4 5.............................................. 5............................................ 6.............................................. 7......................................... 8.3.................................................4.........................................4..............................................4................................................4.3............................................... 3.4.4.............................................. 5.4.5.............................................. 7 3 9 3................................................. 9 3.......................................... 9 3.......................................... 3..3..................................... 3...................................................... 3.3 3............................................ 3.4..................................................... 3 3.5....................................................... 3 4 6 4....................................... 7 4............................................... 7 4.............................................. 9 5 36 5.................................................... 36 5.............................................. 36 5............................................... 37 5............................................. 39 5.3.......................................... 4 5.3. sn................................................. 4 5.3. cn dn............................................ 4 5.3.3.......................................... 43

6 44 x 6. y = b sn b = k c k c.................................. 44 6................................................. 46 6................................................. 46 6................................................ 46 6.3.............................................. 5 6.4..................................................... 53 7 58 3

4

. ϕ y x = a sin ϕ y = b cos ϕ. x a + y b =. s ds = +.3. x = a cos ϕ dϕ y = b sin ϕ dϕ.3 ds = a b k = a A s a cos ϕ + b sin ϕ dϕ = a a b a sin ϕ dϕ. ds = a k sin ϕ dϕ s a = ϕ k sin ϕ dϕ k ϕ Ek ϕ Ek ϕ = ϕ k sin ϕ dϕ..4 ϕ = π Ek k Ek = E π Ek = π k sin ϕ dϕ..5 s = 4aEk. y = ±b x a 5

ds = + = b + a z = sin ϕ s = a z = x a z x a sin ϕ k z Ek ϕ = z dz Ek = x = a a k z z dz k z z. k z z dz.6.. x = aϕ sin ϕ y = a cos ϕ. x = aϕ b sin ϕ y = a b cos ϕ. x + y ax = b x + y. x a y b =. y b = sin x a x = b a cos x a. sin x a = x cos a φ = x a dφ = dφ a ds = + = + b a b sin x a a. = a + b b sin φ dφ = a + b k sin φ dφ k = a + b a a + b = b a + b sφ = s ds = a + b φ = a + b Ek φ. x a = π k sin φ dφ.7.s = a + b Ek.8 6

. Ek ϕ = ϕ k sin ϕ dϕ k = k = ϕ dϕ F k ϕ = k sin ϕ k <.9.9 ϕ = π Kk Kk = F k π = π dϕ k sin ϕ.. k i ϕ π z = sin ϕ dz = cos ϕ dϕ ϕ = π z = z F k ϕ = ϕ π cos ϕ z F k ϕ = cos ϕ = dz k z ii π ϕ π z = sin π ϕ ϕ = π π z = dz k z cos ϕ. sin ϕ = z z = z dz z k z dz = cosπ ϕ dϕ = cos ϕ dϕ = cos ϕ dϕ F k ϕ = z dz k z cos ϕ. π φ π cos ϕ cos ϕ = sin ϕ = z dz F k ϕ = = z k z z z dz z k z. i ii F k ϕ = z dz z k z F k ϕ Ek 7

.. lemnicate r = a cos θ x = r cos θ y = r sin θ x + y = a x y x + y = r x y = r cos θ. x + y = a cos φ x y = a cos 4 φ. + cos φ = sin φ x = a cos φ sin φ y = a cos φ sin φ x = a cos φ sin φ y = a sin φ cos φ a sin φ = 3 sin φ + sin φ dφ = ds = + sin φ 3 + sin φ + ds = a dφ sin φ a sin φ dφ. sin φ sin φ = φ = r = a s = a φ dφ = a F sin φ φ..3 s = 4 a K.4 k = i x = a z + z3 z3 y = az + z4 z z 3.5 Serret 845 z x + y = a x y ds = + = a dz + z 4 x + y = a z + z 4 x y = 4a z 4 8 + z 4

. cos φ = z sin φ = z + z 4 + z 4.6 sin φ = + z + z 4 dφ = + z dz + z 4 dφ = dz. sin φ + z 4.6 cos φ = z + z 4 z dz = + z 4 F φ.5 z x y z.5 x + y = a + z 4 x y x + y = azx + y x az + y az az = az az z z.5 x y ii x = η η = sin ϕ c c = π x 4 = x 4 c φ dϕ sin ϕ c = cos φ c x 4 x 4.8.7 F φ = c dη η η = {F π F } φ = F φ.7 = K..8 = dz x 4 z + z 4.6 c = cos φ = z + z 4 9

dz Gauss + z 4 x 4 fx = x iii x 4 ds = dr + rdθ =. + r r dr dθ = a sin θ = a 4 r 4. ds = + r4 a 4 r 4 dr. dθ dr dr.8.4 s = s = 4 r a dr a4 r 4..9 a 4a = K x 4.3 3 px px px 3 4 f x px. px 3 4 x = sin ϕ F k ϕ = x ϕ x k x = dϕ. k sin ϕ px x 4 px 3 I = c x 3 x = + 3 y y + x = c c = + 3 y y + y = c + 3 c + + 3 y y = c + 3 c + + 3 3y + 3y 3y + 3 3y + 3 = y + = y + = 3 y + I = y 3 y+ + 3 y y+ 3 = 4 3 y y { 3 + + 3 y }

+ 3 y 3 < y y < y = z z z = y y + z = dz z I = z 4 dz z 3 z z { 3 + + 3 z } = 4 z 3 dz z + 3 4 z + 3 + 6 k = = sin ϕ = z cos ϕ dϕ = dz 4 4 I = 4 3 ϕ cos ϕ dϕ sin ϕ k sin ϕ = 4 3 F k ϕ px x 3 4 z = sin ϕ 3 px k z Ek ϕ = z dz = k z z dz = k sin ϕ sin cos ϕdϕ = k sin ϕdϕ ϕ πk n ϕ = dz + nz z k z = cos ϕdϕ = sin ϕ k sin ϕ dϕ + n sin ϕ k sin ϕ dz. z k z k z dz.3 z dz + nz.4 z k z..3.4 3 Legendre-Jacobi k n. x x x 4 x = t x = dt x dt t = sin t = sin x ϕ = π Kk = π π dϕ k sin ϕ Ek = k sin ϕdϕ

