S I.. http://ayapin.film.s.dendai.ac.jp/~matuda /TeX/lecture.html PDF PS.................................... 3.3.................... 9.4................5.............. 3 5. Laplace................. 5.... 5.. 5. Laplace.............. 6........... 6............... 7.3........... 7.3................. 7.3........ 8.3.3 s............ 8.3.4 t............ 8.4.............. 9 x yx y y, y,, y n F x, y, y, y,, y n n yx n n 3 4 3............. 4 3....... 5 3.3............... 6 3.4......... 7 3.5...... 9 3.6................ 3 3.6........... 3 3.6........... 3 3.6.3...... 3
S I. dy fx x fx y fx + C 3 C dy fx 4 x, y dy v C xt y C v e kt k > xt yt gt [ v dt dt v e kt xt v e kt + C k x v + C C k xt v k 3 r r + dr e kt S dt d dt v k e kt } k v k e kt v e kt x± lim xt k < t ± v k g vt yt [ dv dt g dy dt vt vt vt gt + C yt yt gt + Ct + C Sr πr [ S ds πrdr Sr ds πrdr πr + C S C S πr
S I 3 r r + dr V V r 4πr x x x + U Ux 3 λ x x + l m 4 σ r r + dr a M I Ma N [ Nt dn dt dn dt kn dn kdt N log N kt + C N C exp[ kt N N C N N N exp[ kt k log T [ Nt N exp log T t.. dy fx gy 5 x fx y gy gydy fx 6 y x gydy fx + C 7 4 T 5 Lambert x Ix 5 m v Stokes [ Newton m v f m dv mg kv dt k dv mg k v mg log k v v C exp k m dt k m t + C km t + mg k
S I 4 v vt mg exp km } k t v mg k η a k 6πηa ρ m 4πρa 3 /3 v mg/k ρga /9η a Âv..8.6.4. 6 /e -exp-.5*x 4 6 8» t v v vt v 7 v v yt 8 v v vt v g v v 9 Newton T T t T w T 6 v vt [ Newton G M dv dt MG r g r g MG dv dt dr dv dt dr v dv dr Âv [km 4 8 6 4 v dv dr g r v g + C r r v v v g r v ±.7.8.9 g + v g r + v µ r/ : ½.. r v r v r v k
S I 5 g > v v > g v > v rt r u ρ du u g u u dt u v ξ g i ξ > v < g v T gξ t [ ρ + tan ρξ ξ ρξ + tan ξ ξ ξ F ρ gξ t gξ T F ρ ii ξ»t [sec e+8 e+7 e+6 e+5 e+4 e+3.9 g t 3 3.99.8.7 ρ 3 e+ e+ e+ e+ e+3 e+4 µ r/.9999 cx 3/ r t g v rt 7 l λ l x xt g v x x l ± ¹ [ xt vx vx xt dv dt v dv λ l dv dt λl + xg λl xg λgx vx v dv g l x. v, x v g l x + C t x x, v C g dt v l x x l g log x + x x t + C x C l g log x + x x t x
S I 6 x g e l t + e g x x g l l t x cosh g l t hyperbolic cosh t sinh t + d d cosh t sinh t, dt dt x sinh t sinh t cosh t dt v x x x cosh t x x cosh t x sinh t 8 A + B C C x A B a b C [ C A B k kdt dt a xb x ka xb x a x b x b a kt + C a x a b log b x a x b x C e µt µ a b k xt v dv dv dt x cosh t g l x x C a/b x x ab eµt e µt ae µt be µt. T T [ x x T x λ l g l λ l x xt T x [ xt T x T x λ x xg l 3 r.8.6.4. k. 3 4 5» t k.5 k. k. k.5 3 C a., b.9 a > b µ > x b, ẋ a b 4 A X A X X
