y x! > exp x < exp y 3 y x!

17 3 z, p C, z p < exp z exp p expp+z p exp p exp pexpz p exp p exp p expz p exp p z p! exp p z p z p! < exp p z p! exp p z p exp z p < exp z exp p < exp p exp z p, z p exp z exp p 4 exp + +! > exp exp exp > exp exp < y > 0 m < y < m,, exp m < y < exp exp x y x m,, 3 [ m, ],, 3, 4, x exp x R 0, exp : R 0,, exp, exp e 4 exp : R 0,, log : 0, R 43 f : X Y X, Y R f : Y X, f, f : X Y X, Y R f : Y X, f x, y Y, x < y f y f x f y ff y ff x x x < y f x < f y f 44 logexp x x, explog y y x R, y 0, log : 0, R log 0 3 x, y 0,, logxy log x + log y logexp x x, explog y y log exp exp 0 log logexp 0 0 exp 43 log 3 z log x + log y exp z explog x + log y explog xexplog y xy logxy logexp z z log x + log y 45, exp : R 0, exp x expx log 46 exp x + y exp xexp y, exp exp x y exp xy x, exp x x 3 0 < <, exp, >, exp 4 exp : R 0,, log : 0, R log x log x log, exp exp x + y expx + y log expx log + y log expx log expy log exp xexp y exp exp x y expy logexp x expy logexpx log expyx log exp xy exp explog,, exp exp exp exp 0 exp 0, y x exp xexp x exp 0 exp x exp x exp exp, exp x x p q p, q exp x q exp qx exp p p, x p q X q p exp x x 4

18 3 x < y 0 < <, log < 0 x log > y log, exp exp x expx log > expy log exp y >, log > 0 x log < y log exp x expx log < expy log exp y 4 exp log exp y 0, exp log y exp log explog y y, x R log exp x logexpx log log log y log x log log x log exp x R exp x, x+y x y, x y xy 47 log : 0, R 48, b, x, y, z log xy log x+log y, log b z z log b log xy log x + log y log 44 3 log b z logexp b z logexpz log b log z log b log z log b log z log b log 49 f : R R, x, y R fx + y fx + fy f R, x R fx fx x, h R, fx+h fx fx+fh fx fh f+fh f f+h f, f h 0, fx+h fx 0 f x, x fx fx x f f f f0 f0 + 0 f0 + f0 f0 0 f + f f + f0 0 f f f f, f f x x p q p, q g : R R gx fx, qfx fqx fp fp fx fx gx x R lim x { },, f, g, fx f lim lim f lim g g lim gx 40 f : R R, x, y R fx + y fxfy f R, fb 0 b R, x R fx expx log f fc 0 c R, x R fx fx c + c fx cfc 0, fb 0 b R, x R fx 0 fx f x + x f x 0, x R fx > 0, g : R R gx log fx, 37 g R, 44 3, g 49 x R log fx gx gx x log f fx expx log f 4 f : 0, R, x, y R fxy fx + fy f R, x R fx fexp log x g : R R gx fexp x, 37 g R, 6 3, g 49 x R fexp x gx gx fexp x fx fexp log x 5 fx fp fp+h fp 5 X, Y R, p X f : X Y, lim x p lim h 0 h, f p f p, f p 5

19 5, p p r, p + r X r > 0 X,, 53 f : X Y X, f, x X f x f, f : X R 54 exp x e x e x expx+h exp x h exp x exp h h exp x h exp x h! exp x h! exp x h expx+h exp x h exp x exp x h exp xexp h exp x h exp x h! h <! e exp x h, h 0 expx+h exp x h exp x x x x+h x h x k h k x k k h k h k x k h 0 x+h x h x k 55 f : X Y p X, 0 < x p < c fx fp < K x p c, K, f p fx fp lim x p f p c > 0, p c, p+c X 0 < x p < c fx fp x p f p < 0 < x p < c fx fp f px p < x p, fx fp fx fp f px p + f px p fx fp f px p + f px p < x p + f p x p + f p x p, A + f p fx fp x p 56 f : X Y p X lim x p 0 f : X Y p X, lim x p lim fx fp x p f p fx fp lim x p f p 0 lim lim fx fp x p fx fp f px p fx fp x p fx fp x p 0 lim x p fx fp x p fx fp + lim x p +, lim x p fx fp x p, lim x p 0, f p 57 f, g : X R p X, 4, x X gx 0 f + g p f p + g p rf p rf p r R 3 fg p f pgp + fpg p 4 f g p f pgp fpg p gp lim lim gx gp x p f+gx f+gp x p f p + g p fx+gx fp+gp lim x p lim fx fp x p + gx gp x p rfx rfp rfx rfp lim x p lim x p lim r fx fp fx fp x p r lim x p rf p 3 55 g p lim gx gp lim fx fp x p gx + fp gx gp x p lim fgx fgp x p lim fx fp x p + lim fxgx fpgp x p fx fp gx gp lim x p lim gx + fp lim x p f pgp + fpg p 6

20 4 lim g x g p x p g p g p gp lim gx gp x p lim, 3 gx+gp gxgpx p f g gx gp x p lim p f g p f p gx gp lim g p + fp g p gp g p f p gp fp g p gp f pgp fpg p gp 58 f : X Y, g : Y Z p X, fp Y g f : X Z p, g f p g fpf p φx fx fp f px p, ψy gy gfp g fpy fp, y fx g fx g fp gfx gfp g fpfx fp + ψfx g fpf px p + g fpφx + ψfx, g fx g fp x p g fpf p + g fp φx x p + ψfx x p 55 0 < x p < c fx fp < K x p c, K ψy y fp 0, ε > 0, 0 < y fp < δ ψy y fp < ε K δ lim y fp, y fp < δ ψy ε K y fp δ c δ K, 0 < x p < δ fx fp < K x p < Kδ δ ψfx ε K fx fp < ε x p, ε > 0, 0 < x p < δ ψfx x p < ε δ, lim x p 0 ψfx φx g fx g fp, lim x p 0, lim x p g fpf p 59 X, Y f : X Y f, p X, f p 0 f fp, f fp f p f,, x p fx fp lim fx fp x p f x p p 0 lim fx fp f p, ε > 0 0 < x p < δ x p fx fp f p < ε δ 34 f fp δ > 0, 0 < y fp < δ f y p < δ, 0 < y fp < δ f y f fp y fp f f p < ε, y f fp y fp f p lim y fp 6 6 X, Y R, f : X Y, p X r, x p r, p+r X fx fp, f p, fp f, r, x p r, p + r X fx fp, f p, fp f f f, f f 6 f :, b R p, b,, f p 0 f p r, r < p, b p x p r, p + r fx fp p r, b p r, { },,, {b },,, p r, p+r, f, fb fp f fp 0, fb fp p f p lim fb fp r f fp p p f fp r 0, p 33 4 f p 0 f p lim fb fp b p 4, f p 0 f p 0 f p 7

21 63 Rolle f : [, b] R,, b, f fb f c 0 c, b f f, fp, fq p, q [, b], fq f fb fp, fp > f fb, p, b f p 6 f p 0, c p fq < f fb, q, b f q 6 f q 0, c q 3 fq f fb fp, f, c, b f c 0 64 Cuchy f, g : [, b] R,, b gb g, x, b f x g x 0 fb f gb g f c g c c, b F : [, b] R F x fb fgx gb gfx, F 63, F c 0 c, b F x fb fg x gb f x, fb fg c gb gf c, g c 0, gb g 0 f c 0, g c 0, fb f gb g f c g c, gx x 65 f : [, b] R,, b, fb f f cb c, b 66 f : [, b] R,, b x, f x 0, f x, b], [, x] fx f f cx c, x f c 0 fx f f f 67 f : [, b] R,, b x x, b, f x 0 f, f x > 0 f x, b, f x 0 f, f x < 0 f x < y b [x, y] fy fx f cy x c x, y, p, b, f p 0 fx fy, f p > 0 fx > fy 68 X, Y R, p, q R, f : X Y ε > 0, p < x < p + δ p δ < x < p x X fx q < ε δ > 0, q lim fx lim fx x p x p ε > 0, x > N x < N x X fx q < ε N R, q lim fx lim fx x x 33 8