.4.4. r Q yz Q y θ Q, r cos θ, r sin θ xy a, b, P P Q P Q = a + b r cos θ + r sin θ = a + b br cos θ + r = a + b + r 4br cos θ P U = π θ= rdθ P Q = r θ = π ϕ dθ = dϕ U = r π dθ = 4br a +b+r cos θ k = 4bra + b + r π dθ a + b + r 4br cos θ r a + b + r π dϕ 4br a +b+r cos π ϕ U = π 4r a + b + r dϕ k sin π ϕ π = 4r a + b + r dϕ k sin ϕ = k r b Kk.4. m x α β m d x dt = αx + βx3 β = β α β s = m t t y = α x x m xt = x α s m m dt = α x α s d dt = m m α x α s d x dt = α α m β y + y3 d y ds = d ds = y + y 3 ds.5 dt dt d x dt = x + x3.5 = E dt x 4 x4.6

E U = x + 4 x4.6 t = ± x E x x4 x a E = U a 4 + a 4E = a = + + 4E E x x4 = 4E x x 4 = {a x + a + x a x } = a x + a + x x = a cos ϕ = a sin ϕdϕ k = + a ϕ t = ± = ± + a a sin ϕdϕ = ± a a cos ϕ + a a cos ϕ + a ϕ dϕ k sin ϕ a ϕ dϕ a +a sin ϕ T T = 4 kkk a E αx βx4 = 4E αx βx 4 = {αa x + βa + x a x } = a x {α + βa + x } k = ϕ t = ± a sin ϕdϕ m a a cos ϕ{α + βa a cos ϕ} βa α + βa m t = ± α + βa ϕ = ± m α + βa ϕ dϕ βa α+βa sin ϕ dϕ.7 k sin ϕ a = α + α + 4βE T = 4 m β α + βa Kk m m β T = 4 K = π α α.4.3 Ux E m = + Ux = E.8 dt 3

.8 dt = ± {E Ux} x t = ± {E Ux} x = a x = a {E Ux} = x a a xv x.9 x = a x = a a a V x x x = a + a a a cos ϕ.3 x = a + a a sin ϕ.3 = a a sin ϕ = a a sin ϕ ϕ t = ± ϕ = ± a a sin ϕdϕ { a + a a } { a cos ϕ a a a + a + a a cos ϕ } V x dϕ V x.3 i Ux = α x + β 4 x4 a = a a = a a = α + α + 4βE β {E Ux} = E αx β x4 = 4E αx βx 4 = βa x a + x + α β b = a + α β {E Ux} = βa x b + x.33.3.9.33 x = a cos ϕ = a cos ϕ x + aa xv x = βa x b + x V x = β a + xa xb + x x + aa x.3 k = b + a ϕ t = ± dϕ β b + a a sin ϕ a = ± βb + a ϕ = β b + a a sin ϕ dϕ = ± a b +a sin ϕ βb + a 4 ϕ dϕ k sin ϕ

b = a + α β k =.7 ii a a + α β + = βa a α + a β βb + a = βa + α f = αx 3 γx Ux = αx γ x3 a < x < a.9.3.3 E Ux = γx a a xa 3 x.34 γx a a xa 3 x = x a a xv x V x = γa 3 x ϕ t = ± dϕ γ { a 3 a a a sin ϕ } = ± ϕ γa3 a θ = ϕ k = a a < k = a a < a < a 3 a 3 a a 3 a t = ± γa3 a ϕ dθ k sin θ dϕ a a a 3 a sin ϕ a a ϕ π T = 4 γa3 a Kk γ a = a = E α a 3 x = E = γa a a 3 γa 3 α T = 4 K = π α α.4.4 θ α l m l θ t l θ t = v m l dθ dt mgl cos θ E α E m l dθ + mgl cos θ = E.35 dt E = mgl cos α.36 5

.35.36 E l dθ dt = glcos θ cos α cos θ = sin θ cos α = α sin sin α θ sin dθ g dt = l k = sin α dθ g dt = k l sin θ sin α k k sin θ = k sin ϕ.37 dθ g dt = k l sin θ g = l sin α θ sin.37 sin θ = k sin ϕ.38 k k sin ϕ = g l k sin ϕ = g k cos l ϕ sin θ = k sin ϕ dθ.4.4.4.4 dϕ = k cos ϕ k sin ϕ dθ dt dϕ = dϕ = t = dθ dt ϕ l g k cos ϕ k sin ϕ g l θ = α ϕ = π T T = 4 l g π k cos ϕ = l g k sin ϕ k sin ϕ dϕ = l g F k ϕ.39 k sin ϕ dϕ = 4 l Kk.4 g K k k sin ϕ k sin ϕ = a k = X fx = = ax X ax Kk = = π π f n X = n!! n a n ax n+ fx = + ax + 3 8 a X + 5 48 a3 X 3 + + + k sin ϕ + 3 8 k4 sin 4 ϕ + 5 + k 4 + 9 64 k + + 48 k6 sin 6 ϕ + + { n!! k} n + n! 6 n!! n a n X n n! n!! n k n sin n ϕ + dϕ n!

