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1 Landau

2 hyperbolic sin, hyperbolic cos, hyperbolic tan

3 sin θ a c, cos θ b c, tan θ b a cosec θ sin θ, sec θ cos θ, cot θ tan θ inverse arcsin θ sin θ, arccos θ cos θ, arctan θ tan θ. sin θ + cos θ sin sin θ sin θ, cos cos θ cos θ tan tan θ tan θ π sin θ sin θ ± πn, cos θ cos θ ± πn, tan θ tan θ ± πn [ π, π] 4 sin π + θ sin θ sin θ sin π θ cos π + θ cos θ cos θ cos π θ tan π + θ tan θ tan θ tan π θ π sin θ cos θ 3

4 sin 3 cos 4 tan. sin α ± β sin α cos β ± cos α sin β cos α ± β cos α cos β sin α sin β sin θ sin θ cos θ cos θ cos θ sin θ sin θ cos θ sin θ cos θ cos θ + cos θ.3 A sin θ + B cos θ A + B sin θ + φ cos φ A A +B, sin φ B A +B, tan φ B A θ A sin ωt + B sin ωt + θ A + B + AB cos θ sin ωt + φ 4

5 cos φ A+B cos θ, sin φ B sin θ B sin θ, tan φ A +B +AB cos θ A +B +AB cos θ A+B cos θ A sin ωt + B sin ωt + θ A sin ωt + B sin ωt cos θ + B cos ωt sin θ A + B cos θ sin ωt + B sin θ cos ωt sin ωt + θ + sin ωt + φ sin sin ωt + θ sin ωt + φ cos cos ωt + θ + cos ωt + φ cos cos ωt + θ cos ωt + φ sin α ωt + θ, β ωt + φ ωt + θ + φ ωt + θ + φ ωt + θ + φ ωt + θ + φ cos sin θ φ θ φ θ φ θ φ cos sin sin ωt + θ + sin ωt + φ sin α + sin β A α+β, B α β sin α + sin β sin A + B + sin A B sin A cos B + cos A sin B + sin A cos B cos A sin B sin A cos B sin α+β cos α β sin ωt + θ+φ cos θ φ log a b log a b log c b log c a ln b ln a a e t t ln a 5

6 3 3. f a ε >, δ >, a > δ f f a < ε ε δ a δ < < a + δ f a f ε : a M M f [a, b] f a α, f b β α < γ < β γ f c γ a < c < b c Rolle f f a f b f c a < c < b a, b f [a, b] C C f f c f b f a b a a < c < b c a, f a b, f b [a, b] C Cauchy C f, g c f c f b f a g c g b g a a < c < b C m : m C : 6

7 C f, y y f y f 3. Landau lim a u u a v lim v u v o u v u a u v lim v u v O u a u 3.3 f f + f y lim lim 3.4 c u c u u ± v u ± v u v u v + u v u v u v u v u v w u v w + u v w + u v w v Leibniz f, g C m n f g n nc k f n k g k k f g f g + f g f g + f g + f g f g 3 f 3 g + 3 f g + 3 f g + f g 3 7

8 3.5 d sin cos d d cos sin d d d tan cos sec d sin d d cos d d tan d d d e e d d a ln a a d d log a d d ln d d ln f f f ln ln a ln a 8

9 3.7 f, y,, y, : f f +, y, f, y, }{{} : Chain rule d f f f d + dy + y, y, z t t, y t, z t 3 d f f d f dy f dz dt dt + y dt + z dt d dt, dy dt, dz dt t f, y, z a, b, c f a, b, c n, f a, b, c, y f a, b, c z a, b, c t t, y t, z t f t, y t, z t. t a, b, c a, y b, z c. t d f dt f d dt + f y dy dt + f z dz dt t f 9

10 3.8 d f f d d d f f g d dg dg d y f f y d dy dy d y f y d f dy dy d d f dy / t t, y t y dy dy d dyt dt dt d dt y t dy dt dy d d dy dt d dy / dt d dt F, y y y f s F dy d F y df F + F d + F dy y F dy dy y d d : F / F y

11 y f g y f y f, f, f, g y y, g yy y, g yyy y, y f y g f g f f g f f + g f f g f f f f + g f f g f gy f g f f f + g f f g f f f + f f g f f f f 3 g f f f f 3 + 3g f f f f + g f f 3 g f f f f 3 3 f f + f f g f f f f f 5 f f 4 3 g f f f f f 4 + 6g f f f f f + g f f 3 f + 4 f f + g f f g f f f f f f 3 f 3 f f f + f f g f f f f 5 f 3 f 7 + f f f 6 f f 5, y