Ë / S I 7 xt A a X x A X A X k, k 9 x t x s kx k [ dt dt s kxx s kxx x + k s kx s t + C } log x logs kx s s log x s kx C e st x s kx x se st C + ke st s < x C + k < s k.5.5 C e- - C e C S a yt y H [ dt dv avdt v Torricelli v gy dt dv a gy dt dv dv Sdy dv dv a gydt Sdy S y t + C a g y H C S a g yt ga S S y a S a H g t H g t t t C e C e -6-4 - 4 6» 4 k, s A B A B..8.6 -t.4..5.5» 5
S I 8 5 a yt yt n C k + n k n ekαt n kekαt e kαt + I α tanh Iαt tanhx tanh± ± y y 6 y ry yt t ry ytanhx x 7 w H ht g [ ht t + H 3π w gh h n αn α I dn dt I αn nt [ k I/α αdt dn k n k k + n + k n log k + n k n kαt + C k + n k n C e kαt 6 y - tanhx 8 rt 3 cm.5cm 9 Stephan-Boltzmann T T e 4 Boltzmann k Mc dt dt kst 4 T 4 e M c S t T t [ T t T e Boyle Mariotte V p V/p
S I 9 ²T Te.8Te.6Te.4Te.Te.5.5.5 3 3.5 4» t i σ E i + ρ/ t E ρ/ε σ 7 Ω m i log x + a x + a ii x + a a arctan x a a iii x a a log x a x + a a iv a x arcsin x a arccos x a > a x v x + a log + x + a vi ax + bx + c b 4ac log ax + b b 4ac ax + b + b 4ac 4ac b arctan ax + b 4ac b b 4ac>.3 dy y f x 8 u y x, u x du log x fu u + C 9 u y y x x P l O v v B O [ O OP x y Px, y OP x θ dt v B cos θ, dy dt v v B sin θ dy v B sin θ v v B cos θ r x + y, x r cos θ, y r sin θ dy y β x + y β v x v B log x y ux u + x du u β + u du β + u β log u + + u + C x l y u C log l β log x log u + + u l ax + b b 4ac< b 4ac x l β u + + u x } β u + u l u x β x } β l l
S I.5 y l x β x } +β l l.5 À. θ r l c l l θ r tan + π β cos θ 4 x r cos θ β l θ tan x + π 4 β l tan θ x + tan θ + tan θ tan θ tan θ β } β l l x x y x tan θ.5..8.5...4.6.8 t x v v B l 7 v B > v β < O v v B O v B < v l β > x x β y dr dt v B + v sin θ r dθ dt v r cos θ dr dr rdθ dt r dθ dt rθ dr r vb v cos θ tan θ log r v B v log tan θ + π 4 log cos θ + log c.5.5 β.8 β.5 β...4.6.8 8
S I [ t x t l x β+ v B β l β x } +β + β l.4 x px, qx y y β β l 3 yx 4 L 5 i y xy ii 6yy + 9x iii y + y iv y + x + y v x + yy λy vi xy x + y vii xy y x 3 + y viii y y + x y x L[ y dy + pxy qx qx y ce px cwx c x ux u du du wx qx qx wx qx u + c wx Lagrange qx } y e px qx e px + C + 3 A λa B λb C A, B, C N A,, B n B t [ B A λ A C λ B dn B λ A n A t λ B n B t dt dn B + λ B n B t λ A n A t dt n A t λ A n A N A n A dt λ an A n A t N A e λ At e ± px e ± λ B dt e ±λ Bt
ÊB S I n B t e λ Bt } λ A N A e λat e λbt dt + C } e λ λa N Bt A e λ B λ A t + C λ B λ A n B C n B t e λ λ Bt A N } A e λ B λ A t λ B λ A λ AN A λ B λ A e λ A t e λ Bt }.8.7.6.5.4.3.. λ.99 B e λ Bt λ.5 B 4 6 8» λ. B λ A λ. B i Et E } It e t E L e t + C ii Et E sin t } It e t E e t sin t dt + C L Ce t + E L + Ce t + E sin t cos t I i Et E It E e t ii Et E sin t } E It L e + t + sin t cos t..5 9 A B C B Å[A 6 g v 4 L L Et It Et E Et E sin t [ L di di + I Et dt dt + L I L Et Et } It e t Et L e t dt + C, L Et -.5 -....3.4.5» [s L L H, Ω, Hz, E 5 V