Kk = π l l T = 4 g Kk = 4 g l T 4 g Kk = T π l g π + k 4 + l = π g = π Kk = π π + 4 k + 9 64 k4 + = + 4 k + 9 64 k4 + k = sin α α k = sin α k α α { T = + α T 4 9 α } 4 + 64 + = + α 6 + α4 4 +.4.5 i α = π k = k = sin α = sin π =.37 tan θ 4 = x t = l θ g dθ cos θ cos θ = cos θ 4 = tan θ 4 + = x + = x + x x θ dθ = 4 x + θ = θ x = tan θ 4 t = g tan θ 4 l + x 4 g tan θ x + x = 4 l x + g = + x l log + tan θ 4 tan θ 4 tan α = tan θ 4 tan β = = tan π 4 tan α + tan β tanα + β = tan α tan β l t = g log tan θ 4 + tan π 4 l θ tan θ 4 + tan π = log tan g 4 + π l θ + π = log tan. 4 g 4 4 θ = π t = ii.36 cos α = E mgl cos α = α sin sin α = cos α sin α cos α = E = mgl k = sin α E mgl k = mgl k > E > mgl k = E <.37 t = l θ dθ = l θ dθ g k sin θ k g = k sin k l θ dθ g θ k sin θ 7

θ = ϕ θ = θ ϕ = ϕ dθ = dϕ t = k l ϕ dϕ g k sin ϕ = k l g F k l ϕ = k g F π l θ = π T T = k g dθ k sin θ k θ = k l g Kk 8

3 3. 3.. y x x = sin x sin y = x 3. x π y π π y π x = sin y SIN y x = sin yy R π + Nπ y π + Nπ sin y = N SINy + Nπ sin y u x x k x = sn x 3. Kk u Kk k u = sn x k x = sn u = snu k sn u Kk u Kk x = sn u SN u Kk + NKk u Kk + NKk N SNu NKk = sn u x = sn ux R < u < snu, K k = sin u sin u cn u = sn u 3.3 Kk u Kk cn u = sn u cn cos cnu k cn u Kk u Kk x = cn u CN u Kk + NKk u Kk + NKk N CNu NKk = cn u x = cn ux R < u < dn u = dnu k dn u = k sn u Kk u Kk dn u = k sn u dn dnu k dn u Kk u Kk x = dn u DN u Kk + NKk u Kk + NKk DNu NKk = dn u 9

x = dn ux R < u < k = snu = sin u cnu = cos u dnu = sn u cn u 4Kk dn u Kk sn u cn u 4Kk Kk + N + 4Kk u + 4Kk < Kk + N + 4Kk snu + 4Kk = sn u cnu + 4Kk = cn u snu + 4Kk = N+ SN{u + 4Kk N + 4Kk} = N+ SNu NKk = N SNu NKk = N SNu NKk = sn u cnu + 4Kk = N+ CN{u + 4Kk N + 4Kk} = N+ CNu NKk = N CNu NKk = N CNu NKk = cn u dn Kk Kk+N +Kk u+kk < Kk+N +Kk dnu+kk = dn u snu + Kk = N+ SN{u + Kk N + Kk} = N SN{u + Kk NKk Kk} = N SNu NKk = sn u dnu + Kk = k sn u + Kk = k sn u = k sn u = dn u k sn u cn u sin u cos u k = 3. uk = = x x = x { x + } = + x + x log x = tanh x k = sn snu = tanh u cnu = dnu = tanh u = sinh u cosh u = cosh u sinh u cosh = u cosh u cnu = dnu = sech u 3.. sn 3. x = sn u 3.4 3.4 u = x 3.4 3.5 cn d cn u du x k x du = x k x du = x k x = cn u dn u 3.5 d sn u du = cn u dn u = d sn u = sn u d sn u du sn u du = sn u cn u dn u = sn u dn u sn u

dn d dn u du = d k sn du u = k sn u d sn u k sn u du = k sn u k sn u cn u dn u = k sn u cn u sn u = cn u dn u 3.6 cn u = sn u dn u 3.7 dn u = k sn u cn u 3.8 3..3 sn 3. cn dn x = cn u 3.6 3.7 cn u = sn u dn u = cn u k sn u = x k x k = k u = du = x k + k x = x k + k x x x k + k x = x k + k x = cnx k 3. cn dn x = dn u 3.6 3.8 dn u = k dn u sn k u = x x k k = k u = x du = x x k x x x k = x x k = dnx k x 3. 3. x = sin θ = cos θdθ y = θ cos θdθ θ sin θ = dθ x = sin y 3. x = sin φ = cos dφ u = φ cos φdφ = sin φ k sin φ φ dφ k sin φ = sn sin φ 3.9

3.9 am 3. 3.9 φ = am u = amu k x = sin φ = sinam u 3. 3. 3. sn u = sinam u am amplitude 3.3 cn u = sin am u = cosam u am 3. amu = ϕ amnkk = nπ n = am u = amu amu + Kk = amu + π 3. ϕ+π dθ k sin θ = ϕ dθ ϕ+π k sin θ + = u + Kk ϕ dθ ϕ k sin θ = dθ π k sin θ + dθ k sin θ 3.3 3.3 z = sin φ Ek φ = φ k sin φdφ 3. sin φ = sn u π φ π, Kk u Kk 3.3 cos φ = sin φ = sn u = cn u 3.4 3.3 cos φdφ = cn u dn udu 3.4 dφ = dn udu 3. k sin φ = k sn u = dn u 3.3 Ek φ = u dn udn udu = 3.4 z u dn udu ɛu 3.5 dz + nz z k z z = sn w z, Kk w Kk dz = cn w dn wdw = w cn w dn wdw w + n sn w sn w k sn w = cn w dn wdw w + n sn w cn w dn w = dw + n sn w. 3. sn x dn u 3 3. 3

