θ (rgument) θ = rg 2 ( ) n n n n = Re iθ = re iϕ n n = r n e inϕ Re iθ = r n e inϕ R = r n r = R /n, e inϕ = e iθ ϕ = θ/n e iθ = e iθ+2i = e iθ+4i = (
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- こうご あざみ
- 5 years ago
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1 ( ) i i i (Σe i i = e i + e i 2 + ), y n ( ) 2 y y = +iy 2 (, y) = + iy y > (upper hlf plne) y < (lower hlf plne) = + iy (rel prt) y (imginry prt) Re = Im = y = Re + iim = + iy = = ( + iy)( iy) = 2 + y 2 (r, θ) ( = r cos θ, y = r sin θ) = + iy = r cos θ + ir sin θ = r(cos θ + i sin θ) = re iθ = re iθ (polr form) r θ r = = 2 + y 2, tn θ = y
2 θ (rgument) θ = rg 2 ( ) n n n n = Re iθ = re iϕ n n = r n e inϕ Re iθ = r n e inϕ R = r n r = R /n, e inϕ = e iθ ϕ = θ/n e iθ = e iθ+2i = e iθ+4i = (e 2i = cos(2) + i sin(2) = ) 2 m m =,, 2,..., n ϕ = θ + 2m n n n = R /n e i(θ+2m)/n (de Moivre) (cos θ + i sin θ) n = cos(nθ) + i sin(nθ) 2
3 = re iϕ, n = r n e inϕ = r n (cos ϕ + i sin ϕ) n = r n (cos nϕ + i sin nϕ) r = R /n, ϕ = θ + 2m n (e iθ ) n = e inθ (cos θ + i sin θ) n = cos(nθ) + i sin(nθ) 2 = r e iθ r 2 e iθ2 = r (cos θ + i sin θ )r 2 (cos θ 2 + i sin θ 2 ) = r r 2 (cos θ cos θ 2 sin θ sin θ 2 + i cos θ sin θ 2 + i sin θ cos θ 2 ) = r r 2 (cos(θ + θ 2 ) + i sin(θ + θ 2 )) n = (re iθ ) n = r n (cos(nθ) + i sin(nθ)) i n = Re iθ = i = e i/2 (R = ) n = 2 = re iϕ = e i(/2+2m)/2 = e i(/4+m) = cos( 4 + m) + i sin( 4 + m) (r = R/2 = ) m = m = m = : cos 4 + i sin 4 = 2 ( + i) m = : cos( 4 + ) + i sin( 4 + ) = 2 ( + i) m = 2 m = m = 3 m = 2 + i 3 n = Re iθ = + i = 2e i3/4 (R = 2 /2 ) n = 3 = re iϕ = 2 /6 e i(3/4+2m)/3 3
4 3 m =,, 2 D (D ) (D ) D D (regulr) D df d = lim f( + ) D ( ) < < r (r ) ( < < r) ( ) r = = + iy =, y = (i), y = (ii) =, y 2 ( y = = y ) ( ) 2 D, y f( + ) f( +, y + y) f(, y) lim = lim + i y (i), y = f( +, y + y) f(, y) f( +, y) f(, y) f(, y) lim = lim = + i y (ii) =, y f( +, y + y) f(, y) f(, y + y) f(, y) f(, y) lim = lim = i + i y y i y y 2 f(, y) 4 f(, y) = i y
