[ ]. A = IC X n 3 expx = E + expta t : n! n=. fx π x π. { π x < fx = x π fx F k F k = π 9 s9 fxe ikx dx, i =. F k. { x x fx = x >.3 ft = cosωt F s = s4 e st ftdt., e, s. s = c + iφ., i, c, φ., Gφ = lim c + cf c + iφ. s3.4 ft F ω, F ω ft F ω = π ft = π fte iωt dt F ωe iωt dω.., i =,. ft F ω e aω, ft., π e ax dx = a., a >.
ft. ft τfτdτ = e t.5.. fx = x [ π, π] n sinnx n n=. π a π a, x [ π, π] x = a n cosnx cosna + 4 n=. 3, x [ π, π] x 3 π x = n sinnx. n=.6 f, t gt n 3 n F[gt] = gte iπft dt 3 s3 4 s44., i =. mt Mf, e ix = cos x + i sin x, mt cosπf t Mf f Mf + f..7 f, t gt F [gt] = gte iπft dt., i =.,. P t t,. {, t < t P t = t >, t > t 6 s63 P t + 4t + P t 4t t,. 7 s74.8. dx x x + 6 x dx 3 4 log x x dx 9 s9
.9 I n = x n exp x dx a. I = exp x a I n =... n, a. dx a. x n exp x dx a n. a xe x dx. y = x e x,. 9 s9 4 s44 lim x x e x. 5 s53. < x < f x, f x, f 3 x,, f n x, i f x ii i f n+ x = f n t exp[ x t ]dt, n =,, 3,, π ii f x = π exp[ x ]. exp f x exp[ x ]dx = π n a n, b n f n x f n x = π an exp[ x /b n ] a n+, b n+ a n, b n n =,, 3, 3 a n, b n n =,, 3, n 4 I n = f n xdx 6 s637.3 x f x, f x,, f n x, a,b a f x = x b f n x e x = x f n t e t dt 3
.4 fx = I, J, K x I = e t dt, J = t e t dt, K = x x x t e t dt {a n } {b n } f n x f n x = x + a n x + b n n =,,, a n, b n a n, b n n 3 a n, b n n =,,, e t t x dt x > fx + = xfx x n fn = n! 3 f5 8 s83 4 f 5 f = π 9 s93.5 lim x + x x, x cos x sin x lim x x 3 s35.6 < x < fx fx = sin x + x fte t dt + π I, I, I 3, J, J, J 3 I = J = π e t dt, I = cos t dt, J = ft cos t dt te t dt, I 3 = π sin t cos t dt, J 3 = A, B A, B fte t dt = A, π ft cos t dt = B e t sin t dt π t cos t dt 3 fx s36.7 A A = a a a 4
A, A 3 n A n A n 3 A A exp A A =, exp A = n! An exp A.8 A = a a A. A n. 3 A 3A + E = O, a. n= s3,., E. 4 3 a, lim n An. 6 s638.9 [a, b] fx, gx, b a fxgxdx =.. a e [ π, π]., k, l. a cos kx b sin kx c cos kx sin lx d cos kx cos lx k l e sin kx sin lx k l [ π, π] fx,, cos kx, sin kx fx = a + a cos x + b sin x + a cos x + b sin x + a 3 cos 3x + b 3 sin 3x + = a + a k cos kx + b k sin kx k=. a, a k, b k fx. 3 fx [ π, π]. { π x < fx = x π. fx = x e x+. s3 5
y = fx,. lim 3 x fx. fxdx, 9 s933. a dx a > x π x sin xdx 3 s343. log π 3 3 43 + 3x x 3 x dx = log x + t + u e x x 4 dx = w 4 x x dx = π y log xdx = v e 4x sin xdx = x 4 s443.3,. tan 3x x 3 lim = lim x x x e x = 4 lim x cos x log cos x = log x 5 lim = x log cos x.4,. log x 3 lim x log sin x = 9 s943 π 4 x cos xdx = π + π 3 sin x cos xdx = 3 5 e x 4 x dx = 3 x log xdx = 4 e 3 + 6 x x dx = 5 xe x dx = + e 7 x e x dx = e.5 e x dx. 9 s944 s43.6,. arcsin, sin. lim n n = lim x + x + x + n x arcsinx 3 lim = x x = s4 6
.7 fx = xe x,... 3 lim fx lim x x 4. fx..8,. lim x + x 3 x + = x cos x lim x x = n n 3 lim = n n +.9. π.3. lim x x 3 x sinx a dx a < a < π lim x x cos x 4 s443 5 s54 x dx 8 s84 e 3x 3 lim x e x e. 9 s94 p p.3 P = Q = I P I q q P P n Q n = p + q n Q 3 P + q < lim n P n 5 s554.3 x x = + x + x + + x n + R n x x < lim n R nx = x x < x 3 sin x n+sin x n x < π.33 n= 8 s85 7
y = a x yx = dy dx + xy = x + 4 x3.34 fx e t cos xt + x tdt fx f + f x + f x + f f f 8 s85 fx = + x x d Tan x = dx + x y = Tan x y = tan x 3 F x = Tan x < x < 4 y = F x 5 s55.35 y : fy =, < y < + e βy., β.,. β : a β +, b β =, c., x = fy. x = fy y = f x. 3. log xdx, log. 4. gx x f zdz, x x, lim x log x = log =. x 5 gx αx x. x x, α..36 fx x = a. fx fa + n= 6 s65 n! f n ax a n = fa + f ax a + f ax a + 8
, f n x fx n dn f dx n., f x f x fx df dx d f., < x < < x < dx x e x x =. x, gx hx x x n, e x n! xn n= n=. gx = e x, hx = ex x gx x =., hx x = x. 3 hx h x. 4 y = hx < x <,,. 6 s65.37 x, fx fx = x e ax., a. fx f x, f x. x +, fx lim fx. x + 3 fx,,,,, y = fx. 7 s75.38 m =,, lim x xm e x =. 7 s754.39 a =, a n+ = a n {a n }, lim n a n..4 t ft F s = e st ft dt., s Res >. s56 ft, f = f =,.. tf t + 3t f t + t 3ft = f t, f t, F s. tft, tf t, tf t, F s. 3 F s. df s s + + 3F s = ds 4 F s, ft. 9
.4 R 3,. A = s54 A. n v R 3, A v n =,, 3, n, n lim n A v =., x : y : z. x y z 3 s353.4. n= n log n 3 s356.43. I = x e x y dxdy 3 s357.44 t ft F s = e st ftdt., s Res >.., ft f =. f t, e t f t, s, F s. t t e τ f τ dτ, e t e τ f τ dτ, s, F s. 3 t f t + e t e τ f τ dτ = e t, s F s. 4 3 ft. 4 s454
.45 a n = n + 3n n +,, lim n a n..46 z. fx, I[f]z. n =,, f [,, I[f]z. a. I[sin ax]z, I[cos ax]z..47 a n n =,, 3,,. a n, a n. n= n= n= 4 s458 e xz fxdx, 4 s459,. a n a n, a n n =,, 3,., a n. 3 a n, n= n=.48 { } a n,. n= n= an n. { } a n, an. n= a n, n=. 3, a nk < nk a nk < nk n= 7 s758 { an } n= { a nk } k=, a nk < nk,..49 {a n }., 8 s858 a =, a = 3, a n+ 3a n+ + a n = n =,, 3, {b n },. b n+ = b n+ b n n =,, 3,, b b. a b =, b = 7, b 3, b 4, b 5, b 6. b b b d, b n d.