3.4 a + y b = x, y = a sin φ, b cos φ φ = amu k x x = a snu k = a sn u y = b cnu k = b cn u 3.6 k = a cn u dn udu = b sn u dn udu ds = a cn u + b sn u dn udu = a = a a b a ds = + = a a cn u + b sn u dn udu sn u dn udu. cn u + b a sn u dn udu = a sn u dn u k b a sn u dn udu k = a b a 3.7 ds = a k sn u dn udu = a dn u dn udu = a dn udu s.4 s = a u dn udu = aɛu s = aek φ 3.5 3.6 x = a sn u y = b cn u 3.8 k 3.7 e : a b k = = e. 3.9 a F d F P s t Hs F P : P H = e : < e < F P = s + t P H = t e = F P P H s + t e = t e = s + t t 3

a = e e b = e e k = s + t = et e t e t e = e e e e 3.9 e y e e + x =. e e = e e e e e e = e4 + e e e = e 3.: p P 36 3. S S BS = a, OS = a b = ae, b = a e = ae e = e. P a sn u b cn u Sae, P S < e < P S = a sn u ae + b cn u = a e sn u e sn u + = a e sn u = a e sn u. P S = a + e sn u P S + P S = a e sn u + a + e sn u = a =. 3.8 P ρ = ds dθ P a sn u b cn u y = e sn u cn u x + ae cn u x = y = ae cn u y = x = ae cn u cn u e sn u = a sn u a M sn u N ae cn u a MN = sn u + a e cn u = a e sn u a sn u cn = dn u u sn u cn u = a dn u sn u cn u 4

π MN x θ θ < π sin θ = ON MN = sin θ = sin 8 θ ae cn u a dn u sn u cn u = ae cn u sn u cn u a dn u = e sn u dn u sin θ = e sn u dn u e θ = sin sn u π dn u θ < π dθ = e sn u dn u e sn u dn u ρ = ds dθ ds = a dn udu du = e cn u e sn u + e sn u dn u ρ = a dn udu dn u e du = a dn3 u e cn u du = e sn u dn u cn u du = e dn u du 5

4 k k k k snu k sin u cnu k cos u dnu k. k snu k tanh u cnu k dnu k sech u. sn sin sinu + v = sin u cos v + cos u sin v 4. 4. tanhu + v = sinu + v = sin u d sin v dv tanh u + tanh v + tanh u tanh v + sin v d sin u du 4. 4.3 d tanh v dv = cosh v = sech v = tanh v d tanh u = du tanh u d tanh v dv + tanh v d tanh u du cosh u = sech u = tanh u = tanh u + tanh v tanh u tanh v tanh 4. tanh u + tanh v = tanh u d tanh v dv + tanh v d tanh u du tanh u tanh v tanh u d tanh v dv + tanh v d tanh u du tanhu + v = + tanh u tanh v tanh u tanh v = tanh u d tanh v dv + tanh v d tanh u du tanh u tanh. 4.4 v 4.3 4.4 fu sin u tanh u fu + v = fu dfv dv. + fv dfu du k fu fv 4.5 4.3 k = 4.4 k = 4.5 k = dfu fu = sn u = cn u dn u sn du snu + v = sn u cn v dn v sn v cn u dn u k sn u sn v 6

4. 4.. sn snu + v = sn u cn v dn v + sn v cn u dn u k sn u sn v sn u cnc u dnc u + snc u cn u dn u u + v = c F u = k sn u sn c u 4.6 sn c = F u 4.7 c 4.6 u F u 4.7 sn c F = F u/ u = F u u sn u = s cn u = c dn u = d snc u = s cnc u = c dnc u = d F u u F u = N D N = s c d + s c d D = k s s s = c d c = s d d = k s c s = c d c = s d d = k S c N u N = s s s s N = s s + s s s s s s = s s s s. s = c d + c d = s d + c k s c = + k s + k s 3 s = c d c d = s d c k s c = + k s + k s 3 N = s { + k s + k s 3 } s { + k s + k s 3 } = k s s s s D = k s s + s s = k s s c d s s c d = k s s s c d s c d. N F u = = N D ND D d 7

ND = s c d + s c d k s s s c d s c d = k s s s s k s s = N D F u = F u u F u = F = sn c. cn dn cn u cn v sn u sn v dn u dn v cnu + v = k sn u sn 4.8 v dnu + v = dn u dn v k sn u sn v cn u cn v k sn u sn 4.9 v 4.8 s = sn u s = sn v c = cn u c = cn v d = dn u d = dn v 4. cn u + v = sn u + v = k s s c d + s c d k s s. k s s = k s s k s s = c + s d c + s d k s s s c d + s c d = c s s d d + s s c c d d = c c + s s d d cn u + v = c c + s s d d k s s cnu + v = ± c c + s s d d k s s v = cn u 4.8 4.9 dn u + v = k sn u + v = k s s k s c d + s c d k s s. k s s = k s s k s s = k s d k s s k s d k s s = d + k s c d + k s c k s s k s c d + s c d = d d k s s c c d d + k 4 s s c c = d d k s s c c dn u + v = d d + k s s c c k s s dnu + v = ± d d + k s s c c k s s 8

v = dn u 4.9 sn snu + v = sn u cn v dn v + sn v cn u dn u k sn u sn v 4. sn u cn v dn v sn v cn u dn u sn u cn v dn v sn v cn u dn u = sn u sn v k sn v sn v sn u k sn u 4. = sn u sn vk sn 4 u sn v + k sn 4 v sn u = sn u sn v k sn u sn v snu + v = sn u sn v sn u cn v dn v sn v cn u dn u s c d s c d s s snu + v = s s s c d s c d. = s k s s + k s 4 + s k s s + k s 4 s s c c d d s 4 + s s s 4 = s c d s c d s s c c d d = s c d s c d cn u + v = sn u + v cnu + v = s c d s c d s c d s c d 4. 4.. : θ α k = sin α.38 sin θ = k sin ϕ. 4.3.39 ϕ dϕ g g k sin ϕ = l t ϕ = am l t ω = g l sin θ g g = k sin am l t = k sn l t. sin θ = k sn ω t. 4.4 9

4.4 4.4 4.4 θ dθ dt 4.5 cos θ = sin θ = k sn ω t = dn ω t. 4.5 θ cos θ = kω cn ω t dn ω t. θω t = ω k cn ω t dn ω t θ = ω k cn ω t θ = ω k sn ω t dn ω t. 4.6 4.4 4.5 4.6 θ = ω sin θ. 4.4 l α θ p O EOA BB D AD P Q N AD = OA OD = l l cos α = l cos α = l sin α = lk. 4.7 AE = l k = AD AE. 4.: P 7 θ AN = AQ sin QDN = AD sin QDN = lk sin QDN. AN = OA ON = l l cos θ = l cos θ = l sin θ. 3

