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- ゆき ありの
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1 F(s) G(s) H(s) G(s) F(s) H(s) F(s),G(s) H(s) : V (s) Z(s)I(s) I(s) Y (s)v (s) Z(s): Y (s):
2 2: ( ( V V 2 I I 2 ) ( ) ( Z Z 2 Z 2 Z 22 ) ( ) ( Y Y 2 Y 2 Y 22 ( ) ( ) Z Z 2 Y Y 2 : : Z 2 Z 22 Y 2 Y 22 Z,Z 22 Z 2,Z 2 ( I I 2 V V 2 ) ) 2..2 :H(s) H(s) P(s) Q(s) b ns n + b n s n + + b 0 a m s m + a m s m + + a 0 a i,b j : H(s) V out(s) V in (s) CRs + 3: (LPF) s s H(s) 2..3 H(s) H(s) P(s) Q(s) b n a m (s s 0)(s s 02 ) (s s 0n ) (s s p )(s s p2 ) (s s pm ) b n H : (scale factor) a m s 0, s 0n (zero) s p, s pn (pole) H(s 0i ) 0 H(s pi ) 2
3 4: s s, s 2 + (s+i)(s i) s Re Re s n > m s H(s) Hs n m (H: ) H(s) s H(s) (n-m ) n < m H(s) Hs H(s) s m-n m n 2..4 x(t) y(t) X(s) L[x(t)] Y (s) L[y(t)] H(s) Y (s) X(s) y(t) L [Y (s)] L [H(s)X(s)] x(t) δ(t) x(t) δ(t) X(s) L[δ(t)] Y (s) H(s) H(s) y(t) L [H(s)] H(s) K L Ke spit Ke αpit e jωpit s s pi (K : s pi α pi + jω pi ) s pi s s m H(s) K m (s s pi ) m + K m (s s pi ) m + + K (s s pi ) 3
4 K m L (s s pi ) m K m (m )! tm e spit m 2..5 t m s H(s)X(s) ( V V 2 ) ( ) ( Z Z 2 Z 2 Z 22 I I 2 ), ( I I 2 ) ( ) ( Y Y 2 Y 2 Y 22 V V 2 ) Z,Z 22,Y,Y s Z 2,Z 2,Y 2,Y 2...n m + 4
5 2.2.3 H(s) s jω log H(jω) log H(jω) + j arg(h(jω)) H(s) H (s s 0) (s s 0n ) (s s p ) (s s pm ) s jω jω 5: H(jω) H(jω) H d 0d 02 d 0n d p d p2 d pm arg(h(jω)) log H (jω) log H + n m θ 0i i i i θ pi n m log d 0i log d pi i 5
6 (All pass) 6: (All pass) H(s) P(s) Q(s) H (s)h 2 (s) H k (s) H i (s) () H(s) b s + b 0 s + a 0 H s + β s + α b,b 0 LP Low-Pass (b 0) H(s) b 0 s + a 0 s jω 7: LP 2 HP High-Pass (b 0 0) H(s) b s s + a 0 s jω 6
7 8: HP 3 AP All-Pass H(s) s a 0 s + a 0 s jω 9: AP 4 H(s) b s + b 0 s + a 0 s jω 7
8 0: : (2) H(s) b 2s 2 + b s + b 0 s 2 H s2 + ( ωn Q N )s + ωn 2 + a s + a 0 s 2 + ( ω0 Q 0 )s + ω0 2 H(s) s R L s 2 + s R L + LC 8
9 2: LP H(s) Hω 2 0 s 2 + ( ω0 Q 0 )s + ω 2 0 3: LP 2 HP Hs 2 H(s) s 2 + ( ω0 Q 0 )s + ω0 2 4: HP 3 BP(Band-Pass) 9
10 H(s) H( ω0 Q 0 )s s 2 + ( ω0 Q 0 )s + ω 2 0 5: BP 4 H(s) H(s2 + ω 2 N ) s 2 + ( ω0 Q 0 )s + ω 2 0 6: 5 AP ( ) H(s) H(s2 ( ω0 Q 0 )s + ω 2 0) s 2 + ( ω0 Q 0 )s + ω 2 0 0
11 7: AP : 9: (All-pass) Z a Z b R 0