22 69 X, Y R, p, q R, f : X Y lim fx q lim fx q, p { },,, x p x p N, > p < p X lim f q lim fx q lim fx q, { },, lim x x lim lim f q 60 X, Y R, p R, f : X Y M R, 0 < x p < δ x X fx > M fx < M δ > 0 lim fx lim fx M R, p < x < p + δ x X fx > M fx < M δ > 0 lim fx lim fx x p x p 3 M R, p δ < x < p x X fx > M fx < M δ > 0 lim fx lim fx x p x p 4 M R, x > N x X fx > M fx < M N R lim x fx lim x fx 5 M R, x < N x X fx > M fx < M N R lim fx lim fx x x 6 p R, r > 0, f, g, g, g 0 f, g : p, p + r R f, g : p r, p R lim fx lim gx 0 lim fx lim gx 0, x p x p x p x p f lim x x p g x lim f x fx x p g x lim x p gx lim fx f x p gx lim x x p g x lim f x x p g x f, g : p, R f, g :, p R lim gx 0 lim, lim x f x g x lim x x fx lim x f x g x lim x fx gx lim x x fx lim gx 0 x fx gx lim x f lim x x g x 3 f, g : p, p + r R f, g : p r, p R lim fx lim gx lim fx lim gx x p x p x p x p f x x p, lim g x lim f x x p lim g x x p f x g x lim x p 4 f, g : p, R f, g :, p R lim, lim x f lim x x g x x f x g x lim x x fx gx lim x p fx gx lim x p fx lim gx lim x f x g x lim x fx gx lim x x fx f x g x f x g x lim gx x fx gx lim x f lim x p g x A, ε > 0 p < x < p + δ f x g x A < ε δ > 0 fx p < x < p + r gx p < x < p + r f, ḡ : [p, p + r R fx, ḡx, 0 x p 0 x p f, ḡ x p, p + r, [p, x] 64, fx fx f0 gx ḡx ḡ0 f c ḡ c f c g c c p, x p < c < x p < x < p + δ fx gx A f c g c A < ε, lim gx lim f x g x f, g : p r, p R x p fx x p r > 0 r > p, φ, ψ : 0, r R φx f x, ψx g x, lim φx lim f x 0 x 0 x lim fx 0, lim ψx lim g x x 0 x 0 x lim gx 0, x φ x x f x, f x g x 9

23 ψ x x g x lim φ x x 0 ψ x lim f x g x lim fx x gx x 0 f x g x lim f x x g x φx lim x 0 ψx lim φ x x 0 ψ x φx, lim x 0 ψx lim, x 0 f 3 A lim x x p g x, ε > 0 0 < δ < r p < y < p + δ f y g y A < ε f y g y A <, p < q < p+δ q, f q g q A < f q < A +, 0 < δ < q p p < x < p+δ gx > ε fq + gq A + g q p < x < p+δ [x, q] 64, fx fq gx gq f c g c c x, q, x < c < q < p + δ f c g c A < ε gx > ε fq + gq A + fq gx + gq gx A + < ε fq gx gq f c gx g c fq gx + gq gx f c g c < fq gx + gq gx A + < ε fx fq gx gq fx gx fq gx gq fx gx gx fq gx gq f c gx g c fx gx f c g c fq gx gq f c gx g c, fx gx A fx gx f c g c + f c g c A fq gx gq f c gx g c + f c g c A < ε + ε ε 4 r > 0 r > p, φ, ψ : 0, r R φx f x, ψx g x, 3 6 I f : I R p, q I p < q t 0,, tfp + tfq f tp + tq, f I, tfp + tfq f tp + tq, f I, f p, q I p < q t 0,, tfp+tfq > f tp+tq, f I, tfp+tfq > f tp + tq, f I 63 I f : I R I, f I, f, f, f I f I p, q I p < q, t 0,, x t tp + tq, tfp + tfq fx t fx t fp tfq fp, fx t fq tfq fp x t p tq p > 0, x t q tq p < 0 fx t fp x t p fq fp q p fx t fq x t q fx lim t fp fx fp t 0 x t p lim x p x t p f fx p, lim t fq fx fq t 0 x t q lim x q x t q f q t 0 f p fq fp q p f q, f p, q I p < q, t 0,, [p, tp + tq], [ tp + tq, q], f tp+tq fp tq pf c, fq f tp+tq tq pf d c p, tp+tq, d tp + tq, q, tfp + tfq f tp + tq tf tp + tq fp + tfq f tp + tq ttq pf d f c c < d, f tfp + tfq f tp + tq, tfp + tfq > f tp + tq 64 I f : I R I f I, I p x I fx f px p + fp f I, p I p x I t 0, x t tx + tp, tfx + tfp fx t tfx fp fx t fp 0

24 tx p x t p, x < p fx fp x p fx t fp x t p, x > p fx fp x p fx t fp x t p t 0 x t p x < p fx fp f p, x > p fx fp x p f p x p, fx fp f px p I p x I fx f px p+fp p, q p < q I, fq f pq p + fp fq fp q p f p, fp f qp q + fq f q, f p f q, f 63 f fq fp q p I x p 7 7 f : X R f : X R 0 f 0 f f r : X R 0 r <, f : X R X, f f f f, f, f, f 7 x α αα α + x α e x e x log x! x si x si x + π cos x cos x + π α 0, α α αα α +! 73 f, g : X R, fg k0 f k g k k, 57 3 fg r, 57,, 3 r i + r i r+ i fg r r k0 r f k+ g r k + f k g r k+ k k0 r f k+ g r k + k r+ r r r f i g r i+ + f i g r i+ i i i i0 r r r f i g r i+ + f r+ g + fg r+ + i i i r r r fg r+ + + f i g r i+ + f r+ g i i i r r r fg f i g r i+ + f r+ g i i i r r + r+ r + fg r+ + f i g r i+ + f r+ g i i i r f i g r i+ i i0 r k0 r k0 f i g r i+ r k f k g r k r f k g r k+ k r +

25 74 f : [, b] R, b λ, p, x [, b], 0 < θ < fx fp + x p K k f k p k! x p k + f p + θx p θ λ x p! λ λ fx fp x p λ fx fp + k F : [, b] R F t fx ft k f k p k! x p k K, f k p x p k x p λ + K k! λ k f k t k! x t k K x t λ λ, F x F p 0 x < p, F x, p, x > p, F p, x 63, F c 0 c p x θ c p x p 0 < θ <, c p + θx p F f t! t x t + Kx t λ t < x f t! x t Kt x λ t > x, x < p c > x K f c! c x λ f p+θx p! θ λ p x λ fx fp + k f k p k! x p k + f p + θx p! θ λ x p λ x > p c < x K f c! x c λ f p+θx p! θ λ x p λ fx fp + k f k p k! x p k + f p + θx p! θ λ x p λ f p+θx p θ λ x p! λ, R x 75 f : [, b] R, b, f p, b ε > 0, δ > 0 0 < x p < δ fx fp f k t x t k < ε x p k! k f p, δ > 0 0 < t p < δ f t f p < ε!, 74, λ fx fp+ f k p k! x p k + f p+θx p! x p 0 < θ < k 0 < x p < δ 0 < p + θx p p θ x p < δ fx fp f k t k! x t k! f p + θx p f p x p < ε x p 76 f :, b R, f p, b f p f p f p 0, f p 0, f p, f p > 0 f p, f p < 0 f p k