4.3 sin θ = k sin QDN. QDN = ϕ = am u. u = ω t. 4.8 ϕ P Q... 4.: O BPAB P 4. P P P τ B BB P B BP AB. P τ P.P P τ P P. P P P P. 4.3 P P P P R P P P P R P P. 4.3: P P = vdt, P P = v dt P 3 4.3 N N P P DOA m v v P P υ D N m g DN = mv. υ = g DN υ = g DN. P P P P P υ P υ. P P lim = P P P P DN DN 4.9 3

. RP P RP P P P P = P P P P RP = P RP RP P RP P P P P P 4.9 RP lim P P RP = lim P P DN = P P P P DN. 4. 4.4 DA B P P BR B P P 4.4: BB PP BR B P 4 R. P R P R C 4.4 P R = P C CR = P C CB. 4. P C = P O + CO + OC ON = OB + OC + OC ON., 4. OB = DB + OD, CB = DB + CD OB CB = OD CD = OD + CDOD CD = OD + CD OC. 4.3 4., 4.3, 4. P R = OC OD + CD + OC + ON = OC OD + ON = OC ND. 4.4 P R = OC N D P R N D P R = ND. 4.5 4. R R 4. DA. 4.5: P 5 3

4.5 P P P P P P R P P. BB P P P P P P R P P.BB c 4.6 R P P c 4.6: DA B P 5 P P DA. O P P τ. a OC = a. 4.7: AOP = θ AOP = θ ADQ = φ ADQ = ψ P 6 4.7 AOP = θ AOP = θ.ad P N P N Q Q ADQ = φ ADQ = ψ. DQ = AD cos φ ND = DQ cos φ = AD cos φ 4.. k l l = OP α AD = l l cos α = l cos α = l sin α = lk k = sin α. 4.7.4.4 R R P R = a ND = 4alk cos φ. 33

P R = 4alk cos ψ. P R + P R = alkcos φ + cos ψ. P P = OP + OP OP OP cosθ θ = l + l l cos θ θ = l { cosθ θ} = 4l sin θ θ. P P = l sin θ θ = l sin θ θ sin θ θ = sin θ θ kcos φ + cos ψ a kcos φ + cos ψ l = a l =. 4.6 φ = amu ψ = amv g. ω = l 4.8 u = ω t v = ω t τ :. ω τ = u v. sin θ = k sn u sin θ = cos θ = cos θ = k sn v dn u dn v 4.7. 4.6 4.7 4.8 φ = am u cn u = cos φ ψ = am v cn v = cos ψ 4.8 a l = sin θ cos θ + cos θ sin θ k sn udnv + k sn v dn u sn u dn v dn u sn v = = = 4.9 kcos φ + cos ψ kcn u + cn v cn v + cn u τ v =. u w = ω τ. a l = sn w cn w + cn w = cn w = + cn w cn w + cn w 34

. u v = w 4.9 cn u v + cn u v = sn u dn v dn u sn v cn v + cn u = sn u dn v dn u sn v cn v + cn u sn u dn v dn u sn v sn u dn v dn u sn v = = sn u dn v dn u sn v cn v + cn usn u dn v dn u sn v = cn u k + k cn v k + k cn u cn v cn v + cn usn u dn v dn u sn v cn v cn u cn v + cn usn u dn v dn u sn v = cn v cn u sn u dn v + dn u sn v.4. cn u v + cn u v = s d s d = c c. c + c s d + s d cn u v + cn u v = c c s d s d c + c s d + s d. 4.3 A = cn u v B = c c s d s d C = c + c s d + s d. A + A = B C A = C B C + B B = c c s d s d = s c d s c d s c d + s c d C = c + c s d + s d = s c d + s c d + s c d + s c d C B = s c d + s c d C + B = s c d + s c d. A = C B C + B = s c d + s c d s c d + s c d. A = cn u v = s c d + s c d s c d + s c d. v cn u + v = s c d s c d s c d s c d. 4.. 35

5 5. 5.. sn u = = x φ x k x dφ k sin φ x x = sin φ 5. x = sn u K u K. 5. u Kk Kk, u sn u sn Kk u, v Kk Kk u + v Kk u, v sn K u + v K sn sn u + v = sn u cn v dn v + sn v cn u dn u k sn u sn v u + v = {u + v K K} K u, v, u + v K 5.3 sn u + v = sn {u + v K + K} = sn u + v K cn K dn K + sn K cn u + v K dn u + v K k sn u + v K sn K K u + v K. sn K =, cn K =, dn K = k = k sn {u + v K + K} = sn u + v K k + cn u + v K dn u + v K k sn u + v K = cn u + v K dn u + v K. v = K. 8 cn u K >< u K dn u K sn u = cn u + K >: K u dn u + K sn u ± v = ± cn u dn u, 8 >< k sn u K dn u K cn u = >: k sn u + K dn u + K sn u k cn u ± v = k, dn u ± v = dn u dn u u K K u cnu + v K snu + v = dnu + v K 8 k >< dn u = dn u K dn u = k >: dn u = dn u + K u K K u 5.4 36

i u K, v K sn u cn v dn v + sn v cn u dn u k sn u sn v = cn u K dn u K k sn v K dn v K k cn v K dn v K + dn v K k k cn u K dn u K cn v K dn v K sn u K dn u K k dn u K = k sn v K cn u K dn u K k sn u K cn v K dn v K dn u K dn v K k cn u K cn v K = { k sn u K}{ k sn v K} k { sn u K}{ sn v K} = k { k sn u K sn v K} = k { k sn u K sn v K} sn v K cn u K dn u K k sn u K cn v K dn v K k sn u K sn v K = sn {v K + u K} = sn u + v K. K u + v K K i u + v K K { } cn u + v K cn u + v K snu + v K = sn{u + v K K} = = = sn u + v. dn u + v K dn u + v K ii K u + v K cnu + v K snu + v K = sn{u + v K} = = snu + v. dnu + v K i ii sn ii K u, v K iii u K, K v iv K u, K v sn K u + v K 3K u + v 3K, 4K u + v 4K K u + v K 5.., 5. K =, u = x dt t = sin x x = iy iy dt = t sin iy t = iη y dη = i. + η = iv 37 i =.