12 H(s) R 0 Z a (s) R 0 + Z a (s) Z a,z b All-Pass Z in R 0 R 0 ) Z a Z a (s) Z a (s) Z a ( s) H(s) s p R 0 + Z a (s p ) 0 R 0 Z a ( s p ) 0 R 0 Z a (s) s s p s p s p s p H(s) (s s 0) (s s 0n )(s s 0)(s s 02) (s s p ) (s s pm ) s 0 s 0n : s 0,s 02 : (s + s 0)(s + s 02) H(s) (s s 0) (s s 0n )(s + s 0)(s + s 02) (s s p ) (s s pm ) (s s 0)(s s 02) (s + s 0 )(s + s 02 ) H(s) 20: H( ) H(s) /H(s) 2
13 2: H(s) H(s) 2.5 R,L,C 2.5. e(t) i(t) t e(t )i(t )dt Hurwitz Z(s) )s 2)Re(s) > 0 ReZ(s) > 0 Z(s) Z(/s),/Z(s) Z(s) ReZ(jω) 0 s jω s Z(s) (Re(s) > 0) ReZ(s) > 0 s Z(s) 0 Z(s) Z(s) s 3
14 Z(s) ()s (2)Re(s) > 0 Z(s) (3) Re(jω) 0 (4) 2.6 L,C Z(s) P(s) Q(s) P(s) Q(s) Z(s) Z(s) H (s 2 s 2. 0) (s 2 s 2 0n) s (s 2 s 2 p ) (s2 s 2 p(n ) ) 2. H s (s 2 s 2 0) (s 2 s 2 0n) (s 2 s 2 p ) (s2 s 2 pn) Hs (s2 s 2 0) (s 2 s 2 0n) (s 2 s 2 p ) (s2 s 2 pn) H s s p,,s pn, s p,, s pn s 0,,s 0n, s 0,, s 0n (s 2 s 2 0) (s 2 s 2 0(n ) ) (s 2 s 2 p ) (s2 s 2 pn) s s s,0 3 Z(s) Z(jω) jx(ω) () dx(ω) dω > 0 X(ω) 4
15 (2)X(ω) 22: 0 23: 0 24: 0 25: 0 26: 27: 28: : 0 0 5
16 2.6.4 Z(s) Z(s) s5 +6s 3 +8s (s 2 +)(s 2 +3) )Foster s 4 +4s 2 +3 s(s2 +2)(s 2 +4) Z(s) sl sc s /s Z(s) s5 + 6s s s 4 + 4s s + 2 s s s s s s s + 2s + 6 s 30: Foster 2)Foster Y (s) /Z(s) /sl,sc s Y (s) Z(s) s4 + 4s s 5 + 6s 3 + 8s 8 3 s + 4 s 3 s s s s + 4s + 8 s s s 3: Foster 3)Cauer Z(s) Z(s) L s + C s + L 2s+ 6
17 L s C s Z(s) L Z(s) s5 + 6s 3 + 8s s 4 + 4s s(s4 + 4s 2 + 3) + 2s 3 + 5s s 4 + 4s 2 s + 2s3 + 5s + 3 s 4 + 4s s + s + 2 s s+ 3 2 s+ s 3 s 4 +4s s 3 +5s 32: Causer 33: Causer 4)Cauer 2 Z(s) Z(s) C s + L + s C 2 s + 2 Z(s) + 6s2 + 6s 4 2s + 8s 3 ( + 4s2 ) + 2s 2 + 6s 4 2s( + 4s 2 ) 2s + s + s +3s Z(s) s5 + 6s 3 + 8s s 4 + 4s s 2 +s 4 8s+6s 3 +s 5 2s + 2s+8s 3 2s 2 +6s s 7 32 s s s s 34: Causer 2 7
18 35: Causer 2 36: Causer Z(s) Z(s) Z(s) ( Z (s) Z 2 (s) ) Z 2 (s) Z 22 (s) Z Re[Z (s)x 2 + 2Z 2 (s)x x 2 + Z 22 (s)x 2 2] > 0 (Re[s] > 0 ) Z 2 Z 2 x,x 2 Q Z (s)x 2 + 2Z 2 (s)x x 2 + Z 22 (s)x 2 2 Q Q Z Z Q (Re(s) > 0 Q > 0 Z (Y ) s a + bj x,x 2,a,b 4 Z Z Z,Z 22 ( ) Z 2 (Foster ) Z (s) h(0) s + k Z 22 (s) h(0) 22 s + k Z 2 (s) h(0) 2 s + k 2sh (k) s 2 + ωk 2 + h ( ) s 2sh (k) 22 s 2 + ωk 2 + h ( ) 22 s 2sh (k) 2 s 2 + ωk 2 + h ( ) 2 s 8