3 {c n },. a c = /, c. < c <, c n+ = c n n =,, 3, b < c n <. c {c n }, {c n } n..5 n,..5 lim n n.5 n= 9 s953 n+ x n+ + x dx = 3 + 5 7 + + n n + n,. n + n n i i= 9 s958 9 s96 n= p > p np 9 s964.53 lim x sin 3x x lim x + x3 log x 3 lim x cos x x 4 lim x +e x /x log x + sin x 5 lim x x s6.54 fx = x /x x > lim fx, lim fx x + x s63.55 log x ln x sin 3xdx x cos xdx 3 log xdx 4 x dx 5 e x dx s65.56 π tan x < x < π tan x < x < tan x x x dx 6x + 3 n 3 lim n n + n n + + n n + + + n n + n = π 4 s66
.57 i n= n ii n= n x = Taylor ρ x = + x + x + ρ = i + x ii e x iii sin x.58 s67 x 3 e x dx π/4 dx cos x s64.59 Γs s > Γs = Γ x s exp xdx Γs + = sγs s65.6 log + x = n= n+ xn n = x x + x 3 + 3 4 + 5 s67 a.6 a a A = A n a n s6.6 fx = x < x < + x 5 s56.63 x x t sin tdt xe x dx 5 s564.64 fx = e x n x n e x x e t = k! xk + n! x tn dt k= 3
x e t R n x = n! x tn dt x < + R n x [a, b].65 a b b c.66 fx = log x. e ax +bxy+cy dxdy 5 s566 e x = π 5 s568 fx x =, S 3 x. n k lim n k. k= s63.67 d x dt = a x 3. a. 3 at <<, d x dt =. 3 3 t = x = x, dx/dt = v, t. s67.68 t >,.. Γt = e x x t dx. Γt + = tγt. 3 n Γn = n!. 4 s465.69 fx lim x f x = β. h lim fx + h fx x, β h. 5 s563.7 x + y + z φr = aδr,. a, δr 3 δr = π 3 dke ik r. a b 4
φr φr = π 3 dke ik r φk, a φk. dk k x, k y, k z k, θ, φ dk = dk π dθ π dφ J. J. k = k. d k dk d cos θ. π 3 φk, c φr., dx sin x x = π. dφ c d e f 5 s566.7 fx x = a, D. D a I D, x I, x a fx > fa f x = a,, x I, x a fx < fa f x = a.,,.. fx x = a. fx x = a. f x = a, f a =. 3 g, C. g x = x =, < c < g c =. 4 3 g x = x =, g x = x,..7 a n I n a := a 6 s66 dx + x n n =,, 3,. I n = lim a I na, I n. k, n k < n, a,, a k, a k+,, a n. x. a + a x + a x + + a n x n = 7 s76.73. x 3 lim x e x log x lim x x 8 s864 5
.74 < a < b <, fx [a, b], fx. b.75. lim x sin x x a fxdx =, [a, b] c fc =. 9 s96 x cos x lim x x 9 s969.76 {a n } n =,,, a n+ = a n + a n a = a = τ = lim n a n a n a n = a n+ a n a n.77 a = a = a n π π π π sin mx cos nxdx cos mx cos nxdx τ = lim n a n a n 9 s973 m, n m, n > m, n m, n > 3 fx a + [a n cos nx + b n sin nx] n= π fx dx = a π π + [ an + b ] n n= s74.78 fθ = n= n= π n n n! sin + π 4 θ n n n! sin + π 4 f θ f θ fθ 3 fθ fπ = n= π n n! sin n + 4 s7 π 6
.79 r, θ r >, θ < π u r + u r r + u r = i θ u, θ = cos 3θ θ < π ii ur, θ lim ur, θ = θ < π iii r i ur, θ = frgθ i r θ r, θ r, θ c f r g θ 3 m fr = r m c m g θ i u m r, θ = frgθ 4 d m i ur, θ = m d m u m r, θ ii,iii ur, θ s75.8 e x. e z = + z! + z! + z3 3! + =, cos z, sin z. z = fz =. f z = z sin n= z n n! n=, f z = cos z z z, f 3 z = z z c n z n., 3. I = f 3 zdz C, C..8 fx L[x]. e sx fxdx L [fx] = F s. 3 s373 L [e ax fx] = F s a. 3, xe x. 4, L [ e x xe x]. 3 s374 7
.8...,. i +, x < x < i.,., n P n., L.,. lim n nxn = < x < 3 j Q j, Q j+ Q j. 4, Q j x, j..83 t π ft 4 s475 ft = a + a n cosnt + b n sinnt a n, b n a n = π π π n= ft cosntdt, n =,,, b n = ft sinntdt, n =,, 3 π sin A sin B = {cosa + B cosa B}, sin A cos B = {sina + B + sina B} cos A sin B = {sina + B sina B}, cos A cos B = {cosa + B + cosa B} π ft ft ft π π O π π t.84 a > xe ax dx 5 s574 x, y r, θ e ax e ay dxdy 3 e ax dx = π a 4 n x n e ax dx 6 s67 8
.85 fx = x a x, gx = x e x a > fx a x a y = fx y = gx 3 4 a a fxdx a fxdx = gxdx a 8 s87.86 p, q p q A = Ax = λx λ x p q { a n+ = pa n + qb n {a n } {b n } b n+ = pa n + qb n < p <, < q < {a n } {b n }, a, b lim n a n lim n b n p, q, a, b 8 s87.87 n, z n = n n z, z,, z n., n z i arg z i, arg z < arg z < < arg z n < π.,, z, z,, z n P, P,, P n, O.,. 3 P, P, P 3., P P P 3 S 3., 3 P, P, P 3 O, OP P, OP P 3, OP 3 P G, G, G 3. G G G 3 A 3. 3, n P, P,, P n O n G, G,, G n. n P P P n S n n G G G n A n r n = A n S n. 4 3 r n n lim n r n. 9 s973.88 zt = e iωt., t, i, e. ω, zt t, ω,. zt,,. ω a + ib a, b, zt t. zt,,,., b/a. 9
3, z n = zn, n. z n+ z n d n = z n+ z n a, b, n. 4 3 d n, D = N n= d n a, b, N., a b b D. N..89 A =,. s7 A λ, λ u, u., λ. xy P p, p, Qq, q, Rr, r P QR S, p, p, q, q, r, r. A,. α 3 a =, a A n a n = β α n, β n A n., n. α n 4 L = lim,. n β n.9 A = A, A, A. 3/ / / / A.,. 3 A I =., I =, α n β n. s74. 4 t, t gt a, b. a b. t = gtt + at + b 5 A A I. 6 A n e., A = I,! =. n! n=.9 fx = xm + x n fx dx s7., m, n, m < n.., C = C + C + C 3 R, π n. z = x + iy i, C fz., R >.
C fz dz C. 3 R, C C fz dz. 4 i. fx dx y O C3 π n C C s74 R x.9 N n, n, n 3 n + n + n 3 = N. N! n!n!n 3! m N n, n,, n m n i = N W.. 3 W, lim N N log e W. a lim N N log e N =. m b N N = n i n i p i = lim N N i= i=, n i N p i. i =,, m., lim N N log e W p i. k! k. πk k+/ e k+/k+ < k! < πk k+/ e k+/k.93.,. A n fz = n= A n z a n s75., z, a, n. a C,.. fzdz = πia. C gz C D. C gz,. x,. dx + x i
3 x,. 4 x,. dx + x cos x + x dx.94 fx l x l.,, fx, a m = l b m = l l l l l fx = a +.. m= fx cos fx sin mπ x dx m =,,, l mπ x dx m =,, l a m cos mπ l x + b m sin mπ x l 4 s474 fx a m, b m, l = fx. { fx = x + x x < x π x π fx = cos x l = π,,. 3 5 + 35 63 + + m 4m + 5 s57.95, ±, ±, q +, r = q. n, n x n. x = q, x = qr, x = r.,. x =, x 3 =. x n =. 3 n. x n = m < n x m. z = qr u n z = a n z n. n n a n = a i a n i., i= Uz. Uz = u n z., {Uz} Uz z., Uz. n=
4 Uz, a n n. 5 s57.96 3 A λ, λ λ 3 α + βλ + αβλ =., A 3 α + βa + αβa =., α, β, A,, O, E.. A n n 3 A. A n = a n A + b n A + c n E, a n, b n, c n. A D, a n, b n, c n D n = a n D + b n D + c n E n 3. 3 a n, b n, c n. 4 α, β, A i = E + A + A + i= a A + b A + c E, a, b, c. 5 4 a, b, c, E Aa A + b A + c E. 5 s575.97,. f x αf x + α fx = f n x fx n n, α.. x, k e kx, x 3., e. fx, n,. f n+ x αf n+ x + α f n x = f n x, f x fx. 3 fx fx = f m xm m!, m= [ ] f n x, fx = fe αx + f αf xe αx., m, f x fx,! =. 4, f =, f = p p. f x + 4f x + 4fx = e x 5 4 fx, fx = p, p fx. 3
.98.., Iα = 6 s67 e αx dx.. y, {Iα}. {Iα} = = e αx dx e αy dy e αx +y dxdy α ; x r, θ. π Iα = α. 3 4. x, y R C, C, C 3 C. i, z,. e iz dz 4 lim e iz dz =. R C. 5, 6 sin x x dx. y C, ϕ π, sin ϕ ϕ π sin x dx. C 3 C O π/4 C R x 4. 7 s774.99 ; i., z. sin z = z. i i, 3 i. 3 C z + dz C. C.,. 4 z z 3z + < z <, < z. 4
5 x + x 4 dx,.,.. X F x. 8 s874 df dx = F F s, F m = /., m, s., 5. F, F f. f m.,,. 3 E = x fdx.,., x., n x i i =,,, n, r i i =,,, n.. F x 4 ϕ F x = log, x. F x 5 y i = ϕr i, n Q = {y i ϕ F x i } i= m s,. : x = n x j, y = n y j n n j= j= : varx = x x, vary = y y : covx, y = n x j y j x y n j= 9 s97. 3 8, 4,,. A 3, 5 B 3,., A,, 3 X =, X=, B,, 3 Y=, Y =..,. a EX EY. b ρx, Y., A n X = n. a 3. b A, lim α n kβ k., α β n k=. α β,. 5
3 B n Y = r n., r. B, r. 3 s37. A, B A =. B = + +, C C = BAB.., x x T. expx. expx = k=, I, X X k k! A. C. 3 t expat. = I + X + X! 4 3 x f x f x = n k=., n n >, + X3 3! + { exp C πk } T { a x exp C πk } a x n n a 3. a = a a a 3 f x x, b. n x = b k= a. a, a, a 3, n, k.,. b b. 5 4 x, n x a, a, a 3..3 a, a,, a n, lim a n = n n lim a k = n n k= 3 s375 9 s98 6