S I 3 ON Θt t < t > Dirac. δx x,. δx afx fa δx Laplace 7 C C Et It Et E Et E sin t 8 C C It Q 9 Bernoulli dy + pxy qxyn v y n dv n pxv n qx vx.5 c F x, y, c xy c c parameter F x, y, z x +y c c c c y x y F x, y, c y fx, y 3 y fx, y 4 5 xy q [ V r q xy 4πε r q c 4πε x + y c q x + yy y x 4πε x + y 3 y fx, y
S I 4 y fx, y y x y cx 3 λ P,, Q, y + x + } 4x c 3 λ 3 +λ, λ P,, Q, x + c + y c 3
S I 5 Laplace Laplace. Laplace y n + a y n + + a n y + a n y fx 5 Laplace y yt L [ y sl [ y y s L [ y sy y i L [ y i s i L [ y s i j y j j 5 Laplace [ n L a i y n i L [ f i n a i s i L [ y n a i i i j L [ f + L [ y n a i i j i s i j y j i s i j y j n a i s i i s y L [ L [ y.. fx s F s e st ft dt 6 ft Laplace L [ y y [ e st y dt e st y +s e st y dt 7 lim t e st yt s > 8 yt 7 e st y dt y + s e st y dt 9.. 6 ft [ L [ s e st dt s [ e st 7 ft e at [ L [ e at e st e at dt [ e s at s a s a L [ y sl [ y y
S I 6. Laplace s y ft Laplace I F s s s > t n n : n! s n+ s > 3 t x x > Γ x + s x+ x > 4 e at s a s > a 5 cos t s s + s > 6 sin t 7 cosh t 8 sinh t s + s > s s s > s s > 9 t n e at n! s a n+ s > a e at cos t e at sin t t cos t 3 t sin t s a s a + s > a s a + s > a s s + s s + 8 4 a i L [ cos t L [ cos t [ 4 a i L [ e it s i s + i s + s s + + i s + Laplace L [ e it L [ cos t + i sin t L [ cos t + il [ sin t 33 z a + i L [ e zt s z 34 L [ cosh t L [ sinh t 7 8.. 9 y ky [ sl [ y y + kl [ y L [ y y s + k s + k e kt [ y L y ye kt s + k 35 36 y ky + b y + ky e t y + y [ s L [ y sy y + L [ y L [ y sy + y s + s sin t + s s cos t + 37 38 yt y cos t + y y y sin t y + y cosωt 5,6
S I 7 y + P y + Qy ft P Q 3 ft ft 3 Laplace [ Laplace L [ y sy + P y + y s + sp + Q 4 gs s + sp + Q i gs gs s β s β L [ y sy + P y + y gs A s β + A s β A β y + P y + y β β A β y + P y + y β β yt A e βt + A e βt β P, Q ii gs s + P s Q gs ss + P L [ y sy + P y + y gs B s + P + B s B y P B y + y P yt y P e P t + y + y P P.. 3 Γ p Γ Γ p Γ p + u p e u u p > e u du u p e u du [ u p e u + p pγ p p n u p e u du Γ n + nγ n nn Γ n nn Γ n! ft t x L [ t x e st t x dt st u u x s x e u s du Γ x + s x+.3 s x+ u x e u du.3. f t f t α, β L [ αf t + βf t αl [ f t + βl [ f t 5 L [ [ L [ αf t + βf t
S I 8 α e st αf t + βf t } dt e st f t dt + β αl [ f t + βl [ f t e st f t dt.3.4 t ft a ft-a t t Θ t-a Θ t-a ft-a.3. a a ft F s L [ f s s ft [ t L f d s L [ ft [ ft gt t f d L [ ft L [ g t sl [ gt g g... L [ f sl [ g.3.3 s ft F s e at ft L [ e at ft F s a e st e at ft dt e s at ft dt F s a s s a s F s a s F s 39 L [ e at ft F s a 6 9 s 4 t t s F s s s a ft e at ft t t a F s e as t a < t a ft Θ a t t < a Θ a t Θt a t > a ft t ft L [ Θt aft a e as F s 7 3 t [ L [ Θt aft a t a a e st Θt aft a dt e st a sa ft adt e sa e s f d e sa F s 4 Θ a t [ ft t s L [ Θ a t L [ Θ a t e sa L [ e sa s t
