ИПМ им.М.В.Келдыша РАН • Электронная библиотека Препринты ИПМ • Препринт № 23 за 2014 г. Варин В.П. Плоские разложения и их приложения Рекомендуемая форма библиографической ссылки: Варин В.П. Плоские разложения и их приложения // Препринты ИПМ им. М.В.Келдыша. 2014. № 23. 25 с. URL: http://library.keldysh.ru/preprint.asp?id=2014-23 !"#$ ! "# $%&' !(' ) $ (* +" , (- (" - ./ .* ( # , ' 01+ - #*' ( 2 +" " ' . , ( # 3 ( . * ( ( 21 4 ( 56 / ." 7" "' (4 ( ' 3 +' + ( " (". (* ". / # - " 8 (- -. 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" &# 3 & " ' &%& & ## $ % " % & # 4 % ! & # ' ' !" ' 5 & & ! ! p(x, y)dy − q(x, y)dx = 0, )+. ! p q 6 x, y# 2 " )+.# 7! %& ' ' " % & *8 # 9:-# ; ' ' # ! ' ## ' " ' ' ' 6 ! # 7 " ! & !# 2 ! ' %' # 2 ' " ' ! " # $ ! ! & ! " ! # ' ! ! " ! & &# ! ' )+. ' %' & 5 ). " &&% & < ). " *= # ,8:- ## " < " % ! # > ! " )! . ? *@ # ,,A6 x2y = y − x, ),. ! ! y0(x) = ∞ (k − 1)! xk . k=1 )0. > & " ! )# *1-.# B! )' % . & # )0. ?# % )" . ),. y(x) = y0 (x) + Ce−1/x, )8. ! C 6 # )8. " & )0. # 3 % ),. x y ! ! )0. ## y x ' ! # 3 ' ) % . ! ) . ! ## ' # ; % ' ' ! ## &% ) &%. & # ! # 2 " ' ) '. " ' !' *, # 8@-# $ ' %" ' )+. ) . # C " ) %. 6 &% 5 y(x) = ∞ k=0 C k ekR(x) yk (x), )=. ! C 6 % < y0(x) 6 " &% x< yk (x) k = 1, 2, . . . 6 % " &% x ) y1(x) + C .< R(x) 6 )5 . &%' x# ? & ' % % " # # D" yk (x) k = 0, 1, . . . ' % )=. ! ' ' )## " .# > ),.5 y0(x) 6 ' y1(x) = 1 yk (x) = 0 k = 2, . . . E &% " # & y(x) = P (x) , 1 − C Q(x)eR(x) )@. ! P Q 6 R(x) 6 x# ! & )@. C )=. ! ' & y0(x) = P (x)# 3 )@. x C & )@. " )+. &% & )@. %! # / &%& # $ F , ' % " )=.# >' %& " R(x) ! ' # % " )=. # $ F 0 & )=.# yk (x) k = 1, 2, . . . & ' # 2 " F , !& ' ' " ! ' ' yk (x) k = 1, 2, . . . &# $ F 8 ' " E *9-# E ' " ) , ! . ! !# 2" )& ! E. & ! G E# >" E ! ! " ' ! # C E '! ! '# B E % ! " 6 # / ! E ! ! % & ' ' ' # > & & " # 7 E )=. ! & " ' ' # 2 )+. y0(x) " ! ! &% x# ;" y0(x) ! % ! G " & & # / ! y0 (x) ## y0 (x) % ## y0 (x) ≡ 0 )+. 6 && # (1) y0 (x) x (1) ! (5) " y0(x) # # > ' " ! y0(x) & # ! y(x) = y0(x) + z1(x) ! z1 (x) 6 &% )" &% . & x x → 0# H )+. y0(x) z1(x) L(z1 (x)) := p(x, y0(x))z1 (x) − (qy (x, y0(x)) − py (x, y0(x))y0 (x)) z1 (x) = 0, )1. ! py = ∂p/∂y qy = ∂q/∂y# )1. x z1 (x) = Cexp S(t)dt , )9. 