.4 Gx, y, t Gx, y, t = t exp x + y 4t t > G x, G t t > It It =.5 ϕa =.6 I = π Gx, y, t dx dy e x + e x + 3 dx log a cos x + a dx a ϕa = ϕa b ϕa a = { x.7 lim M M } x + ye x+y dy dx y + loga + x x dx a > 3 s383 4 s48 5 s58 xyx + y dxdy 6 s68.8 R f p x+p ftdt = p x ftdt x R. lim n b a sin nx dx = b a π b > a. 8 s88.9 β, γ <..,,. x + y β dxdy x + y γ dxdy <x +y x +y s83. t xt. d x dt = dx dt t, xt. 7
t +, xt +, xt.. n. π sin x cos nx dx. s84 fx [, π]. fx f x a lim n fx cos nx dx =. a. s8.. e x y +xy + x + y dxdy.3 6 d y dx + dy dx. s8 y = y = yx y = 3 lim yx = x + 9 s99.4 D : x, x y x. x + dxdy + x x + x 4 dx, t = x x..5 dy + 3y = cos x dx y = yx. y = yx lim x +.6. xe x + + x dx.7 x F x = F x dy dx + 3y = yx. 3 s93 3 s39 5 s59 t + e xt dt x. F xdx. 6 s69.8 y + e xy = y = e y = yx D 8
dy dx d y x, y x = dx yx, a a lim yx lim x x a.9 3 s33 fz = eiz + z cos x + x dx 3 s39. X, Y, U Nµ, σ, Nµ, σ, Nµ 3, σ 3 a, b, c W = ax + by + cu X µ / Y µ V = 3 P σ 3 X µ 3 σ σ N, Φz = P Z z = π z exp t dt z... 3. Φz.5.843.977.9987 x. fx = Sin x + 3 s3 lim x fx, lim fx, x lim fx x x f x 3 fx 8 s8. π/4 R> Im Γ R Γ R Γ R, Γ R, Γ3 R, Γ 3 R Γ R O π 4 Γ R R Re Γ R dz = Γ R e z 9
G R, P R, C R, S R G R = C R = R R e x dx, P R = cos r dr, S R = P R P R π/4 e R cos θ Rdθ Γ R R e z dz, sin r dr, θ π 4 4 θ cos θ lim π P R = R 3, cos x dx sin x dx π e x dx = 8 s88.3 fz = z4 z 6 +. fz re iθ. z = α fz, fz z = α 6α. 3 fxdx. s5.4 fx = + x x x >. lim x fx, gx = f x fx lim fx. x. gx. 3 gx > x >. 4 fx {fx x > }..5 fz = z z 4 +. fz re iθ. s3 α fz, fz α 4α. 3 fxdx..6 ux, y, t = x +y 4πt e 4t u t u x,, t >. t >,. u x y. 4 s45 3
ux, y, t x.y ux, y, t = a + a x + a y + a x + a xy + a y + a, a, a, a, a, a. 3 lim ux, y, t dxdy. R x +y R.7 fz z z 4 + z +. z 6 = z. 5 s53 H = {x + iy : x, y R, y > } fz α, Resα., i =. 3 I = fxdx. 5 s55.8 fz = eiz z +,., i =,. e fz,. z = Re iθ R >, θ π,. 3 I = fz R cos x x + dx. 8 s85.9 dx dt = x + x t = x = ξ x = ϕt, ξ x = ϕt, ξ t x = ϕt, ξ ξ ξ 3 ξ x = ϕt, ξ t t 4 x = ϕt, ξ < t < t ξ.3 a fx 8 s8 a < x < a fx = a x < a x > a fx = x ± a d y dx y = fx x = ± a dy dx x ± y = 3
yx y x = yx 3 a.3 m, n,. L L sin mπx L 3 s3 nπx cos L dx =. { k π < x < fx = k < x < π, fx + π = fx k >, a, a n, b n. n = 7. fx = a + n= a n cos nπx L + b n sin nπx L,.3 < x <, A = x x x x.. s5 A. P AP P. 3 n, A n. 4 A n, n. 6 s6.33 A, fx fx = detxe A., E 3 3. A = /8 A. fa. 3.34 y = a + A n. n= a k cos kx + b k sin kx. k= x = [ π, π] y = { a : x, : x < 8 s8 a x = [, π] y = a sinnx cosnx n 3 x = [, π] y = x 9 s9 3
.35 n + + 3 + + n = nn + n + 6 lim n n 3 n kk + k= s.36.37 3 + 4 + 3 5 + + n n + + e x +y dx dy n 3 + 4 + 3 5 + + n n + S n lim n S n s s.38 x d y dy + 4x dx dx + y =, yx x= = dy dx = x= i x = e t x [, ] t yxt dy dt = dy dx yxt dx dt d y dt d y dx, d x dt, dy dx, dx dt 3 x = e t i d y dt + 3dy dt + y =, yt t= = dy dt = ii t= 4 ii 5 4 ii i yx.39 x fx i = F ω = fx e iωx dx 3 s3 ω F ω fx F ω fx fx = F ω e iωx dω π 33
F ω ω d dω F ω = d fx e iωx dx dω d dω F ω = d fx e iωx dx = ixfx e iωx dx dω ixfx d F ω dω ω F ω d dω F ω x fx d fx F ω dx 3 d fx dx x fx = n + fx, lim fx = x ± fx F ω n 4 3 3 s34.4 fx x = a lim x x sin x lim h fa + h fa 3h h 4 s4.4 x fx i = F ω = fxe iωx dx ω F ω fx F ω fx fx = F ωe iωx dω π F ω ω d dω F ω = d fxe iωx dx dω d dω F ω = d fxe iωx dx = dω ixfxe iωx dx ixfx d F ω dω 34
ω F ω d dω F ω x fx d fx F ω dx 3 d fx dx x fx = n + fx, lim fx = x ± fx F ω n 4 3 5.4 4 s45 lim x x + 9 3 x lim x + x/x 6 s6.43. lim x x + x lim x x sin x x 3 s.44 fx < x <,. fx = + x t xftdt f, f. fx. fx. s4.45 a >,. Ia = e x+y dxdy D D = {x, y x, y, x + y a} D. Ia. 3 Ia a, < a < +,,,,. s3.46. lim e x x cos x x sin x lim + x x 3 lim 3x 9x 3x x x 3 s3.47,,. 35
lim a n, a n = + n =,,, a = n a n lim x + x 3 x + x 3 lim x cos cos x x 4 5 s5.48 O XY Z. y = z x A = x,, z, y = B. AB λ, AB,, θ, ϕ., 9 < θ < 9, < ϕ < 8. B, OB = OA + AB, B = λ cos θ cos ϕ + x, λ cos θ sin ϕ, λ sin θ + z. E E =, a, h, a >, E B y = z x P. λ P. : λ P Vanishing Point 7 s75.49 x =, A i i =.,, 3, 4. cosx = A + A x + A x + A 3 x 3 + A 4 x 4. logx + e 5x 3 lim x x 4 lim x. 4x + x. 8 s8.5 t, a > ft = sinh at, F s. F s = o ft exp st dt.5., log x, e. 8 s86 a log + x x cos x, 3. b lim x + log + x. x cos x n lim 4n + 3 n. n 3 lim log x + log x log x log x. x sin 4 lim tanh x x. x +.5 a > exp{ ax + y }dxdy 9 s9 36
fa fa = n exp ax dx fa x exp x dx x n exp x dx 3 s336 sin βx.53 Iα, β = exp αx dx α, β x sin βx β x dx = signβπ signβ β ±β +, 5 s533 p p.54 A = p, q q q < p <, < q <,,3 A A n A n 3 A n n lim n An 5 s536.55 fx, y = x + y x + y+, < x, y <. fx, y.56 x, y,. π e x dx = x n e ax dx a >, n.57 d dt x = x x d dt x = x + 3x A A = 3 6 s633 6 s634 A P AP P A 3 x = 4 y = x x y y x A P y = x 3 y, y 5 x, x 6 t = x = x, x t 6 s63 37
.58 e x y dxdy. D, D a y D = {x, y x + y a, x, y } a >., D = {x, y x, y } y dxdy. D e x o D a x 7 s73.59 fx = e x sin x f x x π f x = y = fx 3 fxdx 8 s83.6 x m dx dt = mx t = x = x x K x dx dt = mxk x K t = x = x x x.6 lim x 3 lim x x + 4x + 3 x { logx + + log 3x + 3x.6 X, Y fx, y = sin 3x lim x sin 5x } π e x 8 s83 8 s83 +y / Z = X + Y < x <, < y <.63 lim x + x + x. x fx = sin x. 9 s934 9 s93.64 x xy + y =,.,,. 38
x xy + y = x, π π g, y f f g,. A f g : x x A y y, A A. 3. e x xy+y dxdy {x, y < x <, < y < } D 9 s93.65 fx = x ln x x >,. f x, f x. fx. 3 x + x + fx. 4 fx. y = x ln x x x = a < a < Sa..66 5 Sa. 6 Sa lim Sa. a + : e =.7. kk + = 3 + 4 + 3 5 +. k= k = + 3 + 5 +. k=, k = + + 3 + = π π. 6 k= 9 s934 9 s9335.67 e lim + x = e x x. fx = log x x /x dfx dx = lim logx + x log x x x. s33.68 t = y x,., x >...69,.,. x y dy + y y x y dy dx x + y y x e x dx.7,. fx, s36 e x +y dxdy fx = a + a x + a x + + a n x n + s37, a, a, a,, a n, x., 39
f = a, f = a, f = a. n, f n = n! a n., f n x, fx n, fx n.. 3 fx = f + f x + f x + f x 3 f n + = x n.!!! 3! n! 4 sin x,. sin x. 5 cos x e x,. cos x e x,. 6 θ, e iθ = cos θ + i sin θ,, i..7 e x. n. lim + n n n n= s39 a. + n n < + n+ n + b n! < + n n =,,. n n, e. e.. 3 log + x lim = x x e x lim = x x, {e x } = e x. 4 fx = e x n x =,. x n 5 lim =, e x = n n!.7. n= x n n!. e x dx = π s3 fz = e z,., a >, b >, i = y. e ibx x dx b a O a x s3 4
.73 I = e x dx. s34.74. e x sin x dx. e x sin xdx = e x sin x + cos x + C C 3 s33.75 x fx = e x. f f x. f n f n x, n φ n x, f n x = φ n xe x. 3 n. lim x f n x... 3 s336.76, 3.. n 3 P n.,. n lim P n n k= 3,. 3 3,, + r, 3 + r, + r., r >.,, r, < r < r. r,. 3 s33.77 sin x = x = mπ, ±, ±, ±3,, [ g n x = Cx x + x ] [ x + x ] [ x + x ] π π π π nπ nπ, C x = sin x., n. x = sin x = a + a x + a x + a 3 x 3 +., a k k =,,,. sin x 3, a, a, a, a 3. g n x C 3 g n x 3 n., + + 3 + + + S. n 4
