simx simxdx, cosxdx, sixdx 6.3 px m m + pxfxdx = pxf x p xf xdx = pxf x p xf x + p xf xdx 7.4 a m.5 fx simxdx 8 fx fx simxdx = πb m 9 a fxdx = πa a =

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1 II f Rx = f Lx = fx fx + lim = lim x x + x x f c = f x + x < c < x x x + lim x x fx fx x x = lim x x f c = f x x < c < x cosmx cosxdx = {cosm x + cosm + x} dx = [ ] π sim sim + x + = m 3 m m + cosmx cosmxdx = = { + cosmx} dx [ x + simx m ] π = π m = 4 simx sixdx = {cosm x cosm + x} dx = [ ] π sim sim + x = m 5 m m +

2 simx simxdx, cosxdx, sixdx 6.3 px m m + pxfxdx = pxf x p xf xdx = pxf x p xf x + p xf xdx 7.4 a m.5 fx simxdx 8 fx fx simxdx = πb m 9 a fxdx = πa a = π 6, a =, b = + a = sih π π +, b = 3 a a = δ a, a = a π siaπ a 3 = a = a = a a = siaπ aπ b 4 a b =.6 4 b = π 3 π 5 a = 4, a = π 3 6

3 b = a = π, a = π 7.7 fxe imx dx = c e ix e imx dx = c πδ m = πc m 8 = =.8.9 c = sih a iπ a iπ c = e x e ix dx = e x ix dx + e x ix dx = e π π π π S N x fx cosx S N x..3 S N x = N b six = π dx N fx b = S N x = a N + a cosx + b six 3 = π dx a N fx π + a + b 4 =.4 N N e cosu = R e iu iu/ e in+/u = R e iu/ e iu/ = = = si Nu + u/ siu/ siu/ 5 3

4 .5 b = π cos x si x dx = π [si + x + si x] dx 6 = m b m = 8m π4m 7 = cos x.96 x + cos x x =.6 b = /π.7 c = /iπ.8 a = π si x cosxdx = π si + x + si x dx = π x = π.9 fx = x + 8 x = π cosx 9 = x = π = = x a = π, a = π, b = 3 π x dx = π π + 4 π 3 = π 3..3 = = π

5 . aw = bw si wx fx fx π cos w w siwxdw 34. fx π w siwπ w e iwx dx 35.3 fx..4 fx F w = π e iwx dx = π + iw + w fx e x = fx + f x.6 fx F w = F w = F c w.7 F w = π a + iw 37 w siwπ i π w π F w = i π i w ± w = w = 38.8 x, x F w, w = 4 exp 3 3w + w w 3w.9 iw a e ax / iw.6 F w = iw exp w a a

6 . δ 3 x x.49 lim π fx x + dx = lim π y f y dy = f 4 +. F w = π δxe iwx dx = π e iw = π 4.3 δ ax < ax ax > 43 ax = y / a.4 γ =.55.5 F w = exp w w π a 44 a =.54 F w a.6.86 fx = gx= gx, F w = Gw= Gw.5 fx F w si w w dw = π Xx e λx e λx c e λx + c e λx X =, Xa = c, c λ 3. u, y = cos y, u, y = ux, = ux, = ux, y = π cos A siπy sihπ x, A = sihπ π 4 = 46 ux, = x, u, y = u, y = ux, = ux, y = + B siπx sihπ y, B = sihπ π = 47 6

7 3.3 y =, π Xx = ae x + be x, Y y = siy x e x 3.4 y =, π y cosy ux, y = B e x cosy 48 = x = u, y = B cosy = fy 49 = fy B = π fydy, B = π fy cosydy y = b w δx u π 3.6 x, m, a y,, b 3.7 si cos si 3.8 u, v r, θ r r 3.9 ϕx + ψx =, cϕ x cψ x = fx ϕ x ϕx + ct = c x+ct fsds, ψx ct = c 3. uv, w = ϕu + ψw x ct fsds 5 ϕx + ψx = fx, cϕ x cψ x = gx 5 ϕx + ct, ψx ct 3. A = 4a π u, t = ua, t = ux, = Xx = si πx πct, T t = si a a 54 7