19 h 0 s h (0) 0 h(0) 22 0 h(0) h(0) 22 (h(0) 2 )2 0 h (k) 0 h(k) 22 0 h(k) h(k) 22 (h(k) 2 )2 0 h ( ) 0 h ( ) 22 0 h ( ) h( ) 22 (h ( ) 2 )2 0 Z,2 2 Z 2 k a,b (a,b) ( s 2 +a s 3 +4s b s 3 +4s ( ( a 4 )s s ( a 4 ) s ( b 4 )s s ( b 4 ) s b s 3 +4s s 2 +2 s 3 +4s ) ( b 4 )s s ( b 4 ) s ( 2 )s s ( 2 ) s a 4 0, 2 0, ( a 4 ) 2 (b 4 )2 0 ) a 4 0, 2 0, a 4 2 (b 4 )2 0 a 2 b2, 8 b 2 a, 0 a : a-b 9
20 2.7 RL RC 2.7. RL RC RL Z (s) ψ(s) (R C ) Z LC(s) Z RL (s) ψ(s) Z LC (s) s ψ(s2 ) Z RL (s) s3 + 6s 2 + 8s s 2 + 4s + 3 Z LC(s) s s6 + s 4 + 8s 2 s 4 + 4s Foster RC R L Z RC (s) φ(s) Z LC (s) sφ(s 2 ) 3 ( ) 38: g6 3. ω 0 20
21 LPF LPF,HPF,BPF,BEF Butterworth( ) : Tchebyscheff( ) : LPF ( )F(jω) F(jω) 2 + ϕ 2 (ω) (0 < ω < ), ( < ω) 0 ( log 0 ) ω ϕ 0,ω ϕ ϕ LPF 3.. Butterworth ϕ(ω) ω n F(jω) 2 + ω 2n n Butterworth F(j0) 2 F(j ) 2 0 F(j) 2 2 s jω F(jω) 2 F(jω)F(jω) F(jω)F( jω) F(s)F( s) + ( ) n s 2n + ( ) n s 2n 0 s 2n ( ) n+ ( ) n (e πj ) n e (n )πj e ((n )π+2kπ)j k : s e (n )π+2kπ 2n j 2n 2
22 39: F(s) F( s) s F(s) 2n F(s) F(s) 2 +( ) n s 2n n F (s)f ( s) s 2 ( s)( + s) F (s) + s 3..2 Tchebyscheff ϕ(ω) εv n (ω) n Tchebyscheff V n (ω):n cos(ncos ω) ω V n (ω) cosh(ncosh ω) ω V n (ω) 2ωV n (ω) V n 2 (ω) (V 0 (ω), V (ω) ω) V n (ω) ω n F(jω) 2 F(jω)F( jω) ω + ε 2 V 2 n (ω) ω V n (ω), V n () + ε 2 + ε 2 V 2 n (ω) 22
23 40: g H(jω) H(jω) e jθ(ω) τ(ω) dθ(ω) dω τ(ω) θ(ω) ω 3.2 LC 4: F(jω) V 2 V 0 23
24 0, 0 R R 2 R 0 R ( ) V I ( A C ) ( B D V 2 I 2 ) 42: ( ) ( ) ( ) ( ) V A B A 2 B 2 V 3 I C D C 2 D 2 I 3 ( ) ( ) A A 2 + B C 2 A B 2 + B D 2 V 3 43: C A 2 + D C 2 C B 2 + D D 2 I 3 V 3 V H(s) V 3 V A A 2 + B C 2 44: I I 2 I, V V 2 IZ ( ) ( ) ( ) V Z V 2 0 I I 2 24
25 45: V V 2 V, I I 2 (I I 2 )Z V ( ) ( )( ) V 0 V 2 I Z I 2 LPF ( ) ( ) ( ) ( ) V R 0 V 2 I 0 sc I 2 ( )( ) + scr R V 2 sc H(s) 46: LPF + scr I 2 ( ) ( ) ( ) ( )( ) V sl 0 sl 2 V 2 I 0 sc 0 I 2 ( )( ) s 2 CL + s 3 CL L 2 + s(l + L 2 ) V 2 sc s 2 CL 2 + I 2 ( ) ( ) V 2 I 2 47: Z in, Z out ( ) ( ) ( V A B C D I V AV 2 + BI 2 V 2 I 2 ) 48: I CV 2 + DI 2 25
26 Z in V I AV 2 + BI 2 CV 2 + DI 2 AR 2 + B CR 2 + D ( V 2 I 2 R 2 ) Z in Z in Z in s Cauer Z out Z out DR + B CR + A R R 49: R ( ) ( ) ( ) R A B A + CR B + DR 0 C D C D F(s), T(s) F(s) T(s) V 0 V 2 A + CR N (s I20 2 ) + sn 2 (s 2 ) I 2 0 A C A N (s 2 ),CR sn 2 (s 2 ) Z in AR 2 + B CR 2 + D A + B R 2 C + D R 2 R 2 A C N (s 2 ) sn 2 (s 2 ) R :3 R T(s) s 3 + 2s 2 + 2s + 26