12 F, y, y, y df F F d + y dy F d + F y dy d y, d d dy dy d F y F, d dy F F y d F F d + F y dy F F dy F + F y d d F y Fy d + F yy dy F yy F y dy F y + F yy d F y F y F df d F, df y d F y d d dy d F y d d F F y d d F y F Fy + F yy dy d F dy yf F yy d F dy + F y F + F y F y d F yf + F yy F dy F + F y F + F y d F F y F F + F y F F F y F y F y F F F yf + F yy F y + F y F F y F y y F F y F + F yy F y F + F y F F Fy Fy F F 3, y d y d F F y F + F yy F + Fy Fy d dy F F y F 3 F + F yy F + F y F 3 y Fy

13 3.9 O O s P r s τ s d r ds s r P t χ s d τ ds d r ds χ χ s d τ ds d r ds R R P s t v s t v s : const v t d r dt d r ds ds dt v s τ a t d v dt v s d τ dt v s d τ ds ds dt v s χ v s v τ, a χ τ s d r ds ds dr ds dr P R R 3 χ cs 3

14 y f dy d y R d d dy + d f f + f F, y { } R F + F y F F y F + F y Fy F + Fy F + Fy v c P, y t d v dt, dy dt d + dt dy dt y f y dy dt dy d d dt d + dt dy d d dt + dy d + f g f f t y y t f t t d a dt, d y dt d dt d d d d g dt dt dt dt d d g g d d g g d f d d d f g g f dg d f f f f f + f + f 3 + f 4

15 d y dt d dt dy d d dt dt dt dy d d d d f dt + g dt d f d dt g f d d dt g f + g dt f d f dt + g f d + dt d y dt d + f dt + g f f f + f + f + f + f /R f f f + f + + f + f f f + f + f f f + f R f f + f dy d + dy d d y d dy d F y F, d y d F F y F + F yy F + F y F 3 Fy R + F y F + F y F F F y F + F yy F + F y Fy F + F y { } F F y F + F yy F + F F + Fy y Fy F 3 5

16 λ λ, y y λ, z z λ R λ + y λ + z λ λλ + y λλ + z λλ λ λλ + y λ y λλ + z λ z λλ λ + y λ + 3 λ z v c P, y, z t d + dt dy + dt dz dt dλ dt d + dλ dy + dλ dz dλ dλ dt d + dy dλ dλ + dz dλ G λ : G λ d + dλ λ λ t P λ d dλ, dλ dt λ + y λ + z λ d λ dt d dλ d dt dt dt d dt d dt dy + dλ dz dλ G, G λ + y λ + z λ, G λ λ λλ + y λ y λλ + z λ z λλ G dλ d dt dλ G G G λ G 3 G λ G λ λλ + y λ y λλ + z λ z λλ λ + y λ + λ z d dt d dλ d dt dt dλ λ λ + dλ d dt dt dt dt λ d λ dt λ + dλ dλ dt dt λλ d λ dt λ + dλ λλ dt λ λλ + y λ y λλ + z λ z λλ λ + y λ + z λ λ λλ λ + y λ + z λ λλ λ λλ + y λ y λλ + z λ z λλ λ λ + y λ + z λ λ + y λ + A λ B λλ B z λ d y dt Ay λ By λλ B, d z dt Az λ Bz λλ B R d + dt d y + dt A B A B + B λλ + y λλ + y λλ B 4 d z A λ + y λ + y λ AB λ λλ + y λ y λλ + z λ z λλ + B λλ + y λλ + λλ y dt B 4 λ + y λ + z λ A + B λλ + y λλ + λλ y λ λλ + y λ y λλ + z λ z λλ + B 3 λ + y λ + 3 λ z λλ + y λλ + y λλ 6

17 3. f d a n d n+ an+ + C n d ln a + C a f d ln f + C f e d e + C a d ln a a + C + a d ln + a + C + a d a tan a + C 3. m d n m n m n m n + n n m n i f f f d ln f + C f n n i d ln + + C + + d ln + + C d ln C 7