8 si [ ] πx ct πx + ct cos cos a a = π x ct si πs a a ds x+ct si πs a ds A = a + π, A = a κvx = q x + q πx ux, t = u π x/ κt x = u erf κt e y dy + e y dy e y dy + x/ κt e y dy ux, t = a πκt a a exp a + πκt exp x 4κt x u du 59 4κt C, C Gr = 4π r = 4π r ke i C ikr k k iϵ C dk ke ikr k k iϵ dk πie ik r = e ik r i 4πr Gr = e ir k π 3 ik 4 dk Gr = π π 3 k k dk e ikrµ dµ = k si kr e r 4π r k 4 dk = + 4 8πr si r 63 k si kr = e ikr e ikr /i e πi/4, e 3πi/4 e 5πi/4, e 7πi/4 8

9 4 4. t 4 = x 4 Γ Γ 3 = π 5 Γ =, Γ = 3 3 Γ = 3 π 4 Γ + = Γ = 3 Γ ax = t 4.5 βe = t 4.6 lim zγz = lim Γz + = 67 z z 4.7 Γz + lim z + Γz + = lim = 68 z z z 4.8 Γz = Γz + z = Γz + zz + = = Γz + + zz + z + Γz + + lim z + Γz = lim z z zz + z + =! y = x u u 3 u 4 u 6 Γx + e x x x+/ e u / u4 4x + u6 du = e x x x πx + 8x x 7 4. u Γ + uγ + Γ = e t t x u + u log t + log t dt = e t t x u + log t dt 7 9

10 πz z t z t ζ dt = t+ t t z t ζ dt = t z t ζ dt+ t z t ζ dt ζ Bz, ζ + z 4.3 / x / x / dx = B, 3 = π 5 si 4 θdθ = B, = 3π 8 cos θdθ = B, + = Γ Γ + Γ + = π!!!!!!!! : : / cos m+ θ si + θ dθ =!m! B +, m + = + m +! P x = P x = x log + x log x x = z P ν z xp νz + νν + P ν z = ν + ζ ν ν πi ζ z ν+3 z ν + zζ + z + νζ z dζ = ν + d ζ ν+ ν πi dζ ζ z ν+ dζ 78 ν ν r r = r > + r< r > r < cos θ = r > + r< /r > r > /r < cos θ = r > = r< r > P cos θ

11 4. fx = x 3 3 P 3 x A = 3 fx = x fx = 5 P 3x P x 8 xp x = + + P +x + + P x 8 m = + m = 4. u ϕ 4.6 m = cos θ Q m l ur, θ = α l r l + β l r l P l cos θ 8 l= α l a l + β l a l = l + fxp l xdx α l b l + β l b l = 83 α l, β l 4.3 E. E.3 L z = x i h y i h = i h cos ϕ + si ϕ y x ϕ = i h ϕ L z 4.66 ϕ im L si θ Pl m cos θ = { si θ si θ d } si θ θ θ si θ θ dx P l m x x=cos θ = cos θ d dx P l m x x=cos θ + si θ d dx P l m x x=cos θ = x d dx P l m x + x d dx P l x m x=cos θ m = ll + P m x l x x=cos θ m = si ll + Pl m cos θ 85 θ 4.7 m / si θ / 5! Y = 4π! d dx x = x=cos θ 5 4π x 4 8 = x=cos θ 5 6π 3 cos θ 86

12 ν ν 4.7 cosx si θ x si θ 4. / si k θ cos ν θ dθ !!! = =! x J x d yx = x y xj x J dx J x xyx 89 LX = LY = J x k d k x k + d k x k+ k+ x k+ + x = 9 k!k +! k= k= k= x 4. d x k+ k J ν iz J ν x x ν ν d x ν J ν x = x ν J ν x, dx 88 d x ν J ν x = x ν J ν+ x 9 dx d x ν Y ν x = x ν cosνπjν+ x J ν+ x dx siνπ = x ν cosν + πjν+ x J ν+ x = x ν Y ν+ x siν + π J ν x Y ν x x ix, z ie iθ I ν x