27 3 ( 6 ) R N (s 2 ) 2s 2 + sn 2 (s 2 ) s(s 2 + 2) Z in R Z in 2s2 + s 3 + 2s 2 s s+ 3 2 s 50: 3 R R R 0 R 5: 0 R ( ) ( ) ( ) A B 0 A + B R 2 B C D R 2 C + D R 2 D 27
28 F(s), T(s) F(s) T(s) V 0 V 2 A + B N (s I20 2 ) + sn 2 (s 2 ) R 2 I 2 0 A B A N (s 2 ), B R 2 sn 2 (s 2 ) Z out DR + B CR + A R 0 B A sn 2(s 2 ) N (s 2 ) R 2 3 :3 R 2 Z out s3 + 2s 2s s s s 52: 3 0 R Z out R R R LPF ( ) ω 0 ω c R R LPF LPF,HPF,BPF,BEF 3.3. LPF LPF (cut off Freq) ω c R 28
29 LPF x jx j ω ω c LPF ω ω ω c jxa i j ω ω c a i j a i ω c ω ω c jxb i jωb i j bi ω c ω x ω P64 R R R R R a i a ir ω c b i b i ω c R L C LPF LPF LPF HPF x < x > ω c jx j ω ω c x ω c ω jxa i j ω c ω a i j( ω ca i )ω R j( ω ca i )ω j( ω ca ir )ω C ω ca ir j ω jxb i ω c b i j R ω c b i ω L R ω cb i 29
30 3.3.3 LPF BPF x < ω < ω < ω 2 HPF LPF ω ω x k ω 0 k 0 2 ω ω ω 0 x 0 ω 0 ω ω 2 k k 2 k ω0 ω 2 ω ω b ω b ω 2 ω p64 jx j ω 0 ω b ( ω ω 0 ω 0 ω ) p LPF BEF BEF BPF jx j ω0 ω b ( ω ω 0 ω0 ω ) 53: LPF
31 5. x Z,Y : 54: : { V (x) V (x + x) + Z xi(x) I(x + x) I(x) Y xv (x + x) { V (x+ x) V (x) x ZI(x) I(x+ x) I(x) x 0 () x Y V (x + x) { dv (x) dx ZI(x) di(x) dx Y V (x + x) Z,Y x { V (x) Ae γx + Be γx I(x) Z 0 (Ae γx Be γx ) γ Y Z : Z 0 Z γ : 3
32 (2) γ α + βj Ae γx e αx x Be γx 56: Z 0 57: 0 B 0 V Ae γx I Z 0 Ae γx x V I Z 0 Z 0 Z 0 Z,Y Z jωl Y jωc γ Y Z jω LC jβ L Z 0 C λ 2π β 2π ω LC 32
33 5.3 x V ( l) V,I( l) I,V (0) V 2,I(0) I 2 () { A + B V 2 A B Z 0 I 2 A V 2 + Z 0 I 2, B V 2 Z 0 I : () x l V V 2 cosh γl + Z 0 I 2 sinhγl I V 2 Z 0 sinhγl + I 2 cosh γl (3) 59: Z 2 Z V 2 I 2 Z 2 Z V I Z 0 V 2 cos γl + Z 0 I 2 sinhγl V 2 sin γl + Z 0 I 2 cosh γl Z 0 Z 2 + Z 0 tanh γl Z 0 + Z 2 tanh γl Z Z 2 0) Z s Z Z20 Z 0 tanh γl 33
34 Z 2 ) Z f Z Z2 Z 0 tanh γl Z 0 coth γl Z s Z f Z 2 0 Z 0 Z s Z f, tanhγl Z 0 Z 0 (Z s ) ( ) (Z f Z s Z f { Z s jz 0 tan βl Z f jz 0 cot βl l λ 4 β 2π λ βl π 2 Z s Z f 0 l λ 4 λ 4 l Z s Z f 0 < l < λ 4 L C λ 4 < l < λ 2 C L 5.4 Z L (2) { V Ae γx + Be γx Z 0 I Ae γx Be γx 60: x 0 V,I V Z L I A + B, Z L Z 0 A + B A B Z 0 I A B B A Z L Z 0 Z L + Z 0 s 34
35 s A B Z L Z 0 s 0 B 0 Z L 0 s Z L s Ae γx : Be γx : s(x) Beγx Ae γx B A e2γx : ( ) s(0)e 2γx s(0)e 2αx e 2jβx γ α + jβ s s age 7 3 TEX : 7/3( ) : : 7:40 : 3000 m( )m 35