5 f(, y) f(, y) = u(, y) + iv(, y) u(, y) v(, y) u(, y) v(, y) + i = i + y y u(, y) = v(, y) y, u(, y) y v(, y) = (uchy-riemnn) = u(, y) + iv(, y) u(, y), v(, y) u(, y), v(, y) 2 ( u, v ) = = + iy = iy 2 + y 2 = u + iv (u = 2 + y 2, v = y 2 + y 2 ) u = 2 + y ( 2 + y 2 ) 2 = 2 + y 2 ( 2 + y 2 ) 2 v y = 2 + y 2 u y = 2y ( 2 + y 2 ) 2 v = 2y ( 2 + y 2 ) 2 2y2 2 + y 2 = 2 + y 2 ( 2 + y 2 ) 2 = / = = / u y = / = = ( = ) = ( / ) D r = 2 + y 2 = r 2 r = = = iy = u + iv = (u =, v = y) 5
6 u =, v y =, u y =, v = d ( ) i d = lim N i= N f( i ) i d = d + i dy d = d + idy = u(, y) + iv(, y) d = (u(, y) + iv(, y))(d + idy) = (u(, y)d v(, y)dy) + i (v(, y)d + u(, y)dy), y t d = t2 t f((t)) d dt dt ( ) d 6
7 t b (), (b) () = (b) t t (t) (t ) ( ) (Jordn) () (b) 2 (simply connected) (multiply connected) D ( ) () (b) d = F ((b)) F (()) df () ( =, F () = f(ξ)dξ) d 2 2 d = F ( 2 ) F ( ) 2 ( + g())d = d = d 2 d = d = + 2 d = 2 d = d + g()d d + d 2 d + 2 d (2 ) 4, 2 = + i = + iy ( ) d = ( + iy)(d + idy) = (d ydy + iyd + idy) y (, ) y = y (y y ) 7
8 (d ydy + iyd + idy) = (d d + id + id) = 2i d = i = = = + i 2 ( y ) 2 + i 2 d = (d ydy + iyd + idy) = 2 2 = 2 (d ydy + iyd + idy) + d + ( y + i)dy 2 (d ydy + iyd + idy) = i 2 2 y =, dy = 2 =, d = = + i t ( )(t) (t) = ( + i)t ( t ) d = (t) d(t) dt = dt ( + i)t( + i)dt = 2i 2 = i (t) ( + i)t y (, ) (, ) (t) = (t, t) = (, )t = = + i ( =, y = ) = + iy (t) = ( + i)t t ( + i)t = = + i 2 2 t ) (t) = t 2 (t) = + it ( d = 2 2 d + d = 2 tdt + ( + it)idt = 2 + i 2 = i 2 (t) = + it = y ( ) / r = r, y = = r, y = d = = = i re iθ ireiθ dθ ( = re iθ ) idθ 8
9 = + i = + i (y = ) d == ( iy)(d+idy) = (d+ydy iyd+idy) = (d+d id+id) = 2 d = =, y = = + i ( =, y = ) 2 d = (d + ydy iyd + idy) = (d + ydy iyd + idy) = d + (ydy + idy) = + i y = = y =, y = = + i 3 d = (d + ydy iyd + idy) = 3 3 ydy + (d id) = i (uchy) D D d =, 2,..., n,..., n (,..., n ), 2,..., n (,..., n ) d = n i= i d D D f() = 2i d = sin r > sin sin /( ) 9
10 sin d = 2i sin f (n) () = n! d 2i ( ) n+ (Gourst) f() = 2i d, f( + δ) = 2i δ d (f( + δ) f())/δ = = n= c n ( ) n = c n ( ) n + c n ( ) n n= n= c n = f(ξ) dξ (n =, ±, ±2,...) 2i (ξ ) n+ n= c n ( ) n (principl prt) ( ) = = e / ep e = = n= n n!