0 ! C 6 % S(x) = qy (x, y0(x)) − py (x, y0(x))y0 (x) . p(x, y0(x)) D S(x) ' &% " x# # 2 S(x) = Br xr−1 + . . . r<0 ):. ' % # ' ! ):. ! )9. x ! ) r = 0. ## z1 (x) ! # 2 ! ! S(x) " ' " # B' # I !" )# ( +.# 7 z1 (x) = CeR(x) y1(x), )+A. ! )=.# ( R(x) = Bxr + . . . y1(x) & )B = Br /r # ( ,.# )+. y(x) = y0(x) + k−1 l=1 C l elR(x) yl (x) + C k zk (x) + O(C k ), )++. ## k"& & # ! )++. " )+. ! 7 C # zk (x) L(zk (x)) = ekR(x) Fk (x), )+,. ## zk (x) )1. ! z1 (x) zk (x) eR(x) k ' ' yj (x) j = 0, 1, . . . , k − 1# 2 ! k k " )+. 7 C # 7 Fk (x) 6 # E )+,. " # 2 & zk (x) zk (x) = Wk (x)y1(x) )+,. )1. ! # 7! Wk (x) Wk (x) − R (x)Wk (x) = ekR(x) Fk (x) . p(x, y0(x))y1(x) )+0. 2 )+0. & Wk (x) Wk (x) = ekR(x) wk (x)# wk (x) wk (x) + (k − 1)R(x)wk (x) = Fk (x) . p(x, y0(x))y1(x) )+8. % )+8. & & Ck exp(−(k − 1)R(x)) & & )+A. " # 2 )+8. ' ! # / )+8. ' " %! ! x = u + o(u) R(x) = Bxr + . . . Bur # 7 % # 7! )+8. dwk (u) Fk (x(u)) + B(k − 1)rur−1wk (u) = (1 + O(u)) , du p(x(u), y0(x(u)))y1(x(u)) )+=. ! O(u) 6 u# )+=. # 2 )+=. ! ! us & s# ? " H +# r < 0" B = 0" s $ % dw(x) + Brxr−1w(x) = xs−1 dx xs−mr m−1 xs−r w(x) = (kr − s) = mrm B BrΓ 1− m=1 k=1 ∞ s/r ∈ N (16) # ∞ s −nr −n Γ n + 1 − x B . s r n=0 r )+@. # $ p(x, y) q(x, y) )+. ! )9. ! S(x) # " ! g# 3 )=. C Cxg yk (x)/xkg k = 1, 2, . . . )# 2 @.# D z1 (x) & x " ' %' ' x B r < 0# $ )=. ) & " # *1-. ' & y0(x) & x# & &' x B # 7 & " ' y0(x) " ' x# ! " " # $ 2 + ! )=. " ! ! ! )@. " ! # 2 + % ! ! x y C exp(R(x)) " ! ) . & # $ % ! " ! !# $ ' ' ' # ! −1 x2(x2 + y(x)2) 2 y(x) = x + y(x)2 + C exp((x − 1)/x2) . 2x − y(x) )+1. 2 C = 0 )+1. y(x) = x# 2 C = 0 % % %! C # 7 )=. " ! 2 x3 x4 − y 4 − x2y 2 + 3 xy 3 y = y (6 x6 − 6 x2y 2 + 6 xy 3 + 2 x5 −6 x5y − 3 x4y + xy 4 + 4 x4y 2 − 4 x4 + 3 x3y 2 + 6 x3y − 3 x2y 3 − 2 y 4), )+9. &%! ! )+1. ! √ x−1 a3 a2 3 a4 145 a5 6 − 2+ 3− y(x) = x + a − + O(a ), a = 1/2 −2 C exp . 2x 8x 4x 128 x4 2 x2 )+9. % ' # ; " / # /&% % ! ! ! > & )*= # :8-.# ! > & ! % 5 ! x4 y = y 3 − x3 . )+:. )+:. y(x) x# $" ! y0(x) R(x) = −3/x# / ! F , y = x + 13 x2 + 19 x3 + 2 4 2 5 181 7 760 5 6 8 81 x − 243 x − 243 x − 6561 x − 19683 x + . . . 59 5 775 6 79 426829 8 4 7 +a x2 + x3 + 37 x + x + x − x − x + . . . 54 162 5832 9720 4723920 58 16 782 5956 42716 9 +a2 x3 + 2 x4 + 27 x5 + 9 x6 + 729 x7 + 10935 x8 − 295245 x + ... 142 6 5641 7 57637 8 37718 9 5 +a3 67 x4 + 31 x + x + x + x + x + . . . + ..., 9 27 972 11664 10935 ),A. ! a = C e−3/x# $ &%' ' & !