x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x
[ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + y ),
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信号処理の基礎 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/081051 このサンプルページの内容は, 初版 1 刷発行時のものです. i AI ii z / 2 3 4 5 6 7 7 z 8 8 iii 2013 3 iv 1 1 1.1... 1 1.2... 2 2 4 2.1...
S I. dy fx x fx y fx + C 3 C vt dy fx 4 x, y dy yt gt + Ct + C dt v e kt xt v e kt + C k x v k + C C xt v k 3 r r + dr e kt S Sr πr dt d v } dt k e kt
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213 March 25, 213, Rev.1.5 4........................ 4........................ 6 1 8 1.1............................... 8 1.2....................... 9 2 14 2.1..................... 14 2.2............................
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
S I.. http://ayapin.film.s.dendai.ac.jp/~matuda /TeX/lecture.html PDF PS.................................... 3.3.................... 9.4................5.............. 3 5. Laplace................. 5....
LCR e ix LC AM m k x m x x > 0 x < 0 F x > 0 x < 0 F = k x (k > 0) k x = x(t)
338 7 7.3 LCR 2.4.3 e ix LC AM 7.3.1 7.3.1.1 m k x m x x > 0 x < 0 F x > 0 x < 0 F = k x k > 0 k 5.3.1.1 x = xt 7.3 339 m 2 x t 2 = k x 2 x t 2 = ω 2 0 x ω0 = k m ω 0 1.4.4.3 2 +α 14.9.3.1 5.3.2.1 2 x
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II Karel Švadlenka * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* u = au + bv v = cu + dv v u a, b, c, d R
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II Karel Švadlenka 2018 5 26 * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* 5 23 1 u = au + bv v = cu + dv v u a, b, c, d R 1.3 14 14 60% 1.4 5 23 a, b R a 2 4b < 0 λ 2 + aλ + b = 0 λ =
4 5.............................................. 5............................................ 6.............................................. 7......................................... 8.3.................................................4.........................................4..............................................4................................................4.3...............................................