26 75 ε f p! f p! x p, δ > 0 0 < x p < δ f p! x p < fx fp < f p x p +! f p! x p p < x < p + δ, 0 < x p < δ f p f p f!! x p p < fx fp < + f p!! x p, p δ < x < p f p + f p f!! x p p < fx fp < f p!! x p f p > 0, 0 < x p < δ fx fp > f p! x p > 0, f p f p < 0, 0 < x p < δ fx fp < f p! x p < 0, f p, f p > 0 p < x < p + δ fx fp > f p! x p > 0, p δ < x < p fx fp < f p! x p < 0, f p f p < 0, f p 77 f : [, b] R, b p, x [, b], N, λ > 0, 0 < θ < fx fp + k f k p k!, lim R x lim fx fp + k fp + x p k + f p + θ x p θ λ x p! λ f p+θ x p θ λ x p! λ 0, fx fp + k f k p x p k k! f k p k! x p k R x k 78 α R x < f k p k! x p k f p 0 α x x 0, lim α x, lim α x 0 + lim α + x x <, 79 e x, si x, cos x, + x α α R, log + x 0 fx e x, x R, λ f k 0, R x x! eθx, 7 lim R x 0, e x + x! fx si x, x R, λ f k 0 0, f k+ 0 k, R x x! si θ x + π, 7 lim R x 0, si x x+ +! 0 3

27 3 fx cos x, x R, λ f k 0 k, f k+ 0 0, R x x! cos θ x + π, 7 lim R x 0, cos x x! 4 fx + x α, f k 0 k! α k x + θ x α, λ R x α α θ + θ x α x > i < x < 0, 0 < θ + θ x α θ +θ x + θ x α < + θ x α α 0 < θ + θ x α < + θ x α λ R x α α θ + θ x α x < α x α x 0, λ R x α, 78 lim R x 0 α < 0 < θ < + θ x 0 < θ + θ x α < + θ x α < + x α λ R x α α α θ + θ x α x < αx + x α x, 78 lim R x 0 ii 0 < x <, > α 0 < + θ x α < λ R x α x + θ x α < α x, 78 lim R x 0, x < lim R x 0, + x α f k 0 0 α x 5 fx log + x, f0 0, k! k k k, λ R x x +θ x, λ R x + θ x +θ x i < x < 0, 0 < θ +θ x < +θ < x +x λ R x θ x +θ x < x +x, lim R x 0 ii 0 x, 0 x +θ x < λ R x, lim R x 0, < x lim R x 0, log + x x x +θ x < 8 8 x x 0 8 { x 0 } 0,,, x < x 0 x C, x K > 0, 0,,, x 0 < K, x x 0 x x 0 < K x x 0, x x 0 < K x x 0 0 x 0 83 x x, M {r x < r 0, x, M, x } {0} x 0

28 8, x R 0 R, x < R x, x > R 0 x 0 84 φ, <, lim φ 0 k, lim k 0 <, α log α > 0 e α α j j j! > αk+ k+ k+! j0, k+! 0 lim α k+ k 0 85 φ 0, 0 x R, 0 k < k+!k k+! α k+ k+ α k+ x φx 0 φx S 0 R, S, φ 0, φ x < R, x < r < R r, x r < 84 lim φ x r 0, > N φ x r < N > N φx φ x r r < r r, φx 0 S < R S R φ > N φ > N x < S, > N x < φx, φx, x R < S R S R S x R, f : R, R R fx x, x < R f fx+h fx x lim h 0 h x k k kk k k, x+h x h x h k h k x k h k h k x k h k h k x k h h + x k k k x < R, h < R x h, 85 h + x, fx+h fx h x x+h x h x x+h x h x 0 0 h h + x h h + x lim x fx+h fx h x R, fx x f : R, R R, f 0! 86 f k x k k! + k+ 0 k + x k f k 0 k k! 88 x x R 0 lim +, lim + R lim, lim R 0 5

29 lim + r x C, lim +x+ x lim + x r x, r > 0, x < r x, r 0 x x 0 r > 0 x > r, < c < r x c N > N k > c > N x > c N N x N, x > +x + x N x + k 0 r > 0 N x N c N N x + c N x N c c k N x N+ 0 0 x r r 0 x 0 0 r x C, lim x lim x r x, 3 r > lim 0, x < r 0 x, r 0 x x r > 0 x > r, < c < r x c N > N x > c k > N x > c, x > N x + k c N x + cn+ c ck N 0 0 N+ x r > 0 x r r 0 x 0 89 { } 0,, 0 f :, C fx x, lim x x fx 0 8, x, x ε > 0 0 N N < ε 0,,, N M, ε N M+ε < x < x N < ε M+ε < ε M x fx x N x + 0 x x x N x + xx M x N + ε < ε limx fx x 80 x x R, f : R, R C fz z R, lim fx R x R 0 s R x R, R, fx s x R x R x R 0 j0 R x j R sk k R k 0 j k j k j j0 R x R 0 0 j k 0 0 j0 k j0 j x j R j R x R j k s k s j x R s x k R s x j R R x j R s x j R s x j j R j0 j0 j0 j0 j0 j0 s s j x j R fx s R x R s s j x j R lim s s j 0 gy s s j y j j0 j0 j j0 89 limfx s lim R x x R x R R s s j x j R R lim y gy 0 y j0 9 9 X, Y C f : X Y, f ε > 0, δ > 0 x y < δ fx fy < ε 6

30 9 f : [, b] C, ε > 0 ; δ > 0 x y < δ fx fy ε x, y [, b] δ x y < fx fy ε x, y [, b] {x },,, {y },, v, {x k } k,,, {y k } k,, 0 < < < k < p lim x k, q lim y k x k, y k [, b] k k 4 p, q [, b] x k y k < k k, 4 p q 0 p q f 33, lim fx k lim fy k fp k k, fx k fy k ε, k 0 ε > 0 f : X R, M, x X fx M, f, X S, {y y fx x S } sup x S fx, if x S fx 93 0 b { i } i0,,, [, b], 0,,,,,, i, ξ i [ i, i ] ξ {ξ i } i,,, { i } i0,,,, { j } j0,,,m [, b] 0,,,, 0,,, m [, b], { 0,,, } { 0,,, m}, 3 f : [, b] R [, b] { i } i0,,, ξ {ξ i } i,,, Sf,, ξ i fξ i i i,, ξ f f, [, b] { i } i0,,, m i if x [i, i] fx, M i sup x [i, i] fx, Sf, M i i i, Sf, m i i i f, i 4 f : [, b] R, R, f, R b fxdx ε > 0, δ > 0 < δ [, b] ξ Sf,, ξ R < ε 94 f : [, b] R [, b], ξ Sf,, ξ b fxdx Sf,, ξ b fxdx + {i } i0,,,, ξ {ξ i } i,,,, [ k, k ] f ξ i i k, ξ k [ k, k ] fξ k k k fξ i i i i k Sf,, ξ b fxdx +, f [ k, k ] [, b] [ k, k ] 95 f : [, b] R, { i } i0,,,, { j } j0,,,k [, b] Sf, Sf, Sf, Sf, M sup x [,b] fx, m if x [,b] fx i i Sf, Sf, k M m, Sf, Sf, k M m 3 ξ Sf, Sf,, ξ Sf, 7