4 s43.78 S n = + + 3 + + n n. logn + < S n + log n. S n lim n log n =. 4 s439.79 α. sin x x α dx 4 s436.8 R A, supa, infa.,. A R, supa = inf A., A = { a a A}. A, B R, supa + B = supa + supb., A + B = {a + b a A, b B}. 3 A, B [,, supab = supasupb., AB = {ab a A, b B}..8,..8. lim n n sinh x cosh x.. 3. n! n! n cosh x dx x cosh x dx 4 n =,,,. x n cosh x dx = m= 4 s438 5 s533 n! m + n +! 4
5 s534.83 fx, y = e x y. fx, y.,. D a fx, y fx, ydxdy. D a, a >. D a = { x, y x + y a, x, y } 3 E a fx, y fx, ydxdy a E a.,. 4 3. E a = { x, y x a, o y a } e x dx 5 s536.84 N. A B. t t + t =,,,., A i, i =,,, N, α > i +, β > i, α β i., A B i = N i =,.. N = 3, α = /, β = /. t = i =, t = 6 A. t A i, t A xi, t. N = 3, α = /, β = /, x,. 3 N, i, α, β xi, xi,.. 4, xi,, xi,, xi +,, xi,, α, β. 5 xi, = xi, = x i i =,,,, N, x i x..85 f : R R fx = { x [x] x [x] < x [x] x [x] <. [x] x. x R.. gx = k= fx, x. f k x k x R, gx. 5 s538 43
3 gx. 4 n, g n = n n. 5 gx x =. 6 s635.86 f : X X X, f n : X X, n f n = f f, f }{{} = id X n =X.. f f n X f n X. n= n= f, f f n X = f n X. n=.87 φx = π e x < x <. n= 6 s636 3 φxdx =. e µx φxdx. y µ µy φxdx e dy., µ >. 7 s734.88 fx, y = e x y. I = fx, ydxdy, D = {x, y x, y }., D D a = { x, y x + y a, x o, y }, a >, lim fx, ydxdy a D a r, θ, I. 7 J =.89 n. e t dt. 7 s737 fx, y = n i x i= + n log y < x <, y > y,. n i x = nµ x + nδ i=., µ = n +, δ = fx, y. n i µ i=. n 44
8 s833.9 A : y = x e x,. A, x y =, x = a a < Sa. lim a Sa. 8 s837.9 fx f [, ]. x i = i n i =,,, n [, ],,. I k = x k, x k ] k =,, n S n = s n = n k= n k= n sup x I k fx n inf x I k fx S n, s n n, S = lim n S n, s = lim n s n. S = s f [, ],.,. S = s = fxdx fx = x, a S n, s n, b f [, ]. f [, ]. { x x fx = x x, n k= k = nn + n +. 6 8 s83.9 fx = e x x =. A,. lim n n n! + n + +! n! + n! A {a n } n=,. lim n a n = α, a + + a n lim = α n n 45
3 A..93 fx = πx exp 3 m { log e x µ } fxdx. fxdx = m. xfxdx. < x < ; < µ <. 9 s93 9 s936.94 {a n } n= α R..95 fx = {a n } n=, {b n} n=, lim a n = α R, n lim a nb n = αβ. n lim n b n = β R. 3 I R {a n } n= I n α I, f α I. lim fa n = fα. n 4 f : X Y g : Y Z g f : X Z f. + x + y R lim R R R y fx, ydx dy R.96 θ fx = R R fx, ydx dy n= sin nθ n xn R fx, ydx dy x < f x = sin θ x cos θ + x.97 α x fx = x α sin x > x 9 s938 R R y fx, ydx dy s4 [ sin t = e it e it] i 4 s44 α > fx < x < α fx x = 3 α > f x < x < α 5 s54 46
.98 a > e ax sin xdx 5 s544.99 fx = 3ex + 4 sin x e x + e x lim fx lim x n. {a n } n=. n+ n fx dx 6 s64 n a n, a n <. n=, lim n a n =.,. f. f [,. x [, fx. fxdx <. lim x fx =. 7 s747. fx = + x α,. + x α = + αc i x i i=, αc i. αc i = αα α i + i! i gx x 3. gx =. + x + x 9 s94 + t = t + t t 3 + + n t n + n tn + t t,. log + x = x x + x3 3 x4 xn + + n 4 n + R nx x >, R n x = n x t n dt. + t 3 x R n x n. 4 < x < R n x n. 9 s94.3 { n, D n = x, y R n x, y x }. n α < α <.. dxdy x y α. D n 47
I n, lim n I n. s47.4 x >.. F x = e xt sin t t dt lim F x. x F x., d e xt sin t { e xt } sin t dt = dt dx t x t. 3 lim x + F x. s43.5, ] fx. fx dx < fx dx = [, gx..6 α >. x f α x, gx dx < sup gx = x< x [n, n + f α x = n α., n. N+ f α xdx =. α > n α. α n= n= n α..7 x = x fx,. N n= n α 3δxe ixy dx., i =. s44 3 s34 fxδx x dx = fx δx 48
fx, gx, F y =.8 A = fxe ixy dx, Gy = gxe ixy dx, fxgz xdx e iyz dz,, F y Gy., 3 a fx dx <,.,. gx dx <. A, a ±. a = 4, A. x n+ + x x.9 fx = lim n x n + f, f, f y = fx 6 s643 9 s95 s63. gx = xe x x gx = x gx x 3 gxdx = 4 gx gx x + x + x 3 fx fx = f + xf + x! f + x3 3! f 3 + 4 s467. x + a x + b dx a >, b > s7. + x dx s7.3 f x f x = f n x = 3 Ex = x x f tdt, f x = f n tdt f n x n= x f tdt, f 3 x = x n =,, 3, d Ex dx f tdt 49
s73.4 lim + n n + n n n s74 {.5 S S = a a = p } k, k, p p < k a S a k p S 3 a S x n / S lim n x n = a {x n }.6 z n = lim n z n..7 A = n i n =,,,,. z n. 3 n= / / / / / / 3 s373 nz n. n=,. 9 s974 A λ < λ < λ 3 v, v, v 3. ε u n u n = A n + ε, n =,,,., ε = + ε, A., u n v, v, v 3. 3 x, x = x x. u n,. u n+ b lim n u n a lim u n n d u n n. u n c lim n u n 9 s977.8 ft [,. a, b a, b >, gx, y = fa x + b y.. ft [,. x = r a cos θ [, rθ [, ], y = r b sin θ π ] xy. 5
3 4..9. [, [. Ia, b = gx, ydxdy = [, [. π 4ab ftdt e a x + b y + dxdy. 3, a + b = Ia, b. s76 x e x log + log lim x x.,. fx = xe x [,.. fx = 5 x,. + x + fx = ax + b x + + c a, b, c. x + fx dx. 5 s57 3 fx dx. 7 s77. fz = x x + 3e z,. fz z =, a, a. fz = a k z k C z =.. fz z n dz, n 3. C k=. < x <, yx. yx. yx = x + x tytdt 7 s774 8 s876.3 n, a. xy D a = { x, y R x + y a },. I a n = D a + x + y n dxdy 5
I a. n, I a n. 3 n, lim a I an..4 lim + n. n n 9 s973 6 s68.5. e x + e y dxdy 7 s793.6 lim x log x =. x + fx [, ]., lim n /n /n fx dx n = f. /n.7 x, log + x x x. s94. lim n n k= log + n cos k = n cos x dx 4 s493.8. r dr. r dxdy I n = D n x y. {, D n = x, y x + y } n + n =,, 3,. 3 lim n I n., I n > π n. 5 s59.9 p >. D n = {x, y x n, y n} n =,,, D n dxdy I n p = + x + y p. D n I n p, Ip = lim n I np. x x +.3 lim. x x p, q, p >, p + q =. a, b, ab ap p + bq q. 3 Ipdp. 8 s89, a, b 9 s99 5
.3 dx x x +. x 3 + = a x + + bx + c x x, a, b, c. x + dx 3 x 3 +. 9 s99.3 e k lim λ e λ λ k = n g n λ = 3 lim λ g nλ.33 k= k= k k λ x n e x dx g n λ s3 a k k a [, s4.34 {a n } n= n n + n n n! 3 n a, a s.35 p {q n } n= q {p n } n= lim n q n = p lim n p n = q 3 f { fx = f x x x s.36 { lim x + 4x x + x} x + lim x cos x x log 3 lim + x x x 3 s3 53
.37 π fx fx = n= a n cos nx + b n sin nx fx a n, b n a n = π π π fx cos nxdx, b n = π fx π < x π a n, b n fx = x π π fx sin nxdx 3 s3.38 {a n } a n α n ε > n.39 A = n n n a n α < ε a n α n, a n β n a n + b n α + β n α, β λ, λ, v, v 4 s43 λ, λ λ > λ v, v a, b a n b n = A n a b a n a, b L = lim n b n L 4 s45.4 a > {a n } n= a = a n a n+ = a + a n a < a 3 < a 4 < a {a n } n=, {a n } n= 3 n a n+ a n < + a a n a n 4 lim n a n.4 lim β > + 3 α > n n k= β n n + k 5 s5 dx α > xlog x α dx xlog x α 54
6 s6.4. n a a > a + n= n= 3n 3n + 3 n= 3n 3 n 7 s7.43 n, I n = π sin n tdt.. x, + x e x. x n e nx x e nx + x n < x <. 3 x = cos t, 4 x = tan t, 5 x n dx = I n+. + x n dx = I n n. n In+ e x dx n I n n. n= 7 s73.44 n nn + n + n= 8 s8.45 < x fx = x + C x = p < p xn C n lim fx lim fx x x + 3 fx f x 4 C p n 5 p n 6.46 F x F x =.47, df x dx x e t dt π lim F x = e t dt = x.. x. e t dt F x. 8 s84 3 e t t =, x =. F x. 4. a e ax sin x dx a <. I n = nπ n π e a t dt a b t e t dt s e ax sin x dx, I n I n+. 55
S n = 3 n i= I i, lim n S n I. e ax sin x dx = lim S n. n 4 I, e ax sin x dx..48 {a n } a 3 = 4 3 a 5 = 6 7, a n. n= s5 4 s44.49 fx x =., e, a, b. fx = { e x ax + be x x < x a, b. fx, y = fx. 3 fxdx. 4 s49.5. < a < a, lim n nan =. < a < a,.. 3. 4. n= na n sin 3x 3 sin x lim x x 3 lim x,y, x y x + y 4 s44.5 fx = x e x,. fx x. y = fx. 3. fxdx 5 s53 56