z f(z) f(z) x, y, u, v, r, θ r > 0 z = x + iy, f = u + iv C γ D f(z) f(z) D f(z) f(z) z, Rm z, z 1.1 z = x + iy = re iθ = r (cos θ + i sin θ) z = x iy
z fz fz x, y, u, v, r, θ r > z = x + iy, f = u + iv γ D fz fz D fz fz z, Rm z, z. z = x + iy = re iθ = r cos θ + i sin θ z = x iy = re iθ = r cos θ i sin θ x = z + z = Re z, y = z z = Im z i r = z = z
#A A A F, F d F P + F P = d P F, F y P F F x A.1 ( α, 0), (α, 0) α > 0) (x, y) (x + α) 2 + y 2, (x α) 2 + y 2 d (x + α)2 + y 2 + (x α) 2 + y 2 =
#A A A. F, F d F P + F P = d P F, F P F F A. α, 0, α, 0 α > 0, + α +, α + d + α + + α + = d d F, F 0 < α < d + α + = d α + + α + = d d α + + α + d α + = d 4 4d α + = d 4 8d + 6 http://mth.cs.kitmi-it.c.jp/
x A Aω ẋ ẋ 2 + ω 2 x 2 = ω 2 A 2. (ẋ, ωx) ζ ẋ + iωx ζ ζ dζ = ẍ + iωẋ = ẍ + iω(ζ iωx) dt dζ dt iωζ = ẍ + ω2 x (2.1) ζ ζ = Aωe iωt = Aω cos ωt + iaω sin
2 2.1 F (t) 2.1.1 mẍ + kx = F (t). m ẍ + ω 2 x = F (t)/m ω = k/m. 1 : (ẋ, x) x = A sin ωt, ẋ = Aω cos ωt 1 2-1 x A Aω ẋ ẋ 2 + ω 2 x 2 = ω 2 A 2. (ẋ, ωx) ζ ẋ + iωx ζ ζ dζ = ẍ + iωẋ = ẍ + iω(ζ iωx) dt dζ
) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8)
4 4 ) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8) a b a b = 6i j 4 b c b c 9) a b = 4 a b) c = 7
No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2
No.2 1 2 2 δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i δx j (5) δs 2 = δx i δx i + 2 u i δx i δx j = δs 2 + 2s ij δx i δx j
( ; ) C. H. Scholz, The Mechanics of Earthquakes and Faulting : - ( ) σ = σ t sin 2π(r a) λ dσ d(r a) =
1 9 8 1 1 1 ; 1 11 16 C. H. Scholz, The Mechanics of Earthquakes and Faulting 1. 1.1 1.1.1 : - σ = σ t sin πr a λ dσ dr a = E a = π λ σ πr a t cos λ 1 r a/λ 1 cos 1 E: σ t = Eλ πa a λ E/π γ : λ/ 3 γ =
main.dvi
5 IIR IIR z 5.1 5.1.1 1. 2. IIR(Infinite Impulse Response) FIR(Finite Impulse Response) 3. 4. 5. 5.1.2 IIR FIR 5.1 5.1 5.2 104 5. IIR 5.1 IIR FIR IIR FIR H(z) = a 0 +a 1 z 1 +a 2 z 2 1+b 1 z 1 +b 2 z 2
II 2 II
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N cos s s cos ψ e e e e 3 3 e e 3 e 3 e
3 3 5 5 5 3 3 7 5 33 5 33 9 5 8 > e > f U f U u u > u ue u e u ue u ue u e u e u u e u u e u N cos s s cos ψ e e e e 3 3 e e 3 e 3 e 3 > A A > A E A f A A f A [ ] f A A e > > A e[ ] > f A E A < < f ; >
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Onsager SOLUTION OF THE EIGENWERT PROBLEM (O-29) V = e H A e H B λ max Z 2 Onsager (O-77) (O-82) (O-83) Kramers-Wannier 1 1 Ons
Onsager 2 9 207.2.7 3 SOLUTION OF THE EIGENWERT PROBLEM O-29 V = e H A e H B λ max Z 2 OnsagerO-77O-82 O-83 2 Kramers-Wannier Onsager * * * * * V self-adjoint V = V /2 V V /2 = V /2 V 2 V /2 = 2 sinh 2H
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r(t) t r(t) O v(t) = dr(t) dt a(t) = dv(t) dt = d2 r(t) dt 2 r(t), v(t), a(t) t dr(t) dt r(t) =(x(t),y(t),z(t)) = d 2 r(t) dt 2 = ( dx(t) dt ( d 2 x(t) dt 2, dy(t), dz(t) dt dt ), d2 y(t) dt 2, d2 z(t)
x (x, ) x y (, y) iy x y z = x + iy (x, y) (r, θ) r = x + y, θ = tan ( y ), π < θ π x r = z, θ = arg z z = x + iy = r cos θ + ir sin θ = r(cos θ + i s
... x, y z = x + iy x z y z x = Rez, y = Imz z = x + iy x iy z z () z + z = (z + z )() z z = (z z )(3) z z = ( z z )(4)z z = z z = x + y z = x + iy ()Rez = (z + z), Imz = (z z) i () z z z + z z + z.. z
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(Bessel) (Legendre).. (Hankel). (Laplace) V = (x, y, z) n (r, θ, ϕ) r n f n (θ, ϕ). f n (θ, ϕ) n f n (θ, ϕ) z = cos θ z θ ϕ n ν. P ν (z), Q ν (z) (Fou
(Bessel) (Legendre).. (Hankel). (Laplace) V = (x, y, z) n (r, θ, ϕ) r n f n (θ, ϕ). f n (θ, ϕ) n f n (θ, ϕ) z = cos θ z θ ϕ n ν. P ν (z), Q ν (z) (Fourier) (Fourier Bessel).. V ρ(x, y, z) V = 4πGρ G :.