31 , i i s i s i m i if x [i, i ] fx, M i sup x [i, i] fx, m j if x [ j, j ] fx, M j s i sup x [ j, j ] fx, [ i, i ] [ j, j ] m i js i + m s i +, m s i +,, m s i, M i M s i +, M s i +,, M s i s i m i i i m i j j s i m j j j Sf, m i i i i s i js i + M i j j s i i js i + s i js i + js i + m j j j l js i + m j j j Sf, M i i i M j j j Sf, M i i i i s i i js i + M j j j l M j j j Sf, Sf, Sf, m j M j j {b j } j0,,,k+, m j if x [b j,b j ] fx, M j sup x [b j,b j ] fx, i b si s i i, i, i, s i m js i + j b j b j s i js i + m i b j b j s i js i + M m, i, i s i s i js i + m j b j b j m i i i Mb j b j mb j b j M m i i js i + m j b j b j m i i i 0, k i Sf, Sf, p+ m j b j b j m i i i i s i m j b j b j m i i i k M m i js i + Sf, Sf, k M m Sf, Sf, k M m 3 ξ {ξ i } i,,, m i fξ i M i 96 f : [, b] R 95 {x x Sf, [, b] }, {x x Sf, [, b] }, b fxdx, b fxdx, f, 97 f : [, b] R ε > 0, δ > 0 < δ [, b] b fxdx Sf, < ε Sf, b fxdx < ε L b fxdx, U b fxdx, ε > 0, L ε < Sf,, Sf, < U + ε [, b], 0 95 L ε < Sf, 0 Sf, 0 < U + ε M sup x [,b] fx, m if x [,b] fx, 0 {c k } k0,,,p [, b] { i } i0,,, 95 Sf, 0 Sf, p M m, Sf, Sf, 0 p M m < ε p M m L Sf,, Sf, U < ε + p M m < ε 98 f : [, b] R, b fxdx b fxdx f, b fxdx b fxdx b fxdx f b fxdx b fxdx b fxdx b fxdx b fxdx, R 97, ε > 0, δ > 0 < δ [, b] R Sf, < ε Sf, R < ε, < δ, ξ 95 3 R ε < Sf, < Sf,, ξ < Sf, < R + ε Sf,, ξ R < ε f, b fxdx R 8

32 f, L b fxdx, U b fxdx 97, ε > 0, δ > 0 < δ [, b] L Sf, < ε Sf, U < ε { i } i0,,, < δ [, b] m i if x [i, i ] fx, M i sup x [i, i] fx, i,,, fξ i m i < ε b, M i fζ i < ε b ξ i, ζ i [ i, i ] ξ {ξ i } i,,,, ζ {ζ i } i,,,, Sf,, ξ Sf, fξ i m i i i < ε b i i ε, Sf, Sf,, ζ i i M i fξ i i i < i i ε b i i ε, < δ 0 L Sf, < ε ε < Sf, Sf,, ξ 0 ε < L Sf,, ξ < ε, 0 Sf, U < ε ε < Sf,, ζ Sf, 0 ε < Sf,, ζ U < ε ε < L U < ε, ε L U 99 f : [, b] R, ε > 0, Sf, Sf, < ε [, b] f 98 b fxdx b fxdx 97 ε > 0, Sf, Sf, < ε [, b] ε > 0, Sf, Sf, < ε [, b], Sf, b fxdx b fxdx Sf, 0 b fxdx b fxdx Sf, Sf, < ε ε b fxdx b fxdx 98 f 90,, f : [, b] R 9, ε > 0, δ > 0 x y < δ fx fy < ε b < δ [, b] { i} i0,,, m i if x [i, i] fx, M i sup x [i, i ] fx, M i fp i, m i fq i p i, q i [ i, i ] M i m i < f ε b Sf, Sf, < ε 99 f : [, b] R [, b] { i } i0,,, m i if x [i, i] fx, M i sup x [i, i] fx, m i f i, M i f i Sf, Sf, f i f i i i f i f i fb f, ε > 0, i < ε fb f+ i [, b] Sf, Sf, < ε 99 f f 93 9 f : [, b] R b fxdx lim i b f + b i lim i b f + b i 9 f : [, b] R, fx C b fxdx lim i b C Cb k, f : [0, t] R fx x k t fxdx 0 x [k] xx x x + k,, x k x [k], x k, x k x [k] + k kj x [j] kj, x + [k+] x [k+] k + x [k], x,,,, [k+] 0 i [k] k+ + [k+] i k i [k] + k kj i [j] k+ + [k+] + k kj j+ + [j+] i i i i 9

33 t 0 i fxdx lim i k k +, k+ k+ i t t i k t k+ lim k+ i i k tk+ k+ lim k+ i i k k+ 0 f, g : X R, f g : X R f gx fx gx 0 f, g : X R sup x X f gx sup x X fx sup x X gx, if x X f gx if x X fx, if x X gx x X fx, gx f gx sup x X f gx, if x X fx, if x X gx f gx sup x X fx, sup x X gx sup x X f gx, if x X fx, if x X gx if x X f gx sup x X fx, sup x X gx < sup x X f gx, sup x X fx, sup x X gx < f gx 0 < sup x X f gx x 0 f gx 0 fx 0 gx 0, sup x X fx < fx 0 sup x X gx < gx 0, sup x X f gx sup x X fx sup x X gx 0 f, g : [, b] R, α, β R αf + βg : [, b] R, b αfx + βgxdx α b fxdx + β b gxdx f g, fg : [, b] R c > 0, x [, b] fx > c, f : [, b] R b fxdx A, b gxdx B ε > 0 δ > 0, [, b] < δ ε ε, ξ, Sf,, ξ A < α + β + Sg,, ξ B < α + β + < δ, Sαf + βg,, ξ αa + βb αsf,, ξ A + βsg,, ξ B α Sf,, ξ A + β Sg,, ξ B < ε α + β α + β + < ε αf +βg, b αfx+βgxdx αa + βb x [, b] fx, gx C C [, b], S Sf g, Sf g, Sf, Sf, + Sg, Sg,, Sfg, Sfg, CSf, Sf, + Sg, Sg,, f, S f, Sf, Sf, fx > c > 0 c, ε > 0, Sf, Sf, < Sg, < ε +C ε +C Sg, Sf g, Sf g, < ε Sfg, Sfg, < ε 99 f g, fg Sf, Sf, < c ε S f, S f, < ε 99 f { i } i0,,,, M i sup x [i, i ] fx, M i sup x [i, i ] gx, 0 sup x [i, i ]f gx M i M i, λ, λ,, λ k, µ, µ,, µ k {λ, λ,, λ k } {µ, µ,, µ k } {,,, } sup x [λ s, λ s ] f gx M λs, sup x [µ s, µ s ] f gx M µ s m i if x [i, i ] fx, m i if x [i, i ] gx, p i if x [i, i ]f gx p i m i, m i Sf g, Sf g, k M λs λs λs + k M µ s µs µs p i i i s k M λs p λs λs λs + k M µ s p µs µs µs k M λs m λs λs λs + k M µ s s s m µ s µs µs Sf, Sf, + Sg, Sg, s s i s 30

34 x, y [ i, i ] fxgx fygy fx fygx + fygx gy fx fy gx + fy gx gy C fx fy + gx gy CM i m i + M i m i q i if x [i, i]fgx, Q i sup x [i, i ]fgx, Q i q i CM i m i + M i m i Sfg, Sfg, CSf, Sf, + Sg, Sg, c > 0, x [, b] fx > c x, y [ i, i ] fx fy fx fy fx fy < c M i m i S f, S f, c Sf, Sf, 03 f, g : [, b] R x [, b] fx gx, b fxdx b gxdx g f c [, b], fc < gc b fxdx < b gxdx E, x < y b E [x, y] [, b] x E fx gx b fxdx b gxdx [, b] ξ Sf,, ξ Sg,, ξ, b fxdx gxdx b g f c [, b], fc < gc h g f h c hc > 0 δ > 0 x c δ, c + δ [, b] hx hc < hc < δ 6 [, b] { i} i0,,, ξ {ξ i } i,,, c δ < c, [ k, k ] c δ, c δ 3, [ l, l ] c δ 3, c k, l ξ k, ξ k+,, ξ l c δ, c k i l hξ i > hc, l k > δ 3 Sh,, ξ > l hc i i hc l k > δhc 6 0 ik b gxdx b fxdx b δhc hxdx 6 > 0 c δ < b, [ k, k ], + δ 3, [ l, l ] + δ 3, + δ k, l, b gxdx b fxdx > 0 x E fx 0 b fxdx 0 R b fxdx, ε > 0, δ > 0 < δ 6 [, b] { i} i0,,, ξ Sf,, ξ R < ε ξ i E i,,, ξ {ξ i } i,,,, Sf,, ξ 0 R < ε ε > 0 R 0 04 f : [, b] R f x fx f : [, b] R, b fxdx b fx dx f f f 0 f, x [, b] fx fx fx 03 b fxdx b fx dx c b [, c] { i } i0,,,, [c, b] {b i } i0,,,m, ξ {ξ i } i,,,, ξ {ξ i } i,,,m, c i i 0 i, c i b i i m + [, b] {c i } i0,,,m+ +, ζ i ξ i i, ζ i ξ i + i m + + {ζ i } i,,,m+ ξ + ξ, Sf, +, ξ + ξ Sf,, ξ + Sf,, ξ 05 c [, b] f : [, b] R f [, c] [c, b] f [, b], b fxdx c fxdx+ b c fxdx f [, b] f [, c] [c, b] 94, f [, c], [c, b], M, x [, b] fx M ε > 0 0 < δ ε 4M < δ 3