.5 3 V x θ V. 3 3 Uθ V = Uθ V.53., Uθ Uθ = cos θ sin θ sin θ cos θ.. Uθ det Uθ. Uθ Uθ. 3 K K =, K, K 3, K 4., m, K m K m. 4, X π e X = E + X +! X + + n! Xn + = n= n! Xn., E, X = E. 3, e θk = Uθ. e x cos x dx..54 a = e, a n+ = e n + a n., e. a n n. a n. n=.55 tan x = 6 s67 7 s78 n =,, 3, a n, 7 s79 a n x n π < x < π a n a a 5. n=.56 a > n, + b n n = + a n 7 s77 {b n }., fx fx = + a x.,. 57
b n a n. fx x = f. 3 lim n. n 4 b n b n+. 8 s83.57 A, B, C. A = B = i C =, I 3 3., i i =. A + B + C = ki, k. A. 3, M e M, e M, e iαc =. e iα e iα 4 v φ v φ = sin φ + i cos φ n= n! M n. α., e iαc v φ = v φ, φ...58 nn + = 3 + 4 + 3 5 +.59 A = n= A EA + E,, φ φ 9 s97 4 s4 n E x n x x + a nx + b n a n, b n n 3 A n = a n A + b n E 4 lim n An n.6 lim 3n + n n 8 s84 3 s33 58
.6 T ft,. n =,,, 3, C n θ n, a n b n., ω = π T. ft = a + a n cos nω t + b n sin nω t = C + n= C n cos nω t θ n n= a n b n, ω = π T, a n = T b n = T T/ T/ T/ T/ ft cos nω t dt, n =,,, ft sin nω t dt, n =,, 3,., ft., T ft = t <, t T 3 sin t sin 3 t,. π 4 4 = 3 + 5 7 +. 7 s76.6 t[s] ft F ω = fte jωt dt j = ω [rad/s] a {, < t < a ft = F ω ft F ω, t <, t > a F ω F ω { exp at, t > ft = F ω ft F ω, t < 3 ft a F ωe jωa 4 fat a > ω a F a 8 s85.63 fx = log + x f n x, n =,,, gx = log + x gx gx x a n x n n=.64 ft ft, f x, f x s > gs = e st ft dt e st f x dt = s gs sf f 59 s3
ω ft = sin ωt gs.65 sin x sin x s lim x x sin x 3 s3.66 sinh t = et e t, cosh t = et + e t,. n sinh n t, cosh n t n =,, 3,. { x = cosh t. t, x y. y = sinh t, < t <, x, y..67 DR = {x, y x + y R } R >, I a R = x + y dxdy a > a.. I a R. DR 6 s6 lim R + I ar..68 DR = { x, y : x + y R, y x } R > IR = x + y e x +y dxdy.. IR. DR 6 s6 lim R + IR. 7 s73.69 A = k 3 k 3 k, < k <,. A. A. 3 A n = a n c n b n d n n =,,, lim n a n + d n. 8 s8.7 fx f x = fx, f =.,. e x fx =., fx. fx. 3 lim log n! 3! + 4! + n n!. 6
.7 A = 4.. 9 s9 A. n, 3 n A n. 9 s94.7 a, a,, a n M n log + x a k x k Mx n+ x, a, a,, a n. lim n k= k= n k+ = log. k.73 n, I n. I n = I. 9 s98 x n e x dx n,. I n = e + n I n 3. I n = I n 4. lim n n! n! e,. n k= k! 9 s9.74 n, D n = {x, y R x + y n, x, y } e x y y dxdy. D n e x R n = {x, y R x n, y n}, y dxdy y dxdy =. lim n R n e x 3. e x D n e x dx 9 s9 x, y.75 x = u, y = uv u, v. D x dxdy, D = {x, y x, y x}. + y 3 R >, D R = {x, y x R, y x}. α, lim R + DR x α x + y dxdy.76 x n, R n x.. arctan x = x 3 x3 + 5 x5 + n n xn + R n x., arctan x y = tan x π < x < π. s3., 6
t + t 4 + t n = t n + t, R n x = x R n x n. x t n dt. + t s8.77 tan x π < x < π fx < x <.. d dy tan y = tan y + π < y < π. n =,, 3,. x + f x = < x < x + f n+ x + nxf n x + nn f n x = 3 m =,, 3,, f m+..78 x = u cosh v, y = u sinh v. s u, < v < x, x y 4., cosh v = ev + e v, sinh v = ev e v. x, y x = u cosh v, y = u sinh v u, v. 3 D logx y x dxdy. D = { x, y x, x y 4 }..79 fx, y = e x +y,. f x + f y. D a = {x, y x, y, x + y a }. f lim a D a x + f y dxdy..8. n I n =. x + x dx. s3 3 s33 x n + x n+ dx, I n I n+ 6
3 I n..8 A =, 3 s36 A p i = λ i p i, p i = i =,, 3, λ < λ < λ 3., p p.. λ, λ, λ 3. p, p, p 3. 3 x = x p + x p + x 3 p 3. lim n An x = x, x, x 3..8 x > fx = log x x f x < x. N 3,,. N fn < n=3 N fxdx 4 s4., e < 3. 3 fx. 4 fn..83 A = n= 3 4 5 λ, λ, λ 3 λ < λ < λ 3. 4 s46 i =,, 3, 6 6 λ i u i.,. i =,, 3, λ i u i. v = u + u + 3u 3, x n y n z n = A n v. x n lim. n z n 6 s6.84 x, y, z A x y z = 4x + y + 4z 4x + 3y + z x + y + z 3 3 A.. A. A λ, λ, λ 3 λ < λ < λ 3, p, p, p 3. 63
3 B A, n, B n µ n, µn, µn 3 µ n < µ n < µ n 3. a n = µn µn 3 n =,,, µ n a n. n=..85. 7 s7 dx x + 3x dx x + 3 dx x3 + 7 s76.86,. d x sin x dx e x dx 7 s79.87 λ, t >., D t = { x, y R x, x + y t }.. It = + x + y λ dxdy D t It t. t, It,..88 n, I n. I n = I. 8 s85 x n e x dx n,. I n = e + n I n 3. I n = I n 4. lim n n! n! e n k= k! 8 s84.89 n, D n = { x, y R x + y n, x, y } e x y y dxdy. D n e x { x, y R x n, y n } R n, lim y dxdy y dxdy =. n R n e x 3. D n e x e x dx 8 s86 64
.9 P = v =...9 P... v. 3 P n v. lim n P n+ v lim n P n v. x log + x x x >. + x {a n } lim n a n = a, 3 {b n }. lim n..9 < dx x 3 < π 6. n...93 fx = { e /x I = I n = lim n + b n n 8 s837 + a n n. n n, {b n } e t t n dt n k k! + n+ n +! I n+ k= x > x x >, f x f x. n, lim y yn e y =. 3 f = f = f =. 9 s93 9 s95,. 9 s97.94 3x + dx x x dx 3 + xe x dx s33.95 {a n } {b n } a n n b n n,. a + a + + a n n b n n 5 s538 65
.96 {a n } n=, {a n}, {a n } α {a n } α. 6 s635.97. x + dx.98 e x y dxdy x = r cos θ, y = r sin θ, r, θ. e x dx..99 < a <,. 7 s73 s35 a n e inx x R R., i. n= a n cos nx x R. n= 3 s333.3 ;,. 3 a n = a n. n= n= a n = a n. n= n= a n = a n. n= n= 4 s439.3 I = [, {f n x} n=,,,.,. f n x = n + nx I fx = lim nx. n. 5 s539.3 fx R., x R., a, b. a fx x = x. { } b x y n, lim fy n = fx. n= n 66
.33 { a n } n=, 7 s73, lim sup a n = lim inf a n n n { } a n. n= { }, lim sup a n, lim inf a n a n,. n n n=.34 N. a n n N,, f : N N 8 s833 a n α. n= a fn = α. n= 8 s834.35 fx = + kxe kx x,., k. IX X., X >. IX = X fxdx k =, lim IX. X 3 lim IX, fx. fx X, k..36 k α x d dx fx α d fx + kfx = dx 9 s935 fx dx fx α.37 d xt dt axt = a s47 xt a >, a < xt x = x, dx dt = a > lim t xt =.38 m n 5 s54 67
a b c d π π π π sin mxdx =, π cos nxdx = sin mx cos nxdx = { m n sin mx sin nxdx = π m = n { m n cos mx cos nxdx = π m = n xt T xt xt = a + k= a k cos πk T t + b k sin πk T t a k k, b k k xt 5 s547.39. 4x x lim x 6x + 3x lim n n i i= n 3 lim x cos x x 6 s645.3 m [kg], x, m d x dt = c dx mg dt., g [m/s ], c [kg/m]. dx lim t dt. 7 s74.3. x + 3 x dx e 3x e 3x dx 7 s743.3. x 3x + 5 lim x x lim x x sin x 7 s747.33 fx = log + x x > 4 f x, f x, f x, f 4 x., fx n, f n x,. 3, fx x =,, n= 68
4 3, gx = log x < x <,,..34 fx = x + 4xe x. f x, f x lim fx. x x <, y = fx..35 yx. d y dx + α dy dx + y = 4, α,. s4 s4 dy α = 5, y =, =, yx. dx x, α > yx y p lim yx = y p. y p. x 3 yx, yx α..36. s4 5 x 3 x lim x 5 x + 3 x lim cos x x 3 lim + 8x x x x s43.37 a, b, 3,., r, θ, φ, x, y, z, x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ r, θ π, φ π. x xy + y = 4..38. sin x lim x x.39 A = 3 lim x sin x x. lim x log x x x n 4 lim x e x s4 n s47 A. [ ] [ ] lim n An x = x. n, x = x. 69
.3. lim x sin x x cos x sin x x 4 lim 9n + n 3n n e x lim x + x x n 5 lim n n s49 n + 4n + 3 lim n 3n + 4 s43.3. fx = 5,. a y s. d y dy + a + 5y = fx dx dx b C e αx + C e βx, x a. dy a =, fx = sin x, y =, dx =,. x= α, β 3 s34.3. sin x lim x x =, lim + x = e x ± x sin x lim x + x lim x log e + x x 4 s443.33. fx = x= f n x n n!, f n n. fx = e x n = 4, e. fx = sin x fx = cos x, n = 5. 3 e ix = cos x + i sin x ;, i =..34., α >. dx x + x + α 4 s444.35. + x x lim x 3x 5 s54 7