# D ! %& y )+. " % )=. # 2 ' " ' & # ? ! & # $ (y 3 − x2)y + x3 = 0. ),+. ),+. ' ' y(x) = 1/2 x2 + 1/48 x6 + 1 10 x + O(x14), 320 # $ y0 (x) = x2/3 − 1/2 x2 − x10/3 + O(x14/3), ! p(x, y) ## R(x) = −1/x4/3 # w = x1/3 a = C exp(−1/w4) ! % ),+. y = w2 − 12 w6 − w10 − 55 14 w ... 12 25 −8 409 −4 390377 43060153 4 −12 +a w−16 − 13 w − w − w − − w + . . . 2 24 12 1152 11520 569 −30 1009 −26 8023 −22 2 −38 −34 +a 2 w − 27 w + 6 w − 6 w − 18 w + ... 25649 −48 847471 −44 −52 +a3 6 w−60 − 126 w−56 + 11027 w − w + w + . . . + 12 8 192 ),,. . . . /' ' ' %' ! # /! H + ! n! ? ),.# ! ' )2 + ,.# 2 &% ' " E # ? ' % " ' ' ! " ' ) x → 1/x.# 2' E )# F 8.# % ),0. x3 y = y 2 − x2 . ),0. !# 3! % ! " 2πI0(1/x) + C K0 (1/x) , y(x) = x ),8. 2πI1(1/x) − C K1 (1/x) ! Ij Kj j = 0, 1 6 E ! ! ! ! C 6 # )=. " ),0. ),8. F , ),0.# 2 ),8.5 1 3 3 21 3 x +... + ... y = x + x2 + x3 + . . . + C e−2/x x + x2 + 2 8 4 32 $ ),8. C → 1/C # 2 ! & ! *=-# > # 8A1 " ! &% )JKLMNOJKPQ. # z 2 w = pz + qw, p = 0, q = 0. ),=. / ),=. 5 q w(z) = exp − z p q dt +C , exp t t " 5 ⎞ ⎤ ⎡ ⎛ q ⎣ ⎝ q q2 w(z) = exp − p log z − − . . .⎠ + C ⎦ . ),@. − 2 z 1!z 2!2z ),@. ! ' " + %# > ),@. " # > %5 z = q x w(x) = −p y(x) & ),=. 6 ! ? ),. )8.# 7 ! % " ! # /&% %" # & ? ),. %# ! " ! ' ' R(x) = −1/x # G # 2 ' " ! ! ! % ## x2 y = y − x + a x2 + b x y + c y 2 . ),1. ),1. ! % & &" ! )RST.# 2& ! &% ; *:-# 6 % ! !! ) ! " %' a b c .# B ! U! V# 7! ! # 2 ),1. " ' ' # y = x − (a + b + c − 1) x2 + (a + b + c − 1) (b + 2 c − 2) x3 + . . . +A (1 − 2 c (a + b + c − 1) x +c (a + b + c − 1) (b − 2 + 2 a c + 2 c2 + 2 b c) x2 + . . . ),9. +A2 c − c (b − 2 c + 4 a c + 4 b c + 4 c2 ) x + . . . + . . . ! A = C e−1/x xb+2 c# C ),9. ! " % & % ),1. % c = 0 a+b+c = 1 b + 2 c ∈ Z# $ # ' &'()*+),- .)*'). /0,()*+) 12.-)* -32, /0,()*+),- .)*'). 4)/25.) -3)6 70,832() -0 /0,()*+) W#X# RJYYNQY ' % ' " & 0AA # ? " ! # 4 ' ! " U 4&V# 2 ' ! " ! ! # 2 ! U V *0 # ,1-# ; ! " ' 2 # / )# *0-. " ' % ? *@-# /' ' % )!. & & # ) . %& & " # > E *9- " # 7 & E " & *+A ++ +,- ! $& *+0- E % # C E ! ! )## . " ! +# C # 7 E U V ! ! ' &" ' '# ; ! E ! ' % ) &. # > E y (x) + 2 y(x) y (x) = 0. ),:. ; , % " # $ E *9- # 7& " $ *+0-# $ ),:. x → x/α, y(x) → α y(x) # 2 E ! s = y (0) ' y(0) = y (0) = 0 y(∞) = 1 )# *+8-.