18 i f m f m f f m d m + f m+ + C i d C 3 m d + + d / + 4 d + C d C 8

19 ii-a n : n n a n n a + a + + a n n n n n n n n n i n + c i i n c i i d c i ln i + C ii-a d ln + + ln + C d ln + ln + C ii-b n : n p p n p n a n p n n n a p k k a k + n k + a p k p k a k + + p n k n k + + a n a n n k p n k n n k i p i a i j { n n n i j a i i i } n i p i 3 n g Taylor b n n g g i + g i i + g i! b + b i + b i + + b n i n i + + gn i n! i n 9

20 3 j { n b b a i i j a i i j + i + + b j j i + b } j + b j + b j+ i + + b n i n j i 3 j m i a i m i a i j n i j c c i p + i i + + c p i + c p i + c p i i pi + c pi + i + + c n i n i }{{} }{{} group.b group.a group.a c i i p d p + c i i p+ + C cpi i d c pi ln i + C group.b ii-b d ln f f a + b + + c g + f a g + b g + + c g + a /6, b /6, c /4 g g 57 g 9 g g 7 g 6 g g 78 g 3 g 4 + g 6 g g 4 g g

21 iii : + d tan + C d : unknown 3 + d : unknown / d ln tan + C / { } ln + 4 ln + + tan + C + +

22 f f a n a + a + + a n n n k a k k a k a k k k k k }{{} k n n k l eclude llk a b c a a b a c + b a b b c + c a c b c b p p p n p n l a,l p + l l a,l p n + + l l a n,l n l n p k k l a k,l k l a k,l a b a a b + b a a a b + b a b a 3 b a 3 a b + b a a a b + b a a a b + b a 3 b a b a a b + b a a a b + b a b a b + b a b a b c a a b a c + b a + c a + b a b b c + c a c b c a a b a c 3. sin a d cos a + C a

23 cos a d sin a + C a tan a d ln cos a + C a cot a d ln sin a + C a cos a d sin a d cos a d sin a d sec a d tan a + C a cosec a d cot a + C a sec a d cosec a d ln d ln + C a ln + sin a cos a + C a ln tan a + C a d a sin + C a cos + C π/ ± a d ln + ± a + C a d a + a sin + C a ± a d { ± a ± a ln + ± a } + C 3.3 a φ α b φ β b a f d β α f φ t φ t dt 3.4 f g I C I a, b b a f g d [ f g ] b a b a f g d 3

24 3.5 sin m cos n d e sin d e e cos d e e sin a d e cos a d sin cos + C sin + cos + C e a sin a cos + C a + e a sin + a cos + C a + m n n sin m sin n + m cos m cos n + C m n 4m cos m + C m n sin m cos n d + cos m sin n d sin m + n d cos m + n + C m + n 3.6 β α β α α β d β α3 6 α m β n d n m!n! β αm+n+ m + n +! n β α αm β n d n β n α m+ n m+ m+n m+n αm+n d [ ] β n n m+ n n m!n! m+n+! m+ β αm+n+ m+n m+n α m+n+ m+n+ α 4

25 Bete B p, q p q d p! q! p + q! B, B p, B, p p B p, p p! p! p! B n, B, n n d [ n n n ] n Bete q : B p, q q p B p +, q p : B p, q p q B p, q + Bete B p, q B q, p y B p, q π/ sin p θ cos q θdθ sin θ 5

26 Gamma Γ s s! e s d Γ e d Γ Γ 3 Γ 4 Γ 5 e d e d 3 e d 6 4 e d 4 Γ π Γ e d e t t t dt t e dt t e t dt y e t dt e +y ddy z e +y y z e +y e r r ln z z πr dz r ln z π ln zdz π [z z ln z] π Gamma Γ s + s Γ s Gamma Γ s e r r s dr 3 e r3 r 3s dr r, r 3, 6

27 Gamma Bete B s, t Γ s Γ t Γ s + t Γ s Γ t 4 r e s d π/ θ e y y t dy 4 e +y s y t ddy e r r s+t cos s θ sin t θ drdθ Γ s + t B s, t Gamma Bete π/ cos m θ sin n θ dθ m + B, n + Γ m+ Γ n+ Γ m+ + n+ π/ Γ n + n + n + + π sin n θ dθ case n π/ cos n θ dθ π/ π sin θ dθ Γ n+ Γ n+ π/ cos θ dθ case n π/ sin θ dθ π/ cos θ dθ π 4 case n 3 Fourier π/ sin 3 θ dθ π/ cos 3 θ dθ 3 π π π π cos n d cos n d π π π π sin n d sin n d π π π π π sin n sin m d sin n cos m d π π cos n cos m d δ nm π 7