x x sin x lim x x 5 s545.36 t < ft = ft,. L[ft] =. fte st dt ft = e at ft = t.37. lim x sin x sin 3x cost τ cos τdτ 6 s646 4x 3 3x + lim x + e x 7 s74.38 I = R, e x +y dxdy R : x + y a., a I e x dx..39 y x, y + a = a y,., y =, y =. y I = 7 s746 y dx. I a.,, a I.,..33 dx dt = ax y 6 dy = 5x y + dt 7 s743,., xt, yt xt rt = R., a 5 yt. t rt r R, a., dt = rt r = Xt Y t R, Xt Y t. 7
3 dt, t + a. 4 a = 4, r = rt..33. dx x ln x e x dx.33., ẋ = dx dt. ẋ = x x K, K. 8 s84 8 s84, t, x. t =, x = K. t - x. x O t 3..333 8 s844 x x + dx 6 s65 π.334 4 e x sin x dx dx xx + 8 s854.335 a > Da = {x, y x a, y 3} y Ia = + x + y dxdy Da Da Ia lim Ia a.336. 3 5 D log x x dx dx 5 + x logx + y dxdy, D = {x, y 4 x + y 9} 8 s855 s5 7
.337 fx = e sin x fx = a n x n, a, a, a, a 3..338 lim x x. x n= s55 fx = 4x + 6 x 4x + 3. 3 s35.339 f Maclaurin fx = a n x n. n= fx = e x fx = log + x < x <.34. 4 s454 x 3 + dx 5 s554.34 x x dx log + xdx 3 x.34., a >. I = x < a x a.343 px px = + a x < a x a + x a a a < x e ax dx xe x dx a >.344 dy = yy dx xpxdx,. 5 s56 7 s76 x pxdx y = a., a. 8 s869 yx, lim x yx. 9 s965.345 3. x m x n dx π cos x n dx 3 e x dx s6 73
.346 xyz C l,., P, Q, P Q. x = cos θ C : y = sin θ θ π l : z = cos θ + sin θ x = t y = 3t z = t P C, Q l, P Q θ t. < t < + P C, Q l, P Q,, P Q. s64.347 a. tan x = ax + x,., tan x x, tan =. 3 s36.348 yx y + y + ay =., a. a =,. a >, y lim yx =. x.349 Γt.., Γt = e x x t dx, t >. 3 s367 t > lim x e x x t =. t > Γt + = tγt. n Γn + = n!. 3 Γ, e x x dx e y y dy x, y. 4 x, y r, θ o o { x = r cos θ, y = r sin θ x, y r, θ. 5 4 3 Γ. 74
6 s66.35 m.. Im = x m + dx I, I. r, N,. N r + = n r n + N+ r N+ r + n= 3..35 lim x sin x x lim x x + x x.35 n n= n = + 3 4 + 7 s76 s76 a 3, 5 4, 7 7, 9, 3, b, 4, 6 3, XOY = π/4, P P a OY P OX P P OY P 3, P 4, P 5, a b Y P P 3 a P 5 O X P 4 P 75
3 n= n + 4n + 3 s74.353 + 4 + 6 + + n lim n.354 fx = xe x fx n 3 s377 fx y = fx 3 fx x x = q q > Sq lim q Sq 8 s876.355 ft F s F s = fte st dt Re[s] > { t = a ft = δt = t { t b ft = ut = t < { t t c ft = t ut = t < { d ft = e αt ut = e αt t α > t < δtdt = RC a ν i t ν o t it ν i t t = ν o t = R it C ν o t b ν i t, ν o t it V i s, V o s Is a Gs c ν i t Gs = V os V i s ν i t = ut = ν o t V o s { t t < d V o s ν o t 76
e Gs gt f gt c ν o t 8 s878.356 A, B, 9. A, 9 9. B, 9 m m 9m.. A,. B,. 3 A, B, m = 3 m. 4 A B n n 9. A B,, m n. 9 s975 x.357. lim x + a x x., x. exp sin x 3., x ± a. log x a x + a.358 In, a =. n =, a =, I, = a >, s7 e ax x n dx., n, a e ax dx = e x dx = π. π a. In, a n a., n. s7.359,,.,. 3, 5, 8. a,.,,. b,. c,. 3n, 5n, 8n n., pn. lim n pn. s7 77
.36 a. lim x x + a x + x + =. fx = log x + + x 3 s375.36 x m m > lim x e x = s8.36 U, V 3., U V, V U n, U p n,, q n, r n.. p, q, r. p n, q n, r n p n, q n, r n. 3 p n, q n, r n p, q, r, lim p n = p, n lim q n = q, n lim n r n = r. p, q, r. 7 s785.363 l A, B, A, B x, y. A, B, C, D, C B, D A n, A, B x n, y n,. x n, y n, T. x n y n = T x n y n T. 3 x n, y n, x, y n. 4 n, x n /y n. s8.364 A x x A = A E B A 4 9 E A = EBE E B α β 78
3 E B A n A n n A A = EBE A 4 V V x = dx, y = x y + x y x, x, x 3, x lim dx n, x = n x a = a a A a, Aa, A a, A 3 a, A n.365 fx, y.366 fx, y = { + + tan x + x dx s9 x x, y = y y, 3 9 s994 xy, x + y, x + y > fx, y dx dy s95.367 x + y R A A a > e ax ay dx dy A x = r cos θ y = r sin θ + e ax dx s94.368 lim x tan3x x lim x x + x + x s9 79
.369 Γa = e x x a dx a > Γa + = aγa a Γa = a!.37 n= n + 4n + 3 s93 log + t = t t + t3 3 + n tn +, < t n fx = + x log x 3 x.37 fx [,,., fx. f x x + fx = x + e x f = fx..37 n= 8 s895 s97 n + 5n + 4 3 s396.373 P x, ydx + Qx, ydy =, P x, y = Qx, y, y x. fx, fπ =. y sin x fx dx + fx dy =, x > x,. fx.. 4 s498.374. lim 3 x 6 + x F x F x = x x,.. dt, 3t x >., F x 5 s597 8
.375, X F x α > {F x F x α} dx = α P X = = p >, P X = = q >, p + q = X fx.376 y = fx = 3 x x + 3.377 I = s34 lim {y mx + b} = m, b x ± x = /t y = mx + b 3 fx 4 3 fx x y e αx dx = x e αx dx. π., α >. α.378 x 5 s53 7 s73 fx = 5 e 5 x x >, e 5. y. fx = 5 e 5 x O 5 5 x fxdx 3 8
4 5 lim x x e ax =, a >.379 N 8 n.38 I = 8 s839 N n = 8 N 63 n lim n e αx dx = log N n x e αx dx π α > α 8 s834 8 s839.38.,,. lim x x e x n e an n=.38 px = xe x, qx =, a > { x x <, fx = s33 xy.. pt qt xdt s3.383. 6x 5 x dx 9x e 3x dx s38.384., a a >, e..385 I = xe ax dx s38 π exp ax dx, I I = a., a >. fz = i fn + = n!, x exp x dx. t z e t dt, n ii fn + = n! π n n!. s3 8
.386, 3. 4x + 4x + x + 6 dx 3 t e at dt dx 4 x 4 s43.387 y = fx x = a, x = b x,. b a fxdx = lim n k= n fx k x, fx [a, b], x = b a/n, x k = a + k x.,, lim n xdx.,. n + + n + + n + 3 + + n. 7 s73.388 = dy + aydx y = fx 3..389 Γs =, a a >. f = a,, fx x. gx = xfx, y = gx. 3. lim a gxdx fxdx e x x s dx s > 9 s933 Γs + = sγs Γn = n! n Γ = π e y dy = π 3 s335.39 yρ d yρ dρ + ρ { dyρ dρ + } ll + ρ yρ =., ρ, ρ <, l., yρ yρ = uρ ρ,, uρ d uρ dρ. pρ. + pρuρ = a 83
, a ρ ρ uρ. a ρ pρ, uρ., uρ, λ uρ = e λρ b. λ, uρ. ρ pρ, uρ., uρ, λ uρ = ρ λ. λ, uρ..39 fx = x x + 4. lim fx lim x x fx. fx..39 a. xy C : y = x + a < x <, C : y = log x x > 6 s635 9 s93... a. y = a C. 3 C, C y = a..393 fx = x + xx + x + x x 9 s933 x,,. lim fx lim fx. x x fx. s3.394 fx = x x +. lim fx lim fx. x x fx. s3.395 F n. F n x n e x dx,., n. 84
F. n. F n = n F n.396 fx = x x x + x >. lim fx. x fx..397 I.. I = + + e x dx xe x dx, I. I = + s36 s3 e x +y dxdy 3 x, y r, θ. I, I = π., x = r cos θ, y = r sin θ, dxdy = rdrdθ..398., a, b..399 fx = e ax sinbx dx x + x,. + x + lim fx lim fx. x x fx. s34 s37 3 s33.4. xe ax dx a > x α log x dx α > 3 s335.4., a. π+a π+a exp axdx a sinmx sinnx dx m, n 85
4 s43.4 fx = x + x + x,. lim fx lim fx. x x fx. 4 s437.43 fx = x + x + x >,. lim fx lim fx. x x fx. 5 s53.44. xe x dx x x dx 3 xx + dx 5 s535.45., a. xe x dx dx 3 + x e ax cos xdx 6 s63.46,. n n= n= n n 3 n= n 6 s636.47., a, n. r 3 exp ar dr π/ π/ sin n x cos x dx 7 s73.48 fx = ex e x e x + e x < x <,. lim fx lim x x fx. fx. 7 s737.49 a =, { } a n n=. a n+ = a n n =,, 3,,. { } a n, n= a n <. { } a n. n= 8 s833 86
.4 a, e ax dx = x e ax x dx. π a, 8 s836.4 {a n } n= a =, a n = a n + a n +.. n a, a 3. n a n = a n + a n. 3 lim n a n =..4 x fx = e x sin x,. f, y = fx. f. 3 4 fx dx <. fxdx. 9 s93 9 s933.43 fx = arctan x. x fω = ω ω + 4γ ω., ω, γ. ; a lim fω. ω b ω >, fω ω γ. ω. 9 s934.44 fz fz = z e z z z = z < π fz = n= B n n! zn B n, n =,,, B =, B = e z fz = z B n n n B k = k k= 87