# ( ! E # $ *+0-# C E # > " & % ! y(x) ! ! " # $ & *+0- " $ & # C E $ & # > ! y(x) " # > ' R ≈ 2.845019 )# *9 ++ +,-.# 4 E &% # D y(x) y0(x) = x − b ! b > 0 6 & # > y0(x) = x − b ),:. & y(x) y(x) ≈ y0(x) + y1(x) ! y1(x) 6 # E ' ! & & erfc (x) = 1 − erf (x) # 7## # E ' & & y2(x) " # 2 y2(x) ' ! ! ' ). & y1 (x)# $ E & ) . ' ' " % ' # C E % # ( y(x) 6 ),:. y(x + const) # 2 & y(x) " b y = x ) ' .# / + y(x) = x + C y1(x) + C 2 y2 (x) + C 3 y3 (x) + . . . , )0A. ! C 6 ! !# C C x → ∞# 7 )0A. 6 )=. ) x → 1/x. ! ! y0(x) ! x # ; # D y1(x), y2(x), . . . ' " ! ' )# & E *9-.# C ! # √ y1 (x) = 2 exp(−x2) − 2 π x erfc (x), √ √ √ y2(x) = 6 π exp(−x2) erfc (x) − 2 π x erfc2 (x) − 4 2π erfc ( 2 x). )0+. )0,. D )0+. )0,. ) ' '. E# > y1 (x) exp(−x2), y2 (x) exp(−2 x2), x→∞ ) ! , ),:..# 7 # E ! " ' %' )0+. )0,.# ) !. " & y3 (x) = −144 e3(x) + 21π erfc2 (x) √ exp(−x2) − 4 x π√3/2 erfc3 (x) √ −4 π x erfc (x) exp(−2 x2) +√8 π 2 erfc (x) √ erfc ( 2 x) 2 2 2 +2 π x erfc (x) exp(−x ) + 6 3 π x erfc ( 3 x) − 4 exp(−3 x2), )00. ! e3 (x) = √ π ∞ x 1 5 erfc (t) exp(−2 t ) dt exp(−3 x ) − + ... . 6x2 36x4 2 2 2 % RST )0A. " ) . ),:. & O(C 4) )## y3 (x) exp(−3 x2).# RST & " ' % )0+.6)00. ## ! " xn erfc2 (x)dx, xn erfc (x) exp(−x2)dx, . . . ' ' '# ? &# > *+=-# 2 " # D +#=#0#)+. # 0+ ! 5 xλ exp(−b2 x2) erf (a x)dx = − 21b2 xλ−1 exp(−b2 x2) erf (a x) a + b2 √ π x λ−1 2 2 2 exp(−(a + b ) x )dx + λ−1 2 b2 xλ−2 exp(−b2 x2) erf (a x)dx. D e3(x) ' % & y3(x) " ## )# *+= # 0,- " +#=#0#)+,..# ? )0A. E % & # ' # ( e3(x) ' " # erf () & # 7 )0A. ! ' # 2 E # B ' % " *+0- )0A. # 7 9 (30) ! # # [x0, ∞] C # ( " yk (x) x → ∞ !" # ? yk (x) " ! ! exp(−k x2)# y(x) = p x + q + ∞ x v(x) = y (x), (t − x)v(t) dt, )08. ! p q " y(x) ! ! )08. %# 7 )08. E ),:. v (x) + 2 (p x + q + ∞ x (t − x)v(t) dt) v(x) = 0. )0=. )0=. % )0@. )0A. ## vn(x) = yn(x) n ∈ N# 7 v(x) = C v1(x) + C 2 v2(x) + C 3 v3(x) + . . . , y1 (x) = 4 exp(−x2), % v1(x) )0=. v1(x) = const exp(−p x2 − 2 q x). 2 ' p = 1 q = 0# 7 )0=. v (x) + 2 v(x) x = 2 v(x) ∞ x (x − t) v(t) dt. )01. 2 )0@. vn (x) + 2 x vn(x) = Rn (x) = 2 n = 1 n−1 m=1 2 vn−m(x) ∞ x (x − t)vm (t) dt, n ∈ N. v1 (x) + 2 x v1(x) = 0, )09. v1(x) = 4 exp(−x2)# n > 1 vn (x) = − exp(−x2) ∞ x Rn (t) exp(t2 )dt + Cn exp(−x2), )0:. ! Cn = 0 ! # # 9 (36) ! ! vn(x) exp(−n x2)" #! x > 0 # 7 v1(x) = 4 exp(−x2) )09. √ 8 0 > R2 (x) = −16 (1 − x π exp(x2) erfc (x)) exp(−2 x2) − 2 exp(−2 x2). x 2 )0:. 