13 4.34 ϕ π ϕ cosx si ϕ six si ϕ siϕ cosϕ e ikθ π cosx si ϕ fx, z = exp { x z + x z x z z ν 4.56 z z } + x ν fx, z = {x d z z dx + x d } dx + x ν fx, z ν ν + J ν x = d d x ν x ν+ J ν+ x 94 x dx x dx J x = x 93 d x J x 95 x dx r r k k r r j r, y r V = E = h k m Rr = h k h d m dr + r d dr + h ll + mr Rr 96 m 4.84 j l kr, y l kr j l kr 4.87 ϕ m = si θp j cos θ 4.34 e ikrx P j xdx = a j i j e ikrx P j x dx 97 j + 3

14 x J x = d x J x 98 dx x α j x A j = Jν+ α j α j d x J ν α j x dx = dx α j J ν α j u ρ + u ρ ρ + u z = uρ z vz u d u dρ + ρ du = λ, dρ d v v dz = λ u J α j ρ α j u J α j c = λ = α j uρ, z = A jj α j ρ B j e αjz + C j e α jz j= uρ, = B j = C j 4.37 A j = b J α j b b ρfρj α i ρ dρ π j l r r πr cos r l π = eir i l+ + e ir i l+ r x z H x z! = H x z! = = H x = e x / ϕ x x αx x d mα dx + E h mα4 k h x ψαx = 6 h /4 x α = mk d m dx + h k E x ψαx = 7 4

15 4.35 E = h k m +, ψ αx = e x / H x x H x = xh +x+xh x = 4 H +x+ + H x+ H x x H x c = a j+ = j + ν j + j + 3 a j c = a k = k ν3 ν k ν a k +! k ν a k k k!! k! k k!! k! c = g L x z! L z x! + L x z+ = x L x z!! L x z+! = z = = x z z m ψ m m = m e x x k= z k 4 k= k! L, L z k 5 5

16 4.55 { h d m dr r d ll + + dr r } e Rr = ERr 6 4πϵ r r αr, E βe 4.43 α, β α = 4πϵ h m, β = m 6π ϵ h 7 r = r/α E = m 3π ϵ h, Rr = r l e r / L l+ l r x F,, ; x = x F, b, ; x/b = = = x! = x =!! +!! x = x b x b +! =! b!b!b x = = = = k= x + + k b = log + x x! 9 b k x yt := J t t d y dt + dy + ty = dt d p Y p py+ y + dp dy dp = py p + c Y p = c p + + py p dy dp = 3 4 p 5.3 e pt lim py p = J + = 5 p c 6

17 5. m d ft dt = F δt t 6 mp F p = F e pt 7 ft = L [ F e pt mp ] = F m t t F p = A mp + B mpp + ω = A mp + B mω p p p + ω 9 ft = A m t γ = γ/m γ 4m k m = ω F p = ω ft = x ω B cos ωt 3 mω γ + ω p + γ ω γ ω x 3 p + γ + ω γ + ωe γ ωt γ ωe γ+ωt cos t F p + p F p = p p + 33 F p ft = cos t si t 5.6 ft F p = x = p u F p = p a x a x dx 34 + p u a/ du 35 u F p = π p a siπa/ 36 ft = t a cos πa Γ a 37 7

18 F p = p dx 38 x x + p ±ip F p = π p ft A. A. C. C. s is i ν s e s D. D.5 c D. π/l = w = π/l = w l E. r = E. E.5 3 E.4 3 E.6 πr 3 4πr E.7 E.3 ±r r r d Gxdx = 4 dx d dx Gx d x=r dr x= r Gr = 4 Gr r Gr G r = 4 Gr = r/+cost E. ρ y = y x + y, ϕ y = x x + y 43 x 8

II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2

II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2 II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh

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