Ê S I 9 5 L L t a E It [ Et Θ a te L di dt + I Θ ate di dt + L I Θ at E L sl [ I I + L L [ I E L I e as e as s L L [ I E L s + s E e as L s s + E } e as e as s s + } L s s + e t t It E } e t a Θ a t It E } e t t a [ Y L [ y s Y + 4sY + 3Y s e s Y Y e s ss + s + 3 e s F s /3 F s s / s + + /6 } s + 3 ft L [ F s 3 e t + 6 e 3t t L [ e s F s ft Θ t yt L [ Y ft ft Θ t 3 e t + 6 e 3t t < e e t 6 e3 e 3t < t. 5 4 L It Et V y. a b t 3» t 5 6 y + 4y + 3y gt, y, y gt Θ t Θ t.4 t > p ft + p ft t
S I p ft L [ f p e st ft dt e st f dt + p e st f dt + 3p p p e st f dt + k e as s e as + e as k e as s + e as k as tanh s t + p 3 t + p n t + n p p f + p f, f + p f, L [ f p e st f d + p + p e st f d + + e sp + e sp + e st f d p e st f d e ps ft p 7 L [ f e ps p e st ft dt 8 k ft a a 3a -k t 6 [ p a 8 L [ f e as e as e as a e st ft dt a a } ke st dt + ke st dt a k s e as + k } s e as e as
S I 4 i k ft a ii sin t k ft 4a t π/ π/ 3π/ 4π/ iii sin t k ft π/ π/ 3π/ 4π/ t t k as tanh as k s + e sπ/ k + e sπ s + e sπ/ 8 L V -V vt T T 3T t vt [ vt it + L di dt V s Is + LsIs i} Is i s + + V s s + L, L Is i t t ie t t V s V e T s s + e T s V s s + L V L V L e T s ss + + e T s } s e T s s + e T s L V L V L } s s + e T s e T s n n } s s + n e nt s e n n+t s} gs s s + ht e t t gse nt s ht nt Θt nt i t V n ht nt Θt nt n ht n+t Θt n+t } t N NT t N + T Θt nt i t V N n ht nt ht n+t } n + V N ht NT N V e T n e t T e n V e t T e T N e e T V e T e t e T N e t NT } + ξ t NT ξ T i ξ V e T e ξ + NT e T + N V + N V e T e T + } e ξ e ξ M i s ξ i s ξ N V [ e ξ e T e T + e ξ N V [ e T e T + e ξ 7 T.,. T.,. L
S I L Å.5 -.5 -.5 Å -.5 ½ Ä. 3 4 5» ½ Ä. 4 + e ξ e } ξ T Θξ T V [ T e e T + e ξ V [ e T e T + e ξ T ξ < T T ξ < T ii C V C t it V it + + sint nπ + β n [ exp t n π } Θ t n π + sin β, + cos β.5 -.8.4 7 3 4 5» T.., i s t t Å.6.4..3.. Å T MT T M + T T T i t V M n ht nt ht n+t } + V V n [ ht MT ht M +T Θt M +T e T e T + e t e t MT } ξ t MT ξ T i s ξ V [ e T e T + e ξ 8 -. -. -.4...3.4.5.6 -.»..5 V 4 C vt it + it dt di C dt + i C dv dt Laplace v i L [ i Is sv s s + C