35 [, c], < δ [c, b],, ζ, η Sf,, ζ c fxdx < ε 4 Sf,, η b c fxdx < ε 4 < δ [, b] { i } i0,,, ξ {ξ i } i,,, {b i } i0,,,l, {c i } i0,,, l+ l c l b i i i < l, b l c 0 c, c i i+l i > 0, ζ {ζ i } i,,,l, η {η i } i,,, l+ ζ i ξ i i < l, ζ l η c, η i ξ i+l i >,, [, c], [c, b], < δ, ζ, η, Sf,, ξ Sf,, ζ Sf,, η fξ l fc l l, Sf,, ξ c fxdx b c fxdx fξ l fc l l + Sf,, ζ c fxdx + Sf,, η b c fxdx < Mδ + ε < ε,, { + c i} i0,,,, ξ { + c i} i,,,, { c + b c i} i0,,,, ξ { c + b c i}, i,,,, [, c], [c, b], ξ, ξ, S Sf,, ξ, S Sf,, ξ, {S },,, {S },, R b fxdx, ε > 0, δ > 0 < δ [, b], ξ Sf,, ξ R < ε b, m > δ c < δ, b c < δ + m < δ Sf, + m, ξ +ξ m R < ε S S m Sf, + m, ξ +ξ m Sf, m + m, ξ m + ξ m Sf, + m, ξ + ξ m R + R Sf, m + m, ξ m + ξ m < ε S S m < ε {S },,, {S },, V vi, lim S R, lim S R ε > 0, δ > 0, > b δ S R, S R < ε [, c] < δ, ξ +, + < δ Sf,, ξ R Sf, +, ξ +ξ R + R Sf, +, ξ +ξ + Sf,, ξ R < ε, f [, c], c fxdx R f [c, b], b c fxdx R 06 f : [, b] R, b fxdx b fxdx fxdx, b b fxdx c fxdx + b fxdx, b, c c f [, b] c d b, 05 f [c, b] 05 f [c, d] f [, b] 07 f : [, b] R, c < d b, f p c, d, f [, b] E, c < d b E c, d f : [, b] R,, b f, 06 94, x [, b] fx < M M [, b] {α i } i,, {β i } i,, i,, α i < β i x, y [α i, β i ] fx fy < M i α, β b x, y [α, β ] fx, fy < M fx fy < M α < α < < α i < β i < < β < β, s,,, i x, y [α s, β s ] fx fy < M s α i < u < v < β i u, v, 99, Sf, Sf, < Mv u i [u, v] { j } j0,,, u 0 < < < v, M j sup x [j, j] fx, m j if x [j, j ] fx M m, M m,, M m M j m j, M j m j v u M j m j j j Sf, Sf, < Mv u i M j m j < M i, α i j, β i j α i, β i α i < α i < β i < β i, x, y [α i, β i ] fx fy < M i 3

36 {α i } i,,, {β i } i,,, lim α i lim β i p [ lim α i, lim β i ] i i i i ε > 0 i i p α i, β i δ < p α, β p δ > 0 ε M x p < δ x p δ, p + δ [α i, β i ] fx fp < M i ε f p f :, b R, F :, b R F f, F f f : [, b] R, c [, b] F : [, b] R F x x ftdt c F f p, b F p, F p fp, f, F f 94, M, x [, b] fx M 06, 04, 03, 9 F x + h F x x+h ftdt x c c ftdt x+h ftdt x x+h ft dt x x+h Mdt h M h 0 F x + h F x, F x ε > 0 δ > 0 p δ, p + δ [, b] h < δ fp + h fp < ε F p+h F p h < δ h fp p+h h ftdt p c c ftdt p+h fpdt p h ft fpdt h ft fp dt h εdt ε lim h fp p+h p p+h p p+h p h 0 F p+h F p 3 f, F : [, b] R F, b x, F x fx b fxdx F b F Gx x ftdt G : [, b] R, x, b G x fx F x 66, G F, G 0 b ftdt Gb G F b F 4 f, g :, b R, b x, f, g :, b R p, q, b q p f xgxdx fqgq fpgp q p fxg xdx g :, b R, b x, g :, b R p, q, b g p, q M, m, f : [m, M] R q p fgtg tdt gq fxdx gp f g fg fg, p q 3 I p, q F, G : I R F x x p fgtg tdt, Gx gx ftdt gp, F x fgxg x G x 66 F G F p Gp 0 F G, F q Gq 5 f : [, b] R, φ : [, b] R x [, b] φx 0 b fxφxdx fc b φxdx c, b f, f M, m, x [, b] m fx M φx 0 mφx fxφx Mφx 03 m b φxdx b fxφxdx M b φxdx m b φxdx < b fxφxdx < M b φxdx m < b fxφxdx b < M fp M, fq m φxdx p, q [, b], p q fc b fxφxdx b φxdx c p < q 33

37 p < c < q b, p > q q < c < p b, c, b, b fxφxdx fc b φxdx Z {x x, b, φx 0}, φ, b C b fxφxdx M b φxdx, fc M c, b, c, x, b fx < M 03 x C fxφx Mφx, x Z C Z, b fxφxdx m b φxdx, fc m c, b, c C Z C Z 07, c < d b Z c, d C c, d, x Z fxφx φx 0 03 b fxφxdx b φxdx 0 C Z, c, b b fxφxdx fc b φxdx 6 f :, b R, f :, b R, x, p, b fx fp + i k fx fp + k f i p x p i + i! i! x p x t f tdt f i p i! x p i + x k! p x tk f k tdt 3 fx fp+ x p f tdt, k, k r,, 4 x p x tr f r tdt x p r x tr f r tdt r x pr f r p+ x r p x tr f r+ tdt, k r + 7 x R, f : R, R R fx x 0, x < R x 0 ftdt x+ + x+ R, F 86, F f F x 0 0 ftdt F x I, f : I R I [, b I, b], t I f [, t] [t, b], t b lim fxdx lim t b t t fxdx, f I, b fxdx b t fxdx lim t b fxdx b b fxdx lim fxdx t b t I [, I, b], t I f [, t] [t, b], t b lim fxdx lim t t t fxdx, f I, fxdx b fxdx t fxdx lim t fxdx b fxdx lim b fxdx t t 3 I, b < b, s, t, b f [s, t], c I c fxdx b fxdx, c b fxdx b fxdx c fxdx + b c fxdx 06 b fxdx c 34

38 f : [, b] R, b, f :, b R s, t, b s < t 3 t s f xdx ft fs, f lim fs f, s lim ft fb f [, b], b t b f xdx fb f 3 I, f : I [0,, I I [, b I, b], b {p },,, p < b p >, α b fxdx {α },, f I b fxdx lim α I [, I, b], {p },,, lim p lim p, α b fxdx lim α b fxdx {α },, f I fxdx lim α b {b },,, p < b, β b fxdx {β },, R lim α, ε > 0, > N α > R ε N {p },, {b },, b b N p N+ N > N b b N p N+ β b fxdx pn + fxdx + b p fxdx α N N + + > R ε {β },, R 69 f I b fxdx lim α lim b {b },, 4 I, f, g : I [0,, f I, g I x I fx gx f I fx I [, b I, b], lim x b gx lim fx x gx f I I [, I, b], lim x fx gx lim x fx gx f I I [, b, b b {b },, b < b, α b fxdx, β b gxdx {α },,, {β },, 06 03,, α β {β },, b gxdx {α },, {α },, 69 f b {b },, lim, f I, b], I, b fx lim x b gx C, < c < b c < x < b fx gx C < c < x < b fx < C + gx f [c, b 05 f I I, b], I [,, I, b] 5 f :, [0, fx x, g, h : [, ] R gx x, hx + x g, h :, R g x x, h x +x, x [0, fx g x, x, 0] fx h x g, h [0, ], [, 0] 4 f, 0], [0, p, q > 0, 0 xp x q dx p, q x x p x q [0, ] R, 0 < p < fx x p x q, gx x p f, g : 0, xp ] R g p [0, ] fx g lim x 0 gx lim x 0 xq 4 0 x p x q dx 0 < q < x p x q dx, p, q > 0, 0 xp x q dx 35