= i sinh y iy = sin iv = i sinh v sin sn w = x dt t k t = sn x, k. 5.5 t = sin θ, x = sin φ dt = cos θ dθ = t dθ w = 5.5 x = iy i t = iη φ v dη = dθ k sin θ = sn sin φ, k dt y t k t = sn iy, k 5.6 y dη = i η k η. 5.7 = iv 5.8 η = tan ψ, 5.9 y = tan ϕ 5. dψ + tan cos ψ = ψ + η dψ = dψ 5. cos ψ cos ψ + k η = + k tan ψ = sin ψ + k sin ψ cos ψ = k sin ψ cos. 5. ψ k = k 5.,5. 5.6 v = ϕ cos ψ + η k sin ψ + η ϕ cos ψ dψ = dψ k sin ψ = sn sin ϕ, k sin ϕ = sn v, k. 38

cos ϕ = cn v, k 5. 5.6 5.8 y = tan ϕ = sin ϕ cos ϕ = sn v, k cn v, k. 5.3 iy = sn iv sn iv, k = i sn v, k cn v, k sn 5.4 y < y < +, ϕ π < ϕ < π, v K v K K = π dψ k sin ψ = Kk 5.4 sn iv, k v < v < 5.4 sn iv, k cn iv, k = sn iv, k = + sn v, k cn v, k = cn v, k = cn v, k. 5.5 k sn iv, k dn iv, k = k sn iv, k = + k sn v, k cn v, k = k sn v, k cn v, k = dn v, k cn v, k 5.6 v 5. u v snu + iv = cnu + iv = snu, k cniv, k dniv, k + sniv, k cnu, k dnu, k k sn u, k sn, iv, k 5.7 cnu, k cniv, k sniv, k cnu, k dnu, k + dniv, k k sn u, k sn, iv, k 5.8 dnu + iv = dnu, k dniv, k k sniv, k snu, k cnu, k + cniv, k k sn u, k sn, 5.9 iv, k sniv, k = i snv, k cnv, k, cniv, k = cnv, k, dniv, k = dnv, k cnv, k 5.7 5.8 5.9 snu + iv = snu, k dnv, k + snu, k dnu, k snv, k cnv, k k dn, u, k sn v, k 5. cnu + iv = cnu, k cnv, k i cnu, k dnu, k snv, k dnv, k k dnu, k snv, k, 5. dnu + iv = dnu, k cnv, K dniv, k ik snu, k cnu, k snv, k k dn, 5. u, k sn v, k 39

sn z z sn snz + z = sn z cn z dn z + sn z cn z dn z k sn z sn z 5.7 5.8 5.9 k k z = x + iy, z = x + iy snu + v = sn{x + x + iy + y } x, x, y, y = snx + x cn{iy + y } dn{iy + y } + sn{iy + y } cnx + x dnx + x k sn x + x sn. {iy + y } 5.7 5. snz + z = sn z cn z dn z + sn z cn z dn z k sn z sn z cnz + z = cn z cn z sn z cn z dn z + dn z k sn z sn z, dnz + z = dn z dn z k sn z sn z cn z + cn z k sn z sn z 5.3 5.3. sn sn cnu + K snu + K = sn {u + K + K} = dnu + K = k sn u dn u = sn u, 5.3 k snu + 4K = sn{u + K + K} = snu + K = sn u 5.4 sn 4K sn =, sn K =, sn K =, sn 3K =, sn 4k = 5.5 5.3 5.4 u. sn 4K sn dn u sniv, k = i snv, k cnv, k 5.6 v sn v K 5.3 5.4 sniv + ik, k = i snv + K, k cnv + K, k = i snv, k cnv, k = sniv, k sniv + ik, k = sniv, k. 5.7 k sn ik 5. v = K cnk, k = snik = 5.8 4

5. v = K snk, k =, cnk, k =, dnk, k = k = k 5.9 snu + ik = snu, k cnik, k dnik, k + snik, k cnu, k dnu, k k sn u, k sn ik, k = snu, k dnv, k + snu, k dnu, k snv, k cnv, k k dn u, k sn v, k k snu, k = dnu, k = k snu, k. 5. v = K snu, ik = k snu, k. 5.3 5. s = snk, k =, c = cnk, k =, d = dnk, k = 5.3 snu + ik = snu, k cnik, k dnik, k + snik, k cnu, k dnu, k k sn u, k sn ik, k = d snu, k dnk, k + snu, k dnu, k snk, k cnk, k k dn u, k sn K, k k snu, k = dnu, k = = sn u. k snu, k 5. snu, ik = snu, k 5.3 u sn ik u 5.4 5.34 m n snu + 4K + ik = sn u 5.33 snu + 4mK + nik = sn u 5.34 4K ik 4 ik sn u sn u = ik 5.3 sn 5.3 5.6 sniv + K = sniv = i snv, k cnv, k 5.35 sniv + K = dnv, k 5.36 u = K sn u = K. 4