S I 3 Laplace V s V s Is s + s + e πs V s e πs s + s } s + V e πs + s s + V e πs + s + V e πs + s + V + s + + it V + + cos t n π V s + e πs s s + + + } s s + } s s + + s s + + s + + s s + n s + } s s + } n sin t n π e t n π } Θ e πs n t n π M π t < M + π M n M n M sint nπ + β sint + β n cos nπ sint + β + + e t n π e t M e n π n e t e M+π t e π + it V + sint + β Θ t M + π e t e M+π t e π } + e ξ t M π M+π t Θ t M + π } + iξ V + sinξ + β Θ ξ π } e ξ e Mπ e ξ π e π + e ξ π Θ ξ π } M + i s ξ V + sinξ + β Θ ξ π } e ξ π ξ e π + Θ π } + e ξ sinξ + β e ξ e π e π e π e π ξ < π π ξ < π v c ξ vξ i s ξ V sinξ Θ ξ π } i s ξ i ξ < π ii V V sinξ sinξ + β + + V + V + sinξ γ + + sin γ cos γ +, π ξ < π V + e π e π e e π e π e π e π e e + ξ ξ ξ }
S I 4 ii i i ξ + fξ cosξ γ e π ξ e π e ξ < π fξ ξ ξ ξ v c ξ v c ξ fξ i ii V V π π π 3 3. fx x fx T fx x 9.8.6.5.4.3 T fx T T Å.4... Å -. -. -.4...3.4.5.6 -.» 9.. V T T Å.8.6.4. -. -.4.5.4.3.. -. -.6...3.4.5.6 -.».. V Fourier Å n 9 fx + nt fx x 3 T 3T 4T fx fx gx T hx αfx + βgx 3 T 9 3 3 [ n fx + nt fx + n T + T fx + n T fx + n T fx
S I 5 fx gx T hx + T αfx + T + βgx + T αfx + βgx hx hx T 43 fx x T fax a T/a sinx cosx fx a + a cos π T x + b sin π T x + a cos 4π T x + b sin 4π T x + a k cos kπ T x + k k a k sin kπ T x 3 a n b n T T/k T/k T 44 45 i iv vi / π i fx ii fx iii fx x < x < < x < π cos x < x < < x < π π + x < x < π x < x < π iv fx x < x < π n,,, 3, sin nx / ii x sin nx e x sin nx v vii cos nx / / iii x cos nx e x cos nx x sin nx 3. T π fx a + a k cos kx + b k sin kx 33 k fx a k b k a 33 π fx [ a + a k cos kx + b k sin kx a + k k a k cos kx + b k sin kx πa + a π fx 34 46 k cos kx sin kx a b j 33 cos jx π fx cos jx [ a + a k cos kx + b k sin kx cos jx 35 k a cos jx + [a k cos kx cos jx a j π a j π k b k sin kx cos jx fx cos jx j,, 36 b, b, 33 sin jx π fx sin jx
S I 6 [ a + a k cos kx + b k sin kx cos jx 37 k a sin jx + [a k cos kx sin jx b j π b j π k b k sin kx sin jx fx sin jx j,, 38 j n a n fx cos nx π [ k cos nx + k cos nx π π sin nx k sin nx π n + k n b n fx sin nx π [ k sin nx + k sin nx π π cos nx k cos nx π n k n 4k nπ n, 3, 5, b c a a π a n π b n π fx fx cos nx fx sin nx n,, 39 4k 3 5 sin x + sin 3x + sin 5x + π x π π f k 4k 3 5 7 + + π 47 k j cos kx cos jx πδ kj π k j δ kj k, j k j sin kx sin jx cos kx cos jx πδ kj T π fx a, a, b, fx fx 3 k < x < fx + π fx, fx k < x < π [ a π fx [ π [ kπ + kπ π k + k 3 + 5 7 + π 4 π 3.3 fx fx π x π fx fx x fx x
S I 7 fx x x fx fx fx lim h fx h fx + fx + lim h fx + h x fx h fx lim h h fx + h fx + lim h h f x f - f + 48 49 π fx i fx x < x < π ii fx x < x < π < x < iii fx x < x < π x < x < iv fx π /4 < x < π fx a n, b n kfx ka n, kb n k x 3.4 p. L E sin t p.7 3 y + P y + Qy ft P, Q ft 3 L L Et It Et E sin t [ It di dt + L I L Et It A cos t + B sin t di dt A sin t + B cos t A B L + B cos t + L A sin t E L sin t t sin t, cos t A L + B, B L A E L A, B A E L B E + L + It 5 It E L/ + sin t cos t L L Et It Et E cos t