39 3 s > 0, 0 x s e x dx x 0, ] x s e x x s, x s xs s [0, ] 0 xs dx 4 0 xs e x dx x [,, fx x s e x, gx e x f, g : [, [0, N s, x fx gx xs e x x N e x 84 x lim x xn e 0 lim fx x gx 0 4 x s e x dx x s e x dx 0 3 m R m {x, x,, x m x, x,, x m R m } x x, x,, x m, y m y, y,, y m dx, y dx, y x j y j m m 3 u, u,, u m, v, v,, v m R, u j v j u j m vj, m m m u j vj u j v j u i vj u i u j v i v j i j i<j i<ju i vj u i u j v i v j + u jvi i<ju i v j u j v i 0 3 x, y, z R m, dx, z dx, y + dy, z x x, x,, x m, y y, y,, y m R m, x k y k dx, y m x j y j 3 x x, x,, x m, y y, y,, y m R m m m, x j yj dx, y x x, x,, x m, y y, y,, y m, z z, z,, z m, u j x j y j, v j y j z j m m m j,,,, u j + v j x j z j u j + v j u j + vj m m 78 u j + vj, x k y k m m x j y j m u j + v j m u j m x j y j,, m vj m u j v j 0 m 3 R 3 0, x j dx, 0 dx, y + dy, 0 dx, y + yj m m m m x j yj dx, y yj x j dy, x dx, y, 33 φ : [, b] R m m R m t [, b] φt φ t, φ t,, φ m t, φ j : [, b] R j,,, m, φ j 36

40 34 R m φ : [, b] R m [, b] { i } i0,,, sφ, dφ i, φ i i S φ {x x sφ, [, b] }, φ, S φ φ m φ, φ, φ φ 35 φ : [, b] R,, b, φ :, b R φ, M, x, b φ x M, φ V V Mb { i } i0,,, [, b], φ i φ i φ c i i i c i i, i, sφ, φ i φ i φ c i i i M i i Mb i S φ Mb 36 R m φ : [, b] R m, φ j φ j, [, b], sφ, sφ, [, b], sφ k, sφ, m sφ j, 3 φ, φ 4 φ φ j : [, b] R,, b, φ j :, b R φ { i } i0,,,, { i } i0,,,k, i 0,,, i s i s i s i 3 dφ i, φ i dφ l, φ l, i 0,,, ls i + sφ, sφ, { i } i0,,,, 3 φ k i φ k i dφ i, φ i m φ j i, i,,, sφ k, sφ, m sφ j, i i φ j i 3 φ L,, k,,, m [, b] sφ k, sφ, L S φk φ k φ, φ,, φ m, φ j L j, [, b] sφ, m sφ j, m L j S φ R m φ : [, b] R m, c [, b], φ : [, c] R m, φ : [c, b] R m φ t φt t [, c], φ t φt t [c, b] φ, φ φ, φ, φ, φ L, L, L, L L + L φ L, [, c], [c, b], + [, b], sφ,, sφ, sφ, + sφ, sφ, + L, S φ, S φ, φ φ, φ, φ L, L, L + L L φ φ L, L, [, b] { i } i0,,,, k c k k, [, c], [c, b] {b i } i0,,,k, {c i } i0,,, k+ 37

41 i 0 i k c i 0 b i, c i sφ, dφ c i k i, φ i i+k i k + i k i dφ i, φ i +dφ k, φ k + dφ k, φc + ik+ ik+ dφ i, φ i k i dφ i, φ i +dφc, φ k + dφ i, φ i sφ, + sφ, L + L, S φ L + L, φ, L, L L + L φ, φ, φ, L L + L R m φ : [, b] R m,, p, q [, b] p q φ p,q t φt φ p,q : [p, q] R m L φ p, q, p > q L φ p, q L φ q, p 38 R m φ : [, b] R m, φ j φ j, p, q, r [, b], L φ p, q m L φj p, q L φ p, r L φ p, q + L φ q, r [p, q] { i } i0,,, 3 dφ i, φ i m φ j i φ j i, i,,, sφ p,q, m sφ j p,q, m L φj p, q L φ p, q m L φj p, q L φ p, r L φ p, q + L φ q, r φ : [, b] R, φ φ, φ +, φ : [, b] R φ + x L φ, x, φ x φx L φ, x φ φ + + φ x, y [, b], x < y, 38 φ + y L φ, y L φ, x + L φ x, y L φ, x φ + x, φ x φ y φx L φ, x φy + L φ, y L φ x, y φy φx L φ x, y sφ x,y, {x, y} 0 φ +, φ φ f + g f : [, b] R, g : [, b] R [, b] { i } i0,,,, sφ, i φ i φ i i f i f i + g i g i f i f i + g i g i f i f i g i + g i fb f gb + g i i S φ fb f gb + g 30 90, 3 R m φ : [, b] R m j φ j, j φ j,, b j φ j :, b R, M, x, b j,,, m φ j x M p, q [, b] L φp, q mm q p x [, b] L φ p, x φ, φ j t, b, p [, b], x [, b] L φp, x m t, t φ j t 36 3 φ, L φ p, q m L φj p, q mm q p x, y [, b], L φ p, x L φ p, y L φ x, y mm y x 38

42 , ε > 0, δ > 0 t δ, t+δ, b x t < δ φ j x φ j t < ε m 0 < h < δ h, [t, t + h] { i } i0,,,, 0 L φ t, t + h sφ, < hε, [ i, i ] φ j i φ j i φ j c i i i c i i, i h i i m sφ, h φ j t m m φ j i φ j i φ j t i i i m φj i φ j i φ j t i i i m φ j c i φ j t i i < i i m ε 4m i i < hε m L φt, t + h h φ m j t L φ t, t + h sφ, + sφ, h φ j t < hε L φ p, t + h L φ p, t h m φ j t L φ t, t + h h i m φ j t < ε δ < h < 0, [t + h, t] { i } i0,,,k, 0 < L φ t + h, t sφ, < hε m sφ, + h φ j t < hε, L φ p, t + h L φ p, t h m φ j t L φ t + h, t h m φ j t < ε L φ p, t + h L φ p, t m lim φ j h 0 h t 3 R m φ : [, b] R m φ j : [, b] R x, b, φ,x : [, x] R m L φ, x, x, b L φ, x φ, φ j b, φ lim x b L φ, x x, b, φ x,b : [x, b] R m L φ x, b, x, b L φ x, b φ, φ j, φ lim x L φ x, b M, j,,, m x [, b] φ j x M x L φ, x, lim L φ, x L, x b { i } i0,,, [, b] k < b, [, k ] { i } i0,,,k sφ, sφ, + dφ k, φb L φ, k + m φ j k φ j b L + mm S φ L + mm, φ φ j b, ε > 0 L ε < L φ, p p, b, L φ, p ε < sφ, [, p], L ε < sφ, b [, b] sφ, sφ, > L ε 39