5.3. cn dn sn cn dn cn cnu + K = k sn u dn u snu + K cnu + K = cn{u + K + K} = k dnu + K = k cn 4K cn u dn u k dn u 5.37 = cn u 5.38 cnu + 4K = cn{u + K + K} = cnu + K = cn u 5.39 cnu + K + ik cnu + K = dn = cnu + K = cn u 5.4 u + K cn K + ik 5.39 5.4 cnu + 4K + K + ik = cn u 5.4 cn 4K K + ik dn dnu + K = k dn u dnu + K = dn{u + K + K} = dn K k dnu + K = k k dn u dnu + 4iK = dn u ik sn u cn u dn u dn 4iK 5.43 5.44 5.4 = dn u 5.43 = dn u 5.44 dnu + K + 4iK = dn u 5.45 dn K 4iK m m Ω Ω sn w cn w dn w fw + mω + m Ω = fw 5.46 snk + iv = dnv, k, cnk + iv = ik snv, k dnv, K k snv, K = ik snv, k dnv, k dn = ik snv, k v, k dnv, k, dnk + iv = k cnv, k dnv, k k sn v, k = k cnv, k dnv, k dn = k cnv, k v, k dnv, k, snu + ik = k sn u, cnu + ik ik sn u dn u ik sn u dn u ik sn u dn u = dn = u k sn = u k sn = i dn u u k sn u, dnu + ik = ik sn u cn u dn = ik sn u cn u u k sn u = ik sn u cn u k sn = i cn u u sn u, snu + ik = sn u, cnu + ik cn u i sn u dn u = dn = cn u, u dnu + ik dn u i sn u cn u = dn = dn u, u 5.47 5.48 5.49 4

snu + K + ik = k snu + K = k cn u = dn u dn u k cn u, cnu + K + ik i dnu + K = k snu + K = i k dn u k cn u = ik dn u k cn u, dnu + K + ik i cnu + K = = i k sn u dn u cn u = ik sn u snu + K dn u cn u, snu + K + ik = snu + K = sn u, cnu + K + ik = cn u, dnu + K + ik = dnu + K = dn u. 5.5 5.5 5.3.3 5.46 Ω Ω fw = sn w Ω Ω 4K ik K ik K + ik fw = cn w Ω Ω 4K + ik K K ik K + ik fw = dn w Ω Ω K 4iK K + ik K ik ik ik 43

6 x 6. y = b sn b = k c k c sn.. sn x y = b sn b c = 6. c x s ds = + ds = + = b x x c cn dn. c c b c cn x c dn x. c cn u = sn u = + k dn u ds = + = + b [ + k { c dn x c b dn x c c k + } ] dn x b = + dn x c c c b x b k c k dn4 = c c k b c x k dn c b + c dn x c + b c x k dn4 c x k dn4 c. b = k k c 6. ds k k = k k x k dn > c x k dn c dn x k x + c k k dn4 = 4 x c k dn + c k = dn x c + dn x c k = dn x c x k = dn c + k cn x c k > + k sn x c + k k dn 4 x c ds { x } = k dn c 44

. s x O ds = x k dn c { ds = s = k x k dn c x dn x c x }.. k = k. 6.3 sin φ = sn u 6.4 k sin φ = dn u 6.4 cos φ dφ = cn u dn u du 6.5 6.5 6.6 cos φ = sin φ = sn u = cn u 6.6 dφ = dn u du. εu u dn u du = Ek φ = φ k sin φ dφ. 3.5. s = c x k ε x 6.7 c x = x x = a sn u = u u = Kk. a K = Kk. = K c c = Kk a. 6.8 x = u = x = a u = K sin φ = sn u φ π. E = Ek = π k sin φ dφ = k dn u du = εk 6.9.x = x = a l x = a 6.7 l = c a k ε a = s = c εk a. c k x = K εk = εk, 6.8 c = a K 6.9 E = εk. l = c 4cE E a = k k a = 4aE a Kk 45

6. 6.. sn ω x y f = ω y 6. u = ω y f = du ρ s z l U = ρω y ds 6. y = ys U ρω ρ ω 6.. x x = x = a T T x T cos ψ = T ds x d ds T cos ψ = d T = ds ds T cos ψ = T ds = = T 6. y ds ρds y ω f f = ρω y ds 6.3 46

y F y F y s f = F y s s d ds f = d ds F ys 6.4 s = ds ds 6.4 ds = df : ds = d ds F ys ds 6.5 :ds y T y F y = T sin ψ = T 6.5 ds = d T ds = d T sin ψ ds ds ds ds 6.3 ρω y ds = d T ds = d T sin ψ ds 6.6 ds ds ds y 6. 6.6 ds = ds = p ds ρω y + d ds d ds T = ds T. ds T = ds + ρω y = 6.7 T p = d ds p = dp ds = p dp ds ds = + 6.8 ds = + 6.7 6.9 6.8 ds = + p 6.9 p dp = ρω T y + p 6. 47

6. p p + p p dp = ρω y + p T dp = ρω T y b y + p = t p dp = dt y : b y p : p t : + p +p dt y = ρω y t T b [ t] +p = ρω T [y ] y b + p = ρω y b T + p = + ρω b y 6. T b p = = y y 6. p = = ρω b y { + ρω b y } 6. T 4T < k < 6. = ρω b T = = ρω b T = ρω b T = ρω b T = b ρω b T ρω T ρω b 4T + ρω b 4T = k } { y b + ρω b y 4T b y 4T + ρω b b ρω b y 4T 4T b 4T + ρω b { y 4T b + ρω b { y 4T b + ρω b y 4T b ρω b 4T 4T + ρω b ρω b 4T + ρω b 4T ρω b y b 4T y b 4T + ρω b 4T + ρω b y 4T b k y b } y b } y b = η = b ρω T + ρω b y 4T b k y b dη = b 48 6.3

c = ρω + ρω b = k T 4T k b k = k 6.4 b dη = b η k c η dη η k η = c 6.5 6.5 sn η = η η dη η k η dη η k η x η = sn c = = x c x c x y = b sn c 6.6 6.4 b = k c 6.7 k 6. 6. 6.3 x x = x = a 6.8 c = Kk a 6.8 Kx y = b sn a l 6.9 l = 4a E k K a. l x = a 9 k b 6.7 6.8 6.4 ρω 4T ω π T x ψ tan tan ψ = = b c 6.3 x y = x 6. T cos ψ = T T = T b k = T cos T c + = T k + 49