S I 8 3 ft sin t y + λy + y sin t D 3.. e-4 e-5 e-6 δ π 3π/4 π/ π/4 λ λ λ ³ sin, cos A, B A + λb λa + B yt λ cos t + sin t + λ sint δ tan δ λ + λ D δ D + λ λ δ arctan 3 4 π sin δ 5 CL L C Et sin t It 5 L C Et sin t It ³ 4 Etsin t L C [ yt A cos t + B sin t y t A sin t + B cos t y t A cos t + B sin t A + λb + A cos t + B λa + B sin t sin t 5
S I 9 3.5 y + P y + Qy ft ft a + an cos n t + b n sin n t n n y + P y + Qy a n cos n t + b n sin n t y n yt y t + y t + 33 π y + λy + y ft < t < ft < t < π [ ft fourier ft 4 π n n:odd sin nt n 4 n a n + λnb n + a n cos nt n b n λna n + b n sin nt 4 sin nt nπ sin, cos a n, b n n a n + nλb n nλa n + n b n 4 nπ a n 4 nπ nλ π n + nλ b n n nλ y n t 4 nλ cos nt + n sin nt nπ n + nλ 4 nπ sinnt δ n tan δ n + nλ nλ a n n D n δ n 4 D n nπ n + nλ nλ δ n arctan n 6 3., λ.5 D n n 3 3. n 3 7 n,3,5,7 y + λy + y 4 sin nt n : 4 nπ y n t yt y t + y 3 t + y 5 t + 4.. y n t a n cos nt + b n sin nt y nt nb n cos nt na n sin nt y nt n a n cos nt n b n sin nt. 3 4 5 6 7 8 9 6
Ä S I 3 Ê 3 - - -3 7 3 4 5 6 7 8 9 3 4 5» t 34 L L T it V -V vt T T 3T t vt [ p. it dit + vt it dt L L vt vt 4V π sin πt T + 3πt sin 3 T + 5πt sin 5 T + dit + dt L it 4V nπt sin n 4 Lnπ T i n t yt i t + i 3 t + i 5 t + 4 i n t a n cos nt + b n sin nt L π T di n t dt nb n cos nt na n sin nt 4 nb n + a n cos nt + na n + b n sin nt 4V sin nt Lnπ nb n + a n, na n + b n 4V Lnπ a n LT 4V n + } b n n a n a n, b n 4V sin nt cos nt n i n t } LT n + 4V πn n + sinnt δ n tan δ n n, n : 3 8.5 -.5-8 3 4 5» L T,, L., V 3
S I 3 3.6 Fourie Laplace > fx [ cos x fξ cos xi dξ π sin x fξ sin xi dξ d fx [ a cos x + b sin x d π a fx cos x b fx sin x 43 3.6. T f T x a + a n cos n x + b n sin n x a T a n T b n T n T/ T/ T/ T/ T/ T/ f T x f T x cos n x f T x sin n x n nπ T n+ n n + π T nπ T π T 4 f T x T + π T/ T/ [ n f T ξdξ T/ cos n x f T ξ cos n ξ dξ T/ T/ sin n x f T ξ sin n ξ dξ T/ T n fx fx lim f T x T d gn gd 3.6. fx fx fx 43 fx x 35 x < fx x > [ a b a b fξ cos ξ dξ cos ξ dξ fξ sin ξ dξ sin ξ dξ sin ξ cos ξ sin
S I 3 fx π cos x sin a f a x π a cos x sin 9 d d a..8.6.4. a4 a6 a64 -. - -.5 - -.5.5.5 9 a 3.6.3 43 c a ib c fξe iξ dξ fξe +iξ dξ c fx π π π e[ce ix d fξcos ξ i sin ξ dξ ce ix + c e ix d [ d fξe ix ξ dξ + d fξe ix ξ dξ [ d π [ d π π d + fξe ix ξ dξ dη fξe ix ξ dξ dη fξe ix ξ dξ fξe iηx ξ dξ fξe iηx ξ dξ fx cos x sin + d π π 4 π x x < x < x x sin d π C fxe ix 44 π fx fx Ce ix d 45 π C Six x sin d x