11 e / = n! ( )n n= = ( ) singulrity / = ( R < < R ) = = (isolted singulrity) removble singulrity n n (pole) (essentil singulrity) < < R ( < δ δ < R) = g() = g() ( ) n ( < < δ) n n n lim ( ) n n (residue) = Res(f, ) = d 2i Res(f, ) Res n n d = 2i Res(f, i ) i= n n d = 2i Res(f, i ) i=
12 d = c n ( ) n d n= r ( ) n d (n =, 2,...) n = d = 2 re iθ ireiθ dθ ( = re iθ ) = 2 idθ = 2i () = r = re iθ n ( ) n d = 2 = i r n r n e inθ ireiθ dθ 2 e i(n )θ dθ = i [ r n i(n ) e i(n )θ] 2 = i ( r n i(n ) ) i(n ) = e i2n n n =, 2,... ( ) n (n ) ( ) d = = c n ( ) n d n= c ( ) d = 2ic 2
13 Res(f, ) = c n c = c + c ( ) + c 2 ( ) 2 + c + c 2 ( ) 2 2 c d d (( ) 2 ) = = d d (c ( ) 2 + c ( ) 3 + c 2 ( ) 4 + c ( ) + c 2 ) = = c n ( c = ( (n )! d n ) n ) = d n c ( ) n = n d n ( Res(f, ) = c = ( (n )! d n ) n ) = ( ) ( ) lim e i d = ( > ) R R R R +R +R (ir) R R R = Re iθ lim e i d = R R R R ( R R ( ir) +R ) 3
14 R R ( ) R R R R + = R ( R R R R + R = ) ( ) ( ) I = e i d, I = R R e i d R I I I = e i d = i R e ireiθ f(re iθ )Re iθ dθ ( = Re iθ ) b f() d b f()d b f() f() ( ) f() f() b f() d b b b f() d f()d f()d 2 b f() d b f()d b f() d f() f() b f()d 4 b f() d
15 I = e i d R = i = = = e ireiθ f(re iθ )Re iθ dθ e ireiθ f(re iθ ) Re iθ dθ ep[ir(cos θ + i sin θ) ] f(re iθ ) Rdθ e R sin θ f(re iθ ) Rdθ ( e i = e i e i = ) e R sin θ f(re iθ ) Rdθ ep = f(re iθ ) R θ ϵ I ϵr e R sin θ dθ ϵr e R sin θ dθ = ϵr = ϵr = ϵr = ϵr = 2ϵR /2 /2 /2 /2 /2 e R sin θ dθ + ϵr e R sin θ dθ /2 e R sin θ dθ + ϵr e R sin(θ +) dθ (θ = θ ) /2 e R sin θ dθ + ϵr e R sin θ dθ (sin(θ + ) = sin θ) /2 e R sin θ dθ + ϵr e R sin θ dθ /2 e R sin θ dθ (sin( θ) = sin θ) sin θ θ /2 2θ sin θ ( sin θ sin θ 2θ/) 5
16 I 2ϵR /2 e R sin θ dθ 2ϵR /2 e 2Rθ/ dθ = 2ϵR [ 2R e 2Rθ/] /2 = ϵ ( e R ) R I ϵ ϵ lim I = R I I = e i d = i R 2 e ireiθ f(re iθ )Re iθ dθ ( = Re iθ ) I 2 e ireiθ f(re iθ ) Rdθ = 2 ep[ ir(cos θ + i sin θ)] f(re iθ ) Rdθ = R ϵr = ϵr = ϵr = 2ϵR 2 2 /2 e R sin θ f(re iθ ) dθ e R sin θ dθ e R sin(θ +) dθ (θ = θ ) e R sin θ dθ e R sin θ dθ θ 3/2 2θ/ sin θ sin d = sin 6
17 ( ) n sin = (2n + )! 2n+ = 3! 3 + n= 2, 4,... = = R r sin d = R 2i r = 2i = 2i = 2i = 2i J ( R r ( R r ( R r e i e i d e i R d e i ) r d e i R d e i ) r d e i r d + e i ) R d e = + + = 2i lim lim J = R r sin d (2) r R r R d + d + R r d + d 2 ( ) = r ( ) 2 R ( ) R r +r +r +R +R R J = e i r d = e i R d + e i R d + e i r 2 d + e i d = J = J + e i d + 2 e i d = (3) J r =, R = (3) R 7
18 2 e i d = e ir cos θ e R sin θ dθ e ireiθ Re iθ ireiθ dθ = i e ireiθ dθ = i e ir cos θ e R sin θ dθ = e ir cos θ e R sin θ dθ e R sin θ dθ (3) e i d = i = i re iθ eir cos θ e r sin θ re iθ dθ ( = re iθ ) e ir cos θ e r sin θ dθ r i lim r e ir cos θ e r sin θ dθ = i dθ = i (3) J (2) sin / sin d = 2i lim lim J = R r 2i lim e r i d 2i e lim R 2 i d = 2 sin / sin d = 2 ( sin + sin ) d = ( sin 2 = 2 = 2 = 2 ( ( sin sin + sin d sin( ) ) d sin ) d sin ) d sin d r R sin d + sin R d + r sin d 8