Kx y = b sn a Kx y = b sn 3a 6. 6.3 x x ω T ρ = s = ρds y z ρdsω y ρdsω z x y z d T = ds ds d T + ρω y = 6.9 ds ds d T dz + ρω z = 6.3 ds ds 3 6.9 T ds = T 6.3 6.3 y 6.9 z y d T dz ds z d T ds = y T dz d T dz ds z T dz d + T ds ds ds ds ds ds ds ds ds ds ds ds ds = T y dz ds z = H 6.3 ds dz 6.9 ds + 6.3 ds ds ds d T d ds + ρω y ds + T dz dz ds + ρω z ds ds ds ds ds ds ds = T T d ds + ds ds ds ds ds ρω y + T dz dz T dz d dz ds + ds ds ds ds ds ρω z = T ds T + dz ds ρω y + T dz ds T + ds ρω z { = } dz + + ds ds ρω y + z = T + ρω y + z = λ. 6.33 6.3 6.3 6.33 T H λ y + z = r y ds + z dz ds = dr 6.34 5

6.3 6.3 6.35 ds T T ds = T T = ds T 6.35 T y dz ds z ds y dz ds z ds T y dz z = H = H = H 6.34 y + yz y dz z = H T. 6.36 dz y + z dz = { dr } 4 dz + z = { dr }. 4 6.36 y yz dz y z dz = H T dz + z = H. T y + z + y + z y + z { + dz = { dr } + H 4 T } = { dr } + H. 4 dz T r = y + z { r + } dz = 4 { dr } + H T. 6.37 ds = + + dz 6.37 { r + } dz = r { + dz } { ds = r }. 6.37 { ds r } = 4 { dr } + H T { dr } { ds = 4r } H T. 5

ds 6.3 = T T 6.33 { dr } 4T = r T 4 H T. T + ρω y + z = λ T + ρω r = λ T = λ ρω r { dr } = 4r T λ ρω r 4r 4H T = 4r T 4λ 4λρω r + ρ ω 4 r 4 4r 4H T = 4ρ ω 4 { T r 6 λ } T ρω r4 ρ ω 4 4λ r H ρ ω 4 = 4ρ ω 4 T r b r c r d. 6.38 b c d r b c d r b > c r b c r b r c < 6.38 d > b d > b > r > c dr r = c + b c sin φ = b sin +c sin φ = b sin φ + c cos φ. 6.39 6.4 6.39 k = b c d c = b sin φ cos φ c cos φ sin φ = b c sin φ cos φ dφ. 6.4 b r = b b sin φ c cos φ = b sin φ c cos φ = b c cos φ r c = b sin φ + c cos φ c = b sin φ c cos φ = b sin φ c sin φ = b c sin φ d r = d b sin φ c cos φ = d c b c d c sin φ = d c k sin φ. 6.38 6.4 4b c cos φ sin φ φ dφ = 4ρ ω 4 T r b r c r d = 4ρ ω 4 b c cos φb c sin φd c k sin φ T dφ = ρ ω 4 T d c k sin φ dφ = ρ ω 4 T d c k sin φ dφ = ρω d c T k sin φ dφ = ρω d c T k sin φ dφ k sin φ = ρω d c T dφ k sin φ = ρω T d c x. 5

a ρω d c T = K a φ = am φ dφ k sin φ = sn sin φ K a Kx a = = φ φ dφ k sin φ x dφ k sin φ = sn sin φ. Kx 6.39 a { } Kx Kx Kx Kx r = b sn + c sn = b sn + c cn. a a a a 6.4 y F y, x I = Iy x Iy = F y, 6.4 x I y = yx F x, y, yx = y, yx = y 6.4 y I yx. x x = x x = x y x y. I y = yx yx δy. δy y.δy x. δ Y = yx + δyx 6.43 δyx = δyx = 6.44 Y Y x = yx = y, Y x = yx = y y = yx 6.4 6.43 6.4 I y = yx IY = x x Iy = { F y + δy, x x F + δ y, }. 6.45 6.46 53

I δi 6.45 6.46 x { } x δi = Y x yx = F y + δy, + δ F y, x x x { F = x y δy + F δ }. δ = dδy δ = d δy. δi = x x { F y δy + x F δi = δy + F δy x y 6.44 δi = x δyx F x { F y d d F F d } δy. x x x x { d } F δy. F } δy =. F y = 6.47.. 6.. x ρy ds. ds = + = 6.48 y ds = y +.6.48 α 6.49 I = a y α = 6.49 + 6.5 δi = 6.47 6.5 { } d y α y + = + 54

. d y α y + + { y α d { d + } = 6.5 d y + { + y α + y α + y α + + } 3 } d y s «+ = d y + y { { d { d + + y. + + log + + d y =. β + + { + d y = + + } } y y + + «+ + s «+ = = } d y = { + } = y α d y + y = y α d y + = y + y y α d y = y y α + y y α d y = y y α = 6.5 } 3 d y = log α y + A A 6.5 = βα y 6.53 55

. = y b β = α b 6.53 = α y α b α b = α αy + y 4 α + αb b 4 α b = αy + b y + y 4 + aαb b y b 4 α b = y α b y + b α b y α b = b y α b y α b = α b b y α b + b y α b = α b b y { + b y } α b 6.54. α b = ρω T } = ρω b y { + ρω b y T 4T 6.. 6.54 6.6 6.7. y y + α +.6.5. + = βy + α. + = 6.55 = βy + α 6.56 = βy + α = βy + α = βy + α = βy + α. βy + α = t β = dt. x = t dt β = β βx = log t + t + C βx x = log t + t dt t = log t + t β e βx x e βx x βy + α = βy α. 56 = t + t = βy α + βy + α

e βx x e βx x βy α = e βx x βy α = e βx x + y + α = e βx x + e βx x β y = = e βx x + β = e βx x β cosh βx x α. α = e βx x + e βx x α β x x y = a cosh α. 6.57 a. a = β α x. 57

7 [] 4 [] 97 [3] 975 [4] 4 [5] 3 [6] 993 [7] 987 [8] 986 [9] 996 [] 4 [] IB 997 [] 998 58