n k 4 s4335.45 N N = n pn = n = a n e a /n! < n >= npn = n a S M A L pl = l p = A/S pl = l = [M!/l!M l!]p l p M l a = M/SA.46 Ia = exp[ x ] cos ax dx 4 s4337 x, y r, θ x + y dxdy π I I 3 Ia Ia a.47 w 8 s8334 + w = w n, n= w < w = w = z Log + z z =, Log + z π <Im log + z π log + z z 3 lim + z n = e z n n.48 x > log x =. x lim nlog + z = z n n, a d. 8 s8337 dt, log x. t a log x x = log x + log x x, x > b log x n n m = m log x m, n c log e = e = lim + n n n d d dx log x = x 88
log x expx, e h. e expx + x = expx expx f exp = e n g exp = e m n m m, n h d dx expx = expx s33.49 fx = x a ibx,., x, a b, i =. fx fx = A x z x z, z z,.., A, z, z ζ,, Iζ = lim ε ζ ε Iζ = fx x ζ dx fx x ζ dx + fx ζ+ε x ζ dx., C, C, C 3, Iζ., C R, C ζ ε, C 3,. a C R. b ε, C. c C 3. d, Iζ. y y ζ O x O ε y x O ζ x y ζ C C C 3 O x,, C, C, C 3. s335.4 R x, y t { C : x = t y = t 3 t < t < 4. 89
C x., y. C., C. 3,, t ±, C. 4 C. s33.4 a n A n, n B n. a C A n C, lim. 3. n B n C s33.4 X fx, M X t M X t = p qx = e tx fxdx., px, qx p qx puqx udu. 3.,. X, Y, fxgy., Z = X + Y hz hz = f gz. X, Y, M X+Y t = M X tm Y t. 3 X m X σx,. m X = dm X dt, σx = d M X dmx t= dt t= dt t= s333.43,. gz = e iz gz,. a, gz < z < a gz = c n z n, c n C n = m, m +, m +,, c m, m Z n=m. a,. 3 m, c m, c m+, c m+. 4.,. gz dz z =.44 {a n } a = 3, a n+ = a n + 3 a n 3 n =,, 3, s338 9
n n a n > 3 n n a n+ 3 < an 3 3 lim n a n = 3 4 s433.45, A, B. k k a k %, b k % a k + b k =. k +, [ ] [ ] S = s s a k, x k = s s b k, x k+ = Sx k. [ ] x = 9,. [ ].9. S =...8 [ ] a x n = [ b x 3 = a n b n a 3 b 3 ].. c n. [ ] S =.6.4.4.6. n. 5 s533.46 lim x + x + 3 x x s34.47 lim x x e x s34.48 x + x dx s345.49 n =,,... n! n n! n! n=.43 p > Γp = n= x p e x dx x a lim = a x ex {.43 lim x x log + } x x + 3 s3344, Γp + = pγp 4 s4344 5 s534 9
.43 x x + x + dx 5 s5344.433.434. dx x + x 8 s8347 e x + e x + dx 9 s9343.435 ft F ω F ω = + fte jωt dt., t, ω, j =., ft = { e t t t < ft F ω F ω..436 R, IR = R IR. x 3 e x dx. s345 lim R IR. s34.437 lim x log + 3 lim x x x log + 3x x. log + 3x dx..438 dx x 3 +. logcos x.439 lim x x lim cos n n n.44 I = [ π, π] fx Rk = π π 3 s334 dx x 3 +. 8 s834. 9 s934 fx cos kx dx, k =,,, a < a n fx = a j cos jx j= 9
lim n k= n Rk fx I Rk R, k =,,,.44 I n = x n + dx x I, I I n + I n lim I n = n k 3, = log k k= 8 s835 9 s935.44 fx fx = π sin y x cos y + x dy f fx 3 fx 4 x fx dx s35.443, xt xt = t sint u xudu + t xt 3 s3354.444 a, b >, a b u t + a ut = < t < v t + b vt a ut = < t < u = v = ut, vt vt 3 vt 4 s4353 93
.445 e x dx = π e itx x dx x + x + x + dx 4 s4355.446 fx = x π x π fx = a + an cosnx + b n sinnx n= a n n, b n n n = π 6 n= 4 s4356.447 s, v, A s = E sv t v. E, t v v., v...448 A s s., A s, A s. 3 x x i+ = A s x i i =,,,., x, x i s., x x v., x i x, x i x. n+ n + 3 n n= N S N = N n+ n + 3 n n= 6 s6356 N S N + a N = a N N C j a j + Na + j= NN a N 8 s835.449 [ π, π] ft. {a k } a k := lim a k = k π π ft sin kt dt, k =,,. a k < k= 94
{a k } π lim N π N ft a k sin kt dt = k=.45 N S N = n= n= π π ft dt = k= a k N u n., n= 8 s835 lim N S N = S S. u n = nn+n+3 = 4 + 3 5 + 3 4 6 + 4 5 7 + + nn+n+3 + u n = n + r n = 3r + 4r + 5r 3 + 6r 4 + + n + r n + n= n=, r <.,.45 x, y, z r, θ, ϕ : lim n nrn =. 9 s9355 x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ r >, θ π, ϕ < π,. e x +y +z Iα = x + y + z dx dy dz α, α = I. 3 Iα α. 4 Jα, β = B x + y + z α logx + y + z dx dy dz β α, β. {, B = x, y, z : x + y + z < }. 4 9 s9356.45, n,, n., Q, n. n. i M i. M i n. i, i,, i m n m. M i, M i, M im n m. 3 E P E. A A P A A, A A P A A. P A A = P A + P A P A A. A, A,, A N P A A A N N P A A A N = l S l S l = l= k <k < <k l P A k A k A kl N i ii 95
., ii, N,,, N l k, k,, k l. l = N S = P A j. i. j= 4 Q, n = n j j! j=. 5 lim Q, n. n 9 s9358.453. sin x + x dx 9 s935.454 fx = x 4 π x π fx = a + a n cos nx + b n sin nx, a n n =,,,, b n n =,, 3,. n+ n = π. n= 9 s935 n=.455 X, Y N,, W, W Z., W = X + Y, Z = W. N, fx, Φt Φt = t fxdx W..,. z, P Z < z Φ. 3 Z..456 n, k, I n,k. I n,k = I,k. 9 s935 n k <,. I n,k = n k 3 n k <, I n,k..457 A = 4 x, k C k,. 5 4 n k, I n,k = lim L a.5 a +.5 L x n + x dx k+ I n,k x n x + k+ C k x k n+3 x n + x dx =. k+ s35,., a. Ax = x =, a. x, x, b, b. A x x = b b x x, a, b, b.,. 3 A, A. 96
4 A n., n. 5 x, lim n + An x = a., lim n + An x = A n x. s35 dx = y + ax 3 + xy dt.458 dy = x + ax y + y 3 dt, t =,. a. xt, yt. a =, t. a >, t >. 3 a <, t >, t,..459 a <.. a n cos nx + a n= n= a n cos nx = a cos x a n cos nx n= cosn + x + cosn x = cos nx cos x. + a n a cos nx = a cos x + a 3 π n= s354 cos nx πan dx = a cos x + a a s356.46 {a n } {b n },. a = 3, a n+ = a n b =, b = 4 b n+ = b n + b n+ {a n } N A N. n {A n },. 3 n c n = b n+ b n {c n }. 4 {b n }. 5 n {b n },. s35.46 4y x + yx4e x = x R. t = e x zt = ylog t. 97
zt = c k t k+ρ ρ R, c. k= ρ c k k =,,., ρ = ±/. 3 c =. c k, z t, z t. 4 y x, y x.46 n. e x y x + y x = x R s355 t n I n t = cos x k n k= π n n + dx dx n t R xk. k= I t. I n t. s356.463 [ π, π] fx fx = x sin x π x π..464 A = fx. a n = π b n = π π π π π a n n =,,,,. a b b a a c, A expa. fx cos nx dx n =,,, fx sin nx dx n =,,, + n= a n = π 3 s357 expa., a, b c., E expa = E + n= n! An 98
A. A,. A P. 3 A expa. 4 expa expa..465 n, {a n },, lim n nrn = r <. a n = n k=n k n+ a n = k= k k s35 a n. n= s35.466 f,. f ˆfk., z. uz = ˆfk = ˆfkz k + k= e πikt ft dt k= ˆfk z k r,, uz z r. u = uz, z = x + iy,. u x + u y = 3 r,, z = re πiθ,. uz = r ft r cosπθ t + r dt s359.467 n, {α n } {β n }. α n β n. f, g n= n= fx = n= α n e πinx. gx = β n e πinx n= 99
px = fxgx qx =. n= fx sgs ds c n e πinx c n. n= d n e πinx d n s35.468, n B. i, j B ij. n x = t x, x,, x n, y = t y, y,, y n, n x, y = x i y i t. B det B, B. i= s b >,. πb e ts e t bs b dt = e n y,. e x,y e B x,x dx πn det B dx dx n = e By,y 3 j, k n,. x j x k e B x,x dx πn det B dx dx n = B jk 4 n A,. πn det B Ax, xe B x,x dx dx dx n = TrAB, TrAB AB. s35.469 {a n },., n. a =, a = 6, a n+ a n+ + a n = n b n = a n+ a n {b n }. {a n } 3 S n. 4 n S n. n S n = a m= m 3 s335.47 gx π.. lim m π gx sin mx dx =
p n x = a + n a k cos kx + b k sin kx. n =,, k= π lim m π gxp n mxdx = a gxdx 3 p n x n + px [, π] lim m π.47 {a n }, {b n }, {x n }, {y n } P = x y =, x n+ y n+ gxpmxdx = π = A 3 5 x n y n π pxdx π gxdx 3 s335, A = 4 3 + 3 n. n.. P AP. {a n }. 3 n c n = n b n {c n }. 4 x n y n. n=.47... n= 3, a n b n = P x n y n 4 s435 log + x x x + x3 3 lim x x 3 = a e x + x + x lim x x = b log + n n n + 3n lim = n n 3,,. { lim n n + n n e n + } 4n = 4 s4355
.473 x fx = π cosx, x < π π fx fx = a + {a n cosnx + b n sinnx} n=, a n, n =,,,, b n, n =,,.. n= n 4 s435.474 {a n } A n, n A n a n. A n A n+ : A n+ a n+., a =, a =.. a 3, a 4, a 5, 3, 7 9, 7. a n+ a n+ a n. 3 {a n }, n a n..475. lim R Γ R, R z z + eiz dz = lim R Γ R Γ R = {Re iθ θ π},. 5 s535 z z eiz dz =. 3. I = J = x x sin x dx + x x sin x dx 5 s5356.476 X., k, C k px = C k x k e x, x pxdx =.