43 [, b], c, b c < x < b j,,, m φ j x φ j b < ε m b d c, d < p < b p [, b], b [, p], sφ, sφ, sφ, + dφp, φb L φ, p + m φ j p φ j b < L + ε ε > 0 sφ, L, L S φ 33 R m φ : [, b] R m j φ j j φ j,, b b m, φ j :, b R φ j t dt, b m φ, φ φ j t dt c, b 36 4, x, c, y c, b φ x,c : [x, c] R m, φ c,y : [c, y] R m L φ x, c, L φ c, y, 3 3 L φ x, c c x m φ j t dt, L φ c, y y c m φ j t dt c m b m lim L φ x, c φ j t dt, lim L φ c, y φ j t dt, 3 x y b c c m b m, φ,c : [, c] R m, φ c,b : [c, b] R m, φ j t dt, φ j t dt c 37, 34 φ : [, ] R φt t, t, φ,,, φ t t, φ t t, φ t, φ t t t φ t + φ t dt dt t dt t, 5 33, φ, r, θ, ψ : [, b] R,, b, r, θ, ψ :, b R φ t rt si θt cos ψt, φ t rt si θt si ψt, φ 3 t rt cos θt φ : [, b] R 3 φ t + φ t + φ 3t r t + rt θ t + rt ψ t si θt, φ b r t + rt θ t + rt ψ t si θt dt 35 R m φ : [, b] R m, L, φ, L φ ε > 0, δ > 0 < δ [, b] L sφ, < ε L sφ, 0 > L [, b] 0 0 [, b], sφ, L < sφ, 0 L, sφ, < sφ, 0, 36 L S φ φ ε > 0, L sφ, < ε [, b], L S φ 36 f :, b R, δ > 0 x y < δ fx fy < b δ, c i + ib, x c i δ, c i + δ, b 40

44 fx fx fc i + fc i < + fc i fc, fc,, fc M c δ + b δ, c i+δ c i+ δ δ b δ > 0, c +δ + b +δ b b +δ b, b c i δ, c i + δ, x, b fx < + M f i 37 R m φ : [, b] R m j φ j j φ j,, b, φ j :, b R 35 L , φ φ L, ε > 0, L sφ, < ε [, b], δ > 0 x y < δ φ j x φ j y < ε mb < δ [, b] { i } i0,,, { i } i0,,,k, i 0,,, i s i s i, l,,, k, i,,, φ j l φ j l φ j ξ l l l, φ j i φ j i φ j ζ i i i ξ l l, l, ζ i i, i s i + l s i l, l i, i ξ l ζ i < i i < δ φ j ξ l φ j ζ i < sφ, sφ, ε mb <, 3 i i s i ls i + s i ls i + s i i ls i + s i i ls i + s i i ls i + m φ j l φ m j l φ j i φ j i l m l φ j ξ l s i m s i φ j ζ i l m l φ j ξ m l φ j ζ i m l l φ jξ l φ jζ i l ε l b ε 0 L sφ, L sφ, + sφ, sφ, L sφ, + sφ, sφ, < ε, L , φt t, t R φ : [, ] R f : [, ] R fx x dt 0 t, x > 0 fx φ [0, x], f x fx, π f dt 0, f π t, f f π f : [, ] [ π, π ] φ f : [ π, π ] [, ], ψ : [ π, π ] [0, ] ψx φx, φ0 0, φ π, φ π, ψ0, ψ π ψ π x π, π, φ x f φx φx ψx, ψ x φ xφx φx φ φx, ψ [ π, 0], [0, π ], φ φx x ψx ψ ψx x + φx 4

45 φ0 0, ψ0, x [ π, π ] N φx x+ +! + N ψn+ N θ x x N+, ψx N +! 0 0 < θ, θ < 0 ψ N+ θ i x N φx x+ +! x N+ N N +!, ψx 0 N x N+ N+!, x N N! 0 x [ π, π ], z C φx x+ +!, ψx x x x! + N ψn θ x N!! x N N!! si z i expiz exp iz z+ +!, cos z expiz + exp iz 0 expiz cos z + i si z, x [ π, π ] si x φx, cos x ψx 4 z, w C si 0 0, si π, cos 0, cos π 0 si z si z, cos z cos z, cos z + si z 3 siz + w si z cos w + cos z si w, cosz + w cos z cos w si z si w 0 z si x φx, cos x ψx φ0 0, φ π, ψ0, ψ π si z si z, cos z cos z si z i expiz exp iz, cos z expiz + exp iz, exp iz expiz cos z + si z 3 si z cos w+cos z si w expiz exp izexpiw+exp iw 4i + expiz+exp izexpiw exp iw 4i expiz + w exp iz w siz + w i 4 si x cos x, cos x si x si z cos z si z, cos z cos z, si 3z 3 si z 4 si 3 z, cos 3z 4 cos 3 z 3 cos z 3 si z + π cos z, siz + π si z, siz + π si z, cos z + π si z, cosz + π cos z, cosz + π cos z 4 si π si π 0, cos π, cos π, si π 4 cos π 4, si π 6, cos π 6 3, si π 3 3, cos π w z, si z cos z si z cos z si z, cos z cos z 4 3 w z, si z cos z si z, cos z cos z, cos z + si z si π φ π, cos π ψ π 0 si z + π cos z, cos z + π si z siz + π si z + π + π cos z + π si z, cosz + π cos z + π + π si z + π cos z, siz + π siz + π + π siz + π si z, cosz + π cosz + π + π cosz + π cos z 4 3, 3, 5, 6 z 0 si 0 φ0 0, cos 0 ψ0 si π si π 0, cos π, cos π, z π 4 cos π 4 si π 4, cos π 4 0,! x N 4

46 x [ π, π ] cos x ψx [0, ] cos π 4 cos π 4 si π 4 si π 4 3, 4 z π 6 3 si π 6 4 si3 π 6, 4 cos3 π 6 3 cos π 6 0 si π 6 + si π 6 0, cos π 6 4 cos π 6 3 0, si π 6 φ π 6 > 0, cos π 6 ψ π 6 > 0 si π 6, cos π 6 3, z π 6 si π 3 3, cos π 3 x [ π, π] 4 3 si x cos x π ψ x π [ π, π] si x [ π x [ π, π ] 4 3 si x cos x + π ψ x + π, π] si x [ π, π] si x 0 x 0, ±π si siz + π si z si z + π cos z 43 si x 0 x π, cos x 0 x π + π si [ π, π ] φ,, f f si cos x cos x si π x si [ π, π ], cos [0, π] cos [0, π] cos : [, ] [0, π] 44 si 0 cos 0, si cos 0 π, si π, cos π, si cos 3 π 6, si cos π 4, si 3 cos π 3 x [, ] cos x + si x π 3 x, si x x, cos x x x [, ] si π cos x coscos x x π cos x si x 3 si x x f cos x + si x π cos x x D {x R x π + π Z} 43 D cos x 0 t : D R t x si x cos x 45 x D tx + π t x, t x t x, + t x cos x t x+t y t x x, y, x + y D tx + y t x t y x, x D tx t x 3 t 0 0, t π 6 3 3, t π 4, t π tx+π six+π cosx+π si x si x cos x t x 4 t x cos x si x cos x t x cos x + si x cos x 0, cos x + t x cos x 4 3 six+y si x cos y+cos x si y cosx+y cos x cos y si x si y, cos x cos y x D t x cos x t π, π, lim t x, lim t x x π x π t π, π 3 t : D R π, π t : R π, π t x +x 43

47 4 t x si x cos x si x cos x si xcos x cos x cos x+si x cos x cos x x π, π t x > 0 t π, π 4 x π, π cos x > 0, lim x π cos x lim cos x 0, x π lim x π lim t x x π 3 45 t t x si x, lim si x lim t x, x π x π t x cos x +t x t x +x 47 x, t x x+ + 0 t <,, t, +t t, x <, 0 x t x x lim x t x t π 4 0 π π α t 5, t α 5 0 9, t 4α π t 4α t 4 π 4 +t 4α t π 4 6 t 5 4 t lim x t x x+ + t 0, t α t α t α 5 t α, t 4α 39 4α π 4 t 39 π 6α 4 t t α , x > 0 e x > x! e x k0 5 s, x k k! + eθx +! x+ 0 < θ <, x > 0 e x > x! 0 x s e x dx γ : 0, R, γ :, R γ t t xs e x dx, γ t t xs e x dx γ t t s e t < 0, γ t t s e t > 0 γ, γ 0 < x, e x < γ t t xs dx s ts < s, γ, lim γ t, 0 < s < t 0 0 xs e x dx > s, 5, x > 0 x s e x <!x s γ t t!xs dx! s t! s s γ, lim γ t t x s e x dx 44