28 3.7 u, v, y u, v, y u, v u, v J u y u v y v, y f, y ddy f u, v, y u, v J dudv, y Jacobian J u, v d u du + dv dy v y du + u y dv v u, v, u, v ddy d, dy d dy u v y u y u 5 u, v, w, y, z u, v, w, y u, v, w, z u, v, w u, v, w f ddydz f J dudvdw J J u y u z u v y v z v w y w z w, y, z u, v, w 8

29 u v y y, y, J u v.... u, v,, y, z d du + u v dv + dv dy w y du + u y dv + v y dv w d, dy, dz du, dv, dw ddydz d, d u v w dy, dz dy dz y y y u v w z u z v z w y z u u + v v + w w + u y u z u + v y v + w y w + u + v z v + w z w +. f u, v, w du, dv, dw, y, z f u u u u d + dy + y z dz + f u u + f v v + f w w + f, y, z OK d f f u du + f v dv + f w dw + + f v v v d + dy + v y z dz + + f w f u d+ u y + f v v y + f w w y + d f f d + f y dy + f z dz + w f dy+ u w w d + dy + y z dz + + u z + f v v z + f w w z + dz+ 9

30 r cos ϕ y r sin ϕ r + y ϕ π z z ϕ tan y r r + ϕ ϕ cos ϕ r sin ϕ r ϕ y r y r + ϕ y ϕ sin ϕ r + cos ϕ r ϕ, y r, ϕ det cos ϕ r sin ϕ sin ϕ r cos ϕ r + y + z r + r r + r ϕ + z r sin θ cos φ y r sin θ sin φ z r cos θ r + y + z θ tan + y z φ tan y θ π, φ π y z r r + θ θ + φ φ r y r + θ y θ + φ y φ r z r + θ z θ + φ z φ sin θ cos φ cos θ cos φ + r r θ sin φ r sin θ φ sin θ sin φ cos θ sin φ + r r θ + cos φ r sin θ φ cos θ r sin θ r θ + y + z, y, z r, θ, φ r sin θ r + r r r r sin θ sin θ + θ θ r sin θ φ 3

31 r, θ, φ, r y, θ y, φ y, r z, θ z, φ z r + y + z, tan θ + y z, tan φ y, y, z J r r y r z θ θ y θ z φ φ y φ z sin θ cos φ sin θ sin φ cos θ cos θ cos φ r cos θ sin φ r sin θ r sin φ r sin θ cos φ r sin θ OK 3

32 4 4. f, y z 3 y, y f +, y + f, y + f y, y y + { f, y + f y, y y + f yy, y y } z f, y, y : n : f f y : f y f y f, f y, f + fy + : f y, f, : f, f y, f + f y : f + f y a,, f, b, y, f y y α, β p α a + β b i j k f y a b f f y y y f y y y // f f y α a + β b z α f + β f y y α, β y, f y y, f y, f, f y,c C f f + f y f y 6 3

33 4. f, f y y, y A f, y B f y, y C f yy, y X, Y y F X, Y f, y + f y, y y + f yy, y y X, Y A X + B XY + C Y i D AC B < F A X αy X βy Y X/α Y X/β F F f, y ii D AC B F A X αy Y X/α A F f iii D AC B > X, Y, 7 F A C F > f A C F < f 33

34 Hessian f, y, y H f, y f, y f y, y f y, y f yy, y det [ H f, y ] [ f f yy f y ],y i det [ H f, y ] < ii det [ H f, y ] iii det [ H f, y ] > iii- f > iii- f < F H f X, Y A X + B XY + C Y X Y [ ] A B X X Y X H B C Y f Y det H f λ, µ α α γ H f λ, H β β f δ γ µ δ α γ α γ H f β δ β δ λ H f T µ λ µ T T H f T λ µ α γ T β δ λ T H f T µ H f H t f H f T t T λ µ t T H f T t T t H t f T t T t H f T t 34 :