振動と波動
Report JS0.5 J Simplicity February 4, 2012 1 J Simplicity HOME http://www.jsimplicity.com/ Preface 2 Report 2 Contents I 5 1 6 1.1..................................... 6 1.2 1 1:................ 7 1.3
X G P G (X) G BG [X, BG] S 2 2 2 S 2 2 S 2 = { (x 1, x 2, x 3 ) R 3 x 2 1 + x 2 2 + x 2 3 = 1 } R 3 S 2 S 2 v x S 2 x x v(x) T x S 2 T x S 2 S 2 x T x S 2 = { ξ R 3 x ξ } R 3 T x S 2 S 2 x x T x S 2
) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4
![) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4](/thumbs/92/107866707.jpg ") ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4")
1. k λ ν ω T v p v g k = π λ ω = πν = π T v p = λν = ω k v g = dω dk 1) ) 3) 4). p = hk = h λ 5) E = hν = hω 6) h = h π 7) h =6.6618 1 34 J sec) hc=197.3 MeV fm = 197.3 kev pm= 197.3 ev nm = 1.97 1 3 ev
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1 1 sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω 1 ω α V T m T m 1 100Hz m 2 36km 500Hz. 36km 1
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63 6 6.1 6.1.1 v = v 0 =v 0x,v 0y, 0) t =0 x 0,y 0, 0) t x x 0 + v 0x t v x v 0x = y = y 0 + v 0y t, v = v y = v 0y 6.1) z 0 0 v z yv z zv y zv x xv z xv y yv x = 0 0 x 0 v 0y y 0 v 0x 6.) 6.) 6.1) 6.)
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chap1.dvi
1 1 007 1 e iθ = cos θ + isin θ 1) θ = π e iπ + 1 = 0 1 ) 3 11 f 0 r 1 1 ) k f k = 1 + r) k f 0 f k k = 01) f k+1 = 1 + r)f k ) f k+1 f k = rf k 3) 1 ) ) ) 1+r/)f 0 1 1 + r/) f 0 = 1 + r + r /4)f 0 1 f
1. (8) (1) (x + y) + (x + y) = 0 () (x + y ) 5xy = 0 (3) (x y + 3y 3 ) (x 3 + xy ) = 0 (4) x tan y x y + x = 0 (5) x = y + x + y (6) = x + y 1 x y 3 (
1 1.1 (1) (1 + x) + (1 + y) = 0 () x + y = 0 (3) xy = x (4) x(y + 3) + y(y + 3) = 0 (5) (a + y ) = x ax a (6) x y 1 + y x 1 = 0 (7) cos x + sin x cos y = 0 (8) = tan y tan x (9) = (y 1) tan x (10) (1 +
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ii p ϕ x, t = C ϕ xe i ħ E t +C ϕ xe i ħ E t ψ x,t ψ x,t p79 やは時間変化しないことに注意 振動 粒子はだいたい このあたりにいる 粒子はだいたい このあたりにいる p35 D.3 Aψ Cϕdx = aψ ψ C Aϕ dx
i B5 7.8. p89 4. ψ x, tψx, t = ψ R x, t iψ I x, t ψ R x, t + iψ I x, t = ψ R x, t + ψ I x, t p 5.8 π π π F e ix + F e ix + F 3 e 3ix F e ix + F e ix + F 3 e 3ix dx πψ x πψx p39 7. AX = X A [ a b c d x
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](/thumbs/101/149159646.jpg "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
II (10 4 ) 1. p (x, y) (a, b) ε(x, y; a, b) 0 f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a 2 x a 0 A = f x (
II (1 4 ) 1. p.13 1 (x, y) (a, b) ε(x, y; a, b) f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a x a A = f x (a, b) y x 3 3y 3 (x, y) (, ) f (x, y) = x + y (x, y) = (, )
(u(x)v(x)) = u (x)v(x) + u(x)v (x) ( ) u(x) = u (x)v(x) u(x)v (x) v(x) v(x) 2 y = g(t), t = f(x) y = g(f(x)) dy dx dy dx = dy dt dt dx., y, f, g y = f (g(x))g (x). ( (f(g(x)). ). [ ] y = e ax+b (a, b )