C k, E[e tx ], t. X,, X n, px,, q n t = E[e tx + +X n ], t. 3. lim n q n n 5 s5358.477 m, k. x, xt { x t + xt t m xt k = t >, x = x.. yt = xt k, yt.. t m e k t dt 3 xt, lim t t xt = t xt. xt x. 6 s6354.478 {f k x} k=.. I R.. f k x <, x I k= max x I m f k x k= n f k x m, n k=,, f k x I. k= f k x I.. k= fx = x x [ π, π]. sx = a + a k cos kx + b k sin kx, x [ π, π] sx [ π, π]. 3 sx [ π, π]. k= 6 s6356 3
.479 N. N... n X,, X n,. T = max{x,, X n }, T N. A P A.. P T t = { P X t } n P T = t t =,, N. t =,, N. 3 ET. 4. ET = N N t N t= ET lim N N = n n + n 6 s6357.48 x y = sinh x = e x e x / sinh x. sh x.. sh x. a Sa = a a sh xdx. Sa.. 3 lim Sa. a 4 lim a 7 s7355.48 m 6, n 3 n m. m m,,, 3,, m. m n n., n.,. m = 8,, 8 3. n,, m P n,m. P n,m n m., lim m P n,m. 8 s8353.48 a, b, c b ac >, b >. D = b ac.. dx + bx + cy =, t dt dy ax by =, t dt x = xt, y = yt x =, y = b, c, D t. y = yt, yt > t. 3. zt = xt yt, t dz dt + az + bz + c =, z = 4
dzt 4 lim zt lim b, c, D. t t dt 8 s8354.483 α. cosαx π < x < π.. cosαx = a + { } a n cosnx + b n sinnx n= a n, n =,,, b n, n =,, y sin y.. sin y = y + { } n y + nπ + y nπ n=, π < x < π.484 xt, yt,. dx = x y + cos t dt dy dt = x xt yt. x = y = xt yt. 8 s8356 3 t xt A cosωt + θ, A, ω, θ., A, ω, θ, A >, ω >, θ < π., θ..485 A = 4. 8 s8359 A. A P. 3 A,, A P. 4 3, Ax =. 5 3 P T P = I, n A n x = x n P, n A., n x n 3., P T P, I. 3 8 s835 5
.486 5 dx x + x + 9x + 4 8 s8363.487 z, e z = 3i z., i. a >,. x x 4 + a 4 dx 9 s936.488 n, J n = x + y n/ e x +y dxdy.,.. l, lim t l e t t. = x = r cos θ, y = r sin θ, J n r. J. 3 n, J n J n, J..489,..49. x 4 + x 6 dx = t. t s36 x 4 x + x 4 x + dx s + 3 t ds = t + s369 s + 3 ds. 3 s336.49 Xr, θ = r cos θ, Y r, θ = r sin θ D =, π, π.. A = { Xr, θ, Y r, θ r, }, B = { Xr, θ, Y r, θ θ π, π }., θ π, π, r,. X r r, θ Y r r, θ Jr, θ = Jr, θ., X θ r, θ Y θ r, θ. X r = X r, Y r = Y r, X θ = X θ, Y θ = X θ 3 X r r, θ, Y r r, θ, X θ r, θ, Y θ r, θ. 6
3 s336.49, z.,., i. z 3 = + i,. I =.493 n, I n I n = I. I n+ I n n. 3 I n. 4 lim n I n+ + I n+ + + I n. cos x x 4 + dx 8 s836 x n + x dx,. n+ 8 s8368.494 a =, a n+ = + a n n =,, {a n } {a n } lim n a n 8 s838.495 fx = a x < a fx x = x n lim = < a, x ax lim.496 a =, a n+ = a n + a n x ax x n = < a < a n a n+ n =,, {a n } lim n a n {a n } n =,, 8 s8384 9 s9384.497 lim n n + + n + + + n + n.498 I n =.499 n= x n e x dx I n I n+ I n+ s38 s38 n = s38 7
.5, fx a b, < t < fta + tb tfa + tfb fx, f x fx 3 s338.5 R D R = {x, y x + y R } I R = +y dxdy D R e x lim R I R.5 n n= n + + n + + + n > n = e x dx 4 s438 6 s638.53 x { } n lim + cos n n n x + cos n x + + cos n x = sin x x 6 s638.54 fx x. fx >,. f + f + + fn F x = lim n x ftdt., f + f + + fn F n n fxdx f + + fn + fn. lim n =. fn =., F n 9 s9385.55 {a n } lim a an n. a n+ = n a = a = 4, a n = a n =,, 3,. a <, lim n =. n 3 a > 4, lim n =. n 4 4 > a >, lim n =. n.56 < a < a, A = n A n. a a 9 s9387.,. n S n = E + A + A + + A n., E =, lim n S n, E A.. s387 8
.57.58 I = dx + x 4. x + y + z x + y x., A =.,. n =,,, A n.. n π k A k S n = I +, lim k! S n. n k= s384 s386.59 { } an, n= a n + n =,, 3, a n+ n.,. n a n a n. a n. n= 3 s3383.5 x = xt t C.,. d x dx dt + 4 = dt, x = dx dt =. 4 s4385.5. n. n= m,. m = 3,,, 3,, 9,,, 999, 9 3. 3 n,,,,,,, S.,, S. S = + + + 9 + + + 9 + + 4 s4386.5 z z =, z., z n n = z + z + + zn n + n=.. n n S n = z k. k= 9
n S n z. m > n m, n. m k=n z k m k = S k k + S m k + m S n n k=n 3,, C. ε, N, m > n N m, n m z k k < ε. k=n, C, 3. C 6 s638.53 A = + t,. A, A 3, A 4. B n = n k= B n = x n E + y n A, B = lim x n E + lim y n A n n. E. 3 B = E t..54 x R f n x = min k Z A k k! x k n. Z.. f, f, f. 7 s7383 x R Sx = f n x n=. 3 x R Sx R..55 A = 6, B = lim.,. n k= n ka k. 7 s7387, A E
B. E A B..56 D x, D n x = + n k= cos kx n, F n x = n + 8 s838 n D k x n R {D n } {F n }.,. D n x sin x = sin n + x. F n x sin x = { } cosn + x = n + n + n + sin x. 3 π π F n ydy = π. 4 < δ < π δ lim n δ y π k= F n ydy =. 8 s8385.57 n, n P n, n d H n x = e x e x dx.. H x, H x, H x. H n x n. 3 H x, H x,, H n x P n. 4 m < n m,. x m H n xe x dx =.58, C - fx n, a n = f n n! x f x xfx =, f =.. 8 s8387 a n. x n dx π., x e x n π e x dx =. x 3 n {a n }.,..59 3 s3385 + + + n lim n n lim n + n n s39
.5 a > dx x a dx + x 3 s339.5 y = fx f x. f y x = lim x x = lim fx + x fx x x,. y = log e x., e = lim + x. x x y = x n., n..5. L L nπ mπ cos L x cos L x dx x e ax+b dx, a. 6 s639, L, n, m. 6 s639.53. dx + x 7 s739.54 I = a x b dx, a >, b > I = e ax dx, a > 8 s8393.55 N P N = n = λn n! e λ, λ >, n =,,,., N : e λ λ n = n!.56 Γs =. Γ. n= 9 s939 x s e x dx s, Γs, x, s > Γs = s Γs. s396.57 n sin n+ x sin n x sin n x x π/
sin x = x π/ n =,,, I n = π/ sin n xdx I n = n n I n n 4 n 3 lim n n 3 n 5 s54.58 x lim m lim n cos m!πxn.59 {a n } n=, {b n } n=, {c n } n= a = 3, b =, c =, 5 s54 a n+ = 3a n + b n + c n, b n+ = a n + b n, c n+ = a n + c n n.53 tan x x tan x x dx. 5 s544 π tan x [,. x 7 s74.53 fx [a, b], a, b] F x F x = x ftdt.. x a a F x x F x fx F x. a, b] fx F x. 3 a, b] F x. 4, F x = ex. x.53, F x fx 9 s94 F x = x log + x, fx = F x.,. gx = x tf t ft dt. f x > fx F x > x x > x. 3 gx..533 [, fx.. fxe sx dx = M lim M s4 fxe sx dx s > n,. x n e sx dx = n s x n e sx dx 3
n, 3 s >,. x n e sx dx. x n e sx dx = n! n=.534 dx. ex n, n= x n dx. ex 3 x fx,. x n e sx dx n! s4 fx e x dx 3 s34.535 fx, f n x n N, [a, b], [a, b], i n f n a = fa, ii b lim n a f nx f x dx =,.. fx fa f x. f n x fx. 3 f n x fx. 4 s44.536 a m,n x m, n N a m,n x = cos n m!πx, g m x m N fx sin x g m x = lim n + a m,n x.. fx = lim m g mx x g m x. x fx. 3 fx x =. 4 fx x =. 5 s54.537 n=3 r. nn xn. 4
x < r fx = n=3. f x x. nn xn 3 fx = x log x + x x. 6 s64.538 t log + t t t. n= n k K =, k =. 3 x >,. fx = n= log + x n 4 x, gx = n= { log + x } n..539 f n x n N., c n.. n + cos x f n x = c n π f n xdx = 7 s74 n N π n + cos x sin xdx. n N c n < n +. 3 < x π, lim f nx. n e x x.54 fx fx = x x = e x. a k k =,,, fx = 8 s84.. a k x k. a k k =,,,. k= 3 fx n f n x. f 99. 5