35 X T Y u T v T X Y u v T t X Y u v X Y T u v F X λ H T T µ Y X H f X Y λ X T T Y µ Y u v λ u λu µ v + µv F H f λ, µ λµ Hessian f, y, z, r, y, z, H f r i f r f y r f z r f y r f yy r f yz r f z r f zy r f zz r.... f r ii f r iii f r iv 35

36 5 5. a n a < a < a 3 < : a a a 3 : a n ε >, N >, n > N a n α < ε ε N N n a n α ε Cauchy a n ε >, N >, p, q > N a p a q < ε N a n Cauchy Cauchy Cauchy : f lim a f ε >, δ >, < α a < δ, < β b < δ L Hopital f α f β < ε f f a g a f f lim a g c c lim a g c f a g a 5. sin θ tan θ lim lim θ θ 36

37 log cos lim cos log lim lim + lim ln + f ln ln t / f ln/t t + t ln t t ln t t ln t < t ln t t ln t lim t t < lim t lim t ln t t < t f ln + lim lim lim e e t t a ln a 5.3 a f b f a + f a b a + f a! R n+ f n+ a + θ b a n! f f + f + f! b a + + f n a n! b a n+, < θ < + + f n n + R n+ n! b a n + R n+ R n+ f n+ θ n+, < θ < n! 37

38 D i i i i Dy j y j j y j }{{} i }{{} i y y y }{{} y y y j }{{} j D i D j D i+ j D i yd j y D i+ j y D i D j y D j yd i i y j i i j y j f f + }{{} y y y }{{} i }{{} y y y j }{{} i j f +! n f + n! Dn f n n n! n n f [ f, y f, + + y ] f, + y! f, y n [ + y y + [ 3 3 3! y y + ] 3y 3 + y3 f, + y y 3 n n f, y n! n n n [ n ] D + D y f, n! n n n! n n m m n m ] + y f, y y m m m! y m f, n y m n+m n!m! n f, ym n!m! Dn D m y f, f,,, i n n n i n [ D + D + D n ] i f,,, n! n!n! n i! Dn D n D n i i f,,, 38

39 n n! α + βn n m n!m! αn β m n n! α + β + γ + + ωn n m l ω n!m! w! αn β m ω w n n! α + βn n n! n n! k! n k! αk β n k k case : n + α + β case : n + α + αβ + β case : n + α 3 + 3α β + 3αβ + β 3 case : n 3 + α a β b a, b C a,b α a β b C a,b a b k a n k b n a + b case 5.4 C a,b a + b! a + b! a!b! a!b! + α + α + e ! n! n + sin 3! 3 + 5! 5 7! n cos! + 4! 4 6! n n n! + n n! + ln n n < n α α + α α α 3 α α α n < 3! n! 5.5 e iθ cos θ + i sin θ 39

40 5.6 n e ik k n n cos k + i sin k k k + e i + e i + + e ni sin + sin n + sin + i cos cos n + sin n cos k + cos + cos + + cos n k sin + sin n + sin sin n + sin + n sin k + sin + sin + + sin n k cos cos n + sin cos n + sin cos sin n e ik k sin { n + } } sin { sin n n sin Laue S + e i + e i + + e n e i S e i + e i + + e n + e n+i e i S e n+i S en+i e i eni e i e i eni e i/ e i/ e i/ e i/ cos n + i sin n cos + i sin cos i sin cos + i sin cos i sin 4

41 cos n + i sin cos n cos sin n sin cos + i sin n cos + cos n sin + sin i sin cos + sin n + sin + sin sin + sin n + sin + i cos cos n + sin n cos k, k n sin k k n cos k, k S n e ik k sin + sin n + sin + cos { sin + sin sin n + } { + sin n + + cos cos 4 sin cos n + cos n + cos n+ + sin n+ n k 4 sin sin sin n sin k k cos k + cos + cos + + cos n + e cos n + sin cos n + + cos n + sin n+ sin sin n+ sin + e i + e i S e ni + + e i + e + e i + e ni + e i + e i } + + e ni + e ni S en+i e ni e i e i S e n i + + e n+i eni e i/ e ni e i/ e i/ e i/ sin n + sin S sin n + sin + S n sin k sin + sin + + sin n e i + e i k i + e i + e i i is e ni e i + e i + + e ni e i is e n i e + e i + + e n+i is en+i e i e + e ni e i en+i + e ni e i ei + e e i eni e i/ + e ni e i/ e i/ e i/ cos n + cos S cos n + + cos i sin i sin sin sin + + e ni + e ni i ei/ + e i/ e i/ e i/ 4