e a b a b b a a a 1 a a 1 = a 1 a = e G G G : x ( x =, 8, 1 ) x 1,, 60 θ, ϕ ψ θ G G H H G x. n n 1 n 1 n σ = (σ 1, σ,..., σ N ) i σ i i n S n n = 1,,
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高校生の就職への数学II
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(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0
1 1 1.1 1.) T D = T = D = kn 1. 1.4) F W = F = W/ = kn/ = 15 kn 1. 1.9) R = W 1 + W = 6 + 5 = 11 N. 1.9) W b W 1 a = a = W /W 1 )b = 5/6) = 5 cm 1.4 AB AC P 1, P x, y x, y y x 1.4.) P sin 6 + P 1 sin 45
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A (1) = 4 A( 1, 4) 1 A 4 () = tan A(0, 0) π A π
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量子力学 問題
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zz + 3i(z z) + 5 = 0 + i z + i = z 2i z z z y zz + 3i (z z) + 5 = 0 (z 3i) (z + 3i) = 9 5 = 4 z 3i = 2 (3i) zz i (z z) + 1 = a 2 {
04 zz + iz z) + 5 = 0 + i z + i = z i z z z 970 0 y zz + i z z) + 5 = 0 z i) z + i) = 9 5 = 4 z i = i) zz i z z) + = a {zz + i z z) + 4} a ) zz + a + ) z z) + 4a = 0 4a a = 5 a = x i) i) : c Darumafactory
A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B
9 7 A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B x x B } B C y C y + x B y C x C C x C y B = A
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3 215 4 27 1 1 u u(x, t) u tt a 2 u xx, a > (1) D : {(x, t) : x, t } u (, t), u (, t), t (2) u(x, ) f(x), u(x, ) t 2, x (3) u(x, t) X(x)T (t) u (1) 1 T (t) a 2 T (t) X (x) X(x) α (2) T (t) αa 2 T (t) (4)
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1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0 0 < t < τ I II 0 No.2 2 C x y x y > 0 x 0 x > b a dx
chap03.dvi
99 3 (Coriolis) cm m (free surface wave) 3.1 Φ 2.5 (2.25) Φ 100 3 r =(x, y, z) x y z F (x, y, z, t) =0 ( DF ) Dt = t + Φ F =0 onf =0. (3.1) n = F/ F (3.1) F n Φ = Φ n = 1 F F t Vn on F = 0 (3.2) Φ (3.1)
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z f(z) f(z) x, y, u, v, r, θ r > 0 z = x + iy, f = u + iv C γ D f(z) f(z) D f(z) f(z) z, Rm z, z 1.1 z = x + iy = re iθ = r (cos θ + i sin θ) z = x iy
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微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.
微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)
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A B A.70g/cm 3 B.74g/cm 3 B C 70at% %A C B at% 80at% %B 350 C γ δ y=00 x-y ρ l S ρ C p k C p ρ C p T ρ l t l S S ξ S t = ( k T ) ξ ( ) S = ( k T) ( ) t y ξ S ξ / t S v T T / t = v T / y 00 x v S dy dx
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50 2 I SI MKSA r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq
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