0 < v2(x) exp(−2 x2) ## C2 = 0 ## v2 (x) 6 &%# 7 t > x ! )09. Rn(x) < 0 ## D vk (x) k = 1, 2, 3# > " yk (x) # 2 ) . ' & ' E# 7 )0@. ! vn(x) & & " ) ' ! .# ' ! # / v(x) = w(x) exp(−x2) )01.# 2" w (x) = 2 w(x) ∞ (x − t) w(t) exp(−t2)dt. x )8A. C )8A. $& *+0- " ! w(x) ! # ! " # / $ # P2 : (w1, w2) → 2 ∞ x w1(u) ∞ u 2 (t − u)w2(t) exp(−t )dt du + A, )8+. % A > 0 &% ' C[x0, ∞]# 2 % S(B) = {y(x) : 0 ≤ y(x) ≤ B} ∈ C[x0, ∞] P (w) = P2 (w, w) &% A# 0 < P (y) < P (B) y ∈ S(B)# 2 P (y) ∈ S(B) P (B) < B ## A < B − 1/4 H(x) B 2, ! H(x) = √ )8,. π 1 + 2 x2 erfc (x) − 2 x exp(−x2). C H(x) 6 &% " # 2 )8,. x = x0 x0 < x# B A )8,. # B P S(B) # 2 P &% ## % ρ < 1 ||P (y1 ) − P (y2 )|| < ρ||y1 − y2||, y1 , y2 ∈ S(B). ( ||.|| C[x0, ∞]# y1 = y2 + h ! |h(x)| ≤ ||h|| ≤ B # 7! A = 0 ' P |P (y1 ) − P (y2 )| ≤ |P2 (y2, h)| + |P2 (h, y2)| + |P2 (h, h)| ≤ P2 (B, |h|) + P2 (|h|, B) + P2 (B, |h|) ≤ (2 P2(B, 1) + P2 (1, B)) ||h|| = 3 B P (1) ||h|| < ρ||h||. 2 3/4 B H(x) < ρ, )80. ! ! B &! 0 < ρ < 1 &! x0 ∈ R# )80. 0 < rhs(42) ## B H(x) < 4# 7 &% ## % w = P (w) S(B) ' % A B # ? " wn+1 = P (wn) &! ! w0 ∈ S(B)# 2 A = 4 C w0 = 0# 7! w1 = 4 C, w2 = 4 C + C 2 v2 exp(x2), . . . 2 & wn exp(−x2) )0@. C n & # 2 wn → w n → ∞ )0@. ' # )0@. $ vn(x) < B exp(−x2) ' ∞ ! ## x (.)dt # C )0A. ' C > 0 yn (x) & ) ' ! .# /' C < 0 # Z ' % )0A. & '" ! # ; )# *++ +,-. y(x) 0 ' " ! ! ! ! ' ! # ? ' U& ! V# B ' # $ ! '" )0A. ! y(x) # 7 ! ' E# ! ' " ! y(x) & ' )0A.# ( ' )8,. )80. )0A. & " ## ' # B& B ' C< 2 1 , 9 H(x0) )88. ' )0A. x0 → −∞ O(1/x20)# 2 ' ) 0, .# ; s ≈ 0.664114672430392 & ≈ 30 " ' *+8- ) ,.# $ y(0) = y (0) = 0 y (0) = s ! ),:. x1 = 16# 2 y (x1) ≈ 1.33 × 10−100 ## ! # 2 b = x1 − y(x1 ) ≈ 0.860393828760251. 2! ),:. x1 = 4# ( b )0A.5 y(x) ≈ x + C y1(x) + C 2 y2(x) C = C∗ ≈ 0.11686381064255. ' & ) ! . " *+, # 8#+- ! B = −2 b κ = s/2 Q = 2 C ' S = 2 R# > Q "!# 7 x0 = 0 C∗ < 2/9/H(x0) ≈ 0.125375463 ## E ! ' ! ) b.5 x∗ = R − b ≈ 1.9846# 2 2/9/H(x∗) ≈ 151.4# 7 ! )# !. ## ' )0A. [−b = x0, ∞]# C ! )0A. C 3# B " C = C∗ ! E ) b . )# # +. " # ! ! ' & x = x0 = −b# b C s # C ' U" & EV ## [x0, ∞] C )0A. )&% & . ' y(x0) = y(x0) = 0# ? ' ' x0 C # / % & [J\PQ C ≈ 0.11742437764239, b = −x0 ≈ 0.88673978127115. 2 ! s = y(x0) ≈ 0.57468729191270# 7 )0A. 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