48 53 p, q, 0 x p x q dx β : 0, R, β :, R β t t x p x q dx, β t x p x q dx β t t p t q < 0, β t t p t q > 0 β q 0 < q <, β m 0 < x <, xq < m q β t t m x p dx m p t p < m p p, β p, lim β t t 0, 0 < p < 0 x p x q dx m p 0 < p < p < x <, xp < m β t t m x q dx m q t q < m q q, β q, lim β t, 0 < q < x p x q dx t R I, J, I J {x, y x I, y J} xy- I J 54 Γ : 0, R, B : 0, 0, R Γs 0 x s e x dx, Bp, q Γ, B 0 x p x q dx 55 Γ, s > 0 Γs + sγs Γ!, Γ e x dx [ e x ] 0 0 m > s m, 5 x > 0 0 < x s e x < m! x lim m s x xs e x 0 Γs + x s e x dx [ x s e x ] 0 0 sx s e x dx sγs 0 56 p, q Bp, q ΓpΓq Γp + q 0 < s < < t, xy- E s,t E s,t { u, v s u v t u, t u v } s u φ : E s,t [s, t] [s, t] φu, v uv, u uv, φ, φ φ x, y x + y, [ E s,t [s, t] t s s s+t, x x+y φ u, v v v u u, φ t x p e x dx y q e y dy s t s t s [s,t] [s,t] x p y q e x y dx dy x p y q e x y dxdy uv p u uv q e u ududv E s,t v p v q u p+q e u dudv E s,t ] t s+t, 0 < v <, u > 0 v p v q u p+q e u, 45

49 t s t x p e x dx y q e y dy s s < r < s + t E s,t [ E s,t ] [ s t s, s, ] s t t x p e x dx y q e y dy s 0 < s < < t t s s s u p+q e u du s s [ r, rt r s [s,t] [ s s+t, t t s t s t s+t s s+t s+t] vp v q u p+q e u dudv v p v q u p+q e u dv u p+q e u du t s+t s s+t du v p v q dv Γp + qbp, q ] [ s r, ] s r r s [ ] [ s t s, s, ] v p v q u p+q e u dudv s t s s t s s s s t v p v q dv u p+q e u du s v p v q u p+q e u dv s s du v p v q dv t x p e x dx y q e y dy Γp + qbp, q s s 0, t, Γp + qbp, q, ΓpΓq, Γp + qbp, q ΓpΓq 57 α, β > π 0 si α x cos β xdx α + B, β + t si x si α x t α, cos β x t β, dt dx si x cos x t t π 0 si α x cos β xdx 0 t α t β t t dt 0 t α+ t β+ dt α + B, β +,!! 4 4!, +!! Γ π, Γ +!! π +!!! 56, 55, 57 Γ B, π Γ 0 dx π Γ > 0 Γ π 55 Γ 3 Γ π Γ +!! π Γ Γ 3!! π Γ + 3!! π!! π 46

50 6 6 p 0, p,, p, q 0, q,, q, p 0, p,, p, P x c 0 + c x + + c x i 0,,, P p i q i c 0, c,, c p j i i, j + A V der Mode A p j p i, p 0, p,, p A 0 0 i<j A, c 0, c,, c f : [, b] R p < q b, r p+q, ρ q p, P f : R R i i P f r + ρi f r + ρi 6, 6 f, g : [, b] R α, β R P αf+βg αp f + βp g, f, i i P f r + ρi f r + ρi P f i i αp f +βp g r + ρi αp f r + ρi + βpg r + ρi αf r + ρi + βg r + ρi 63 f + 6, fx x r k k 0,,, + αf + βg r + ρi, Pαf+βg, q p fxdx q p q p fxdx P f x c j x r j + fr i,,, P f xdx q p P f xdx j ρi c j k ρi, j k ρi ρi j c j k j k k c j c, c,, c, f P f, 0 j k q p q p fxdx P f xdx k + i,,, j ρi c j 0, ρi j i, j V der Mode, j,,, c j 0 P f x c j x r j q p P f xdx 0 q p x r + dx P f x c j x r j + fr i,,, j ρi c j f r + ρi fr, j ρi j c j f r ρi fr 47

51 j ρi c j f r + ρi + f r ρi fr, x j ρ j c j, i,,, j i x j f ip + + iq + f + ip + iq f p + q 6 64 i j i, j A, A A A 4, V der Mode i<j j i i<j j ij + i + i0 i! + i! i! A! + k k i i<j ki + k j i ++ + i i k ki + k i i! + i! i!, j,,,, A i j A i, j, 6 Crmer v i x j A f i+j A i, j f i ip++iq q p q p + f P f xdx P f xdx q p A ip + + iq + f +ip+ iq, c j ρ j+ j + i + ρfr q p i+j A i, j j + v i f + ip + iq x j j + + f p + q f p + q + A f p + q p + q 48

52 δ i i+j A i, j j + 65 q p A q p i P f xdx δ i f ip + + iq + f 4 i i 4 3 i+ i+ 4 j 4 j i j 4 j i 5 j+ 4 i+ j j + i+ 4 j 4 j + ip + iq x, x,, x D,j x, x,, x x k x j x k x j D,j x, x,, x x k i x j x k x j x j+ x j+ i x j+ x j+ + A p + q δ i f i x l x x l x i x l i x i x l x σ k x, x,, x x, x,, x k, σ k x, x,, x k 0,,,, 66 X x X x X x k σ k x, x,, x X k k0 D,j x, x,, x σ j x, x,, x D,j x, x,, x x x j x x i x i x j i i x x j, x j p<q x q x p x + x x j+ x x j i x i + x x j+ i x i x j x + x x j+ x x x i D,j x, x,, x + x D,j x, x,, x i D,j x, x,, x x x i D,j x, x,, x + x D,j x, x,, x i 49

53 , p<q G,j x, x,, x D,jx, x,, x x q x p x q x p p<q G,j x, x,, x G,j x, x,, x + x G,j x, x,, x V der Mode D,0 x, x,, x x x x p<q x q x p, D,j x, x,, x p<q x q x p, G,0 x, x,, x x x x, G, x, x,, x, j 0,,, G,j x, x,, x σ j x, x,, x X x X x X x k G, k x, x,, x X k X x k0 X x X x X x k G, k x, x,, x X k X x k0 k G, k x, x,, x X k + k0 k x G, k x, x,, x X k X + k G, k x, x,, x + x G, k x, x,, x X k k + G,0 x, x,, x G, x, x,, x X + k G, k x, x,, x X k + G,0 x, x,, x k k G, k x, x,, x X k k0, k σ k x, x,, x X k k0 k k G, k x, x,, x X k k0 X j G,j x, x,, x σ j x, x,, x j 0,,,, G,j x, x,, x σ j x, x,, x, 67 δ i k! + k! + k! k0 i+j j σ j,, i, i +,, j + i! + i! 50

54 A i, j j j+ j j+ i j i j+ i j j+ i+ j i+ j+ i+ j j+ j j+ j j+ j 4 j i j 4 i j i i + j 4 i + j i + i i+! i + j j 4 j A i, j! i + j D,j,, i, i +,, D,j,, i, i +,, σ j,, i, i +,, p<q p,q i A i, j q p i k p<q i k ki+ + j i! + i! k0 p<q p,q i q p i k! + k! i! + i! k! k i k q p, k! + k! σ j,, i, i +,, k! δ i i+j A i, j j + 68 K, j σ j,, j + j j + 67, + δ A +! k0 k! + k! K k! A, j δ + j! k0 k0 k! + k! k! k! + k! σ j,,, k! +j j + + j! σ j,, 5