42 5.7 hyperbolic sin, hyperbolic cos, hyperbolic tan sinh e e, cosh e + e, tanh sinh cosh e + e e e cosech, sech, coth sinh + 3 3! + 5 5! + 7 7! i sin i, sinh i i sin 9! cosh +! + 4 4! + 6 6! cos i, cosh i cos 8! e cosh + sinh cos i + i sin i e cosh sinh cos i i sin i cosh sinh sinh cosh cosh sinh sinh α + β sinh α cosh β + cosh α sinh β cosh α + β cosh α cosh β + sinh α sinh β arcsinh sinh ln + + arccosh cosh ± ln + d + arcsinh + C e i cos + i sin e cosh + sinh 4

43 sinh cosh. tanh sinh 9 cosh tanh sinl ln ln + ln + ln + cosl ln + ln + ln ln tanl sinl ln ln + ln cosl ln + ln + ln + + ln + ln + sinl d cosl d ln : sinl + cosl sinl cosl sinl + cosl + cosl sinl? d d d 43

44 3 sinl - ln - + ln + /.5..5 cosl -ln - - ln + / sinl cosl sinl+cosl -ln tanl -ln - - ln + / -ln - + ln sinl cosl 4 tanl 5.8 : a, d : a, r : a n nr n a n a + n d, S n n a + a n na + a n a r n, S n a r n r r S n r + r + 3r 3 + 4r 4 + nr n n n d 44

45 r { n + r n + nr n+} r rs n r + r 3 + 3r 4 + 4r n r n + nr n+ S n rs n r + r + r 3 + r 4 + r r n nr n+ r rn r nrn+ r r r r { { rn n r r n } r n nr n + nr n+} r S n { r r + n r n + nr n+} r n n + r n n + n + 6 { } r 3 n n + r r + 3 n n + n +, r r + r + n n + n + n r r + r + { r r r + r + + r r + r + r + 3} 4 r r + n n odd even + n n n π 6 n n n n n n π 4 π π 8 45

46 f z u, y + i v, y : f f z f z z lim z z z z u v y, v u y u / v v, y / u y 46

47 7 7. L f f a + n a n cos nπ L a L f d L L + b n sin nπ L a n L f cos nπ L L L d b n L f sin nπ L L L d f cos sin cos sin 5 L/n cos 6 L/n sin f a, a n, b n OK 47

48 : π π d π π π cos d π π sin d π π cos sin d cos sin π π cos n cos m d π π sin n sin m d π π cos cos d π π sin sin d π L L cos mπ L L L cos mπ L mπ L cos L d L mπ sin L d sin mπ L mπ sin L d L m n m n 48

49 49

50 cos n + m d cos n cos md sin n sin md [ π, π] π π π cos n cos md π sin n sin md sin n cos n cos md cos md n [ ] sin n sin n cos m cos m d n n [ ] sin n cos m + m sin n sin md n n [ π, π] π π cos n cos md m n π π sin n sin md n m π π cos n cos md π π sin n sin md n m π π cos n d π π sin n d π cos.5 7 cos 5

51 f 4 π sin 3 sin 5 sin 7 sin f 4 π cos 3 cos 5 cos 7 cos

52 f 4 π sin sin 3 sin 4 sin 5 sin f 4 sin + π sin sin sin

53 f 4 π cos cos 3 cos 4 cos 5 cos δ f + cos + cos + cos 3 + cos 4 + cos 5 + π 7 δ f + sin + sin + sin 3 + sin 4 + sin 5 + π 53

54 8 f f f π 4 π cos 3 cos 5 cos 7 cos π 8 9 f f 4 π cos 3 cos 5 cos 7 cos

55 f 4 π cos cos 3 cos 4 cos 5 cos 6 cos f 4 π sin sin 3 sin 4 sin 5 sin 6 sin

56 y f π π f π cos cos 3 cos 4 cos 5 cos 6 cos π 6 56

57 7.3 [ L, L] f f α n ep i nπ L n α n L f ep L L i nπ L d cos nπ L nπ e L i + e nπ i L sin nπ L i e nπ L i + e nπ L i [ L, L] f g n ep i nπ L f α n g n n f f, α, α, α,, α n, g n n g n,,,,,, g n f L L g n f d α n f f f f f L L f f d Ψ α n α n n 57

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