19 r sin / R, r ( ) R lim R R sin d = lim lim R r ( r R sin d + sin R d + sin ) r d = lim lim R r sin d R +R 2 d, 2 d 2 2 ( R) R 2 e i d =, 2 e i d = sin d = = 2i = 2i sin d + e 2i 2 i d 2i 2 ( ei + e i e i )d + 2i d 2i e i 2 e i d e i d 2i d 2 e i d + = + 2 = R +R R +R + + = ( ) + e i d = 2ie = 2i sin d = ( 2i ei e i )d = 2i + e i d 2i e i d = sin d = 2 lim lim sin R r d = 2 9
20 + + ( ) f() lim ϵ ( ϵ b ) f()d + f()d +ϵ (principl vlue) pv b ( ϵ f()d = lim ϵ b ) f()d + f()d +ϵ PV p.v. vp P ( ) d ( ϵ) R R R R ( R, ϵ ) ϵ d = R R d + +ϵ d + d + 2 d (R, ϵ ) /( ) Res(f/( ), i ) n d = 2i f Res(, i) i= = ϵ R d = pv R d + lim ϵ d + 2 d ϵ 2 R /( ) R 2 () lim ϵ d = i lim ϵ f( + ϵe iθ ) ϵe iθ ϵe iθ dθ = i lim 2 ϵ f( + ϵe iθ )dθ = if()
21 R, ϵ pv n d = if() + 2i f Res(, i) i= = f() = e i e i d ( = ) pv e i d = i f( = ) = e = = e i 2
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
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
f f x, y, u, v, r, θ r > = x + iy, f = u + iv C γ D f f D f f, Rm,. = x + iy = re iθ = r cos θ + i sin θ = x iy = re iθ = r cos θ i sin θ x = + = Re, y = = Im i r = = = x + y θ = arg = arctan y x e i =
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
n=1 1 n 2 = π = π f(z) f(z) 2 f(z) = u(z) + iv(z) *1 f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x
n= n 2 = π2 6 3 2 28 + 4 + 9 + = π2 6 2 f(z) f(z) 2 f(z) = u(z) + iv(z) * f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x f x = i f y * u, v 3 3. 3 f(t) = u(t) + v(t) [, b] f(t)dt = u(t)dt
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b n c n d n d n = f() d (n =, ±, ±, ) () πi ( a) n+ () () = a R a f() = a k Γ ( < k < R) Γ f() Γ ζ R ζ k a Γ f() = f(ζ) πi ζ dζ f(ζ) dζ (3) πi Γ ζ (3)
[ ] KENZOU 6 3 4 Origin 6//5) 3 a a f() = b n ( a) n c n + ( a) n n= n= = b + b ( a) + b ( a) + + c a + c ( a) + b n = f() πi ( a) n+ d, c n = f() d πi ( a) n+ () b n c n d n d n = f() d (n =, ±, ±, )
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Z: Q: R: C: sin 6 5 ζ a, b
Z: Q: R: C: 3 3 7 4 sin 6 5 ζ 9 6 6............................... 6............................... 6.3......................... 4 7 6 8 8 9 3 33 a, b a bc c b a a b 5 3 5 3 5 5 3 a a a a p > p p p, 3,
9 5 ( α+ ) = (α + ) α (log ) = α d = α C d = log + C C 5. () d = 4 d = C = C = 3 + C 3 () d = d = C = C = 3 + C 3 =
5 5. 5.. A II f() f() F () f() F () = f() C (F () + C) = F () = f() F () + C f() F () G() f() G () = F () 39 G() = F () + C C f() F () f() F () + C C f() f() d f() f() C f() f() F () = f() f() f() d =
() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)
0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()
1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ =
1 1.1 ( ). z = + bi,, b R 0, b 0 2 + b 2 0 z = + bi = ( ) 2 + b 2 2 + b + b 2 2 + b i 2 r = 2 + b 2 θ cos θ = 2 + b 2, sin θ = b 2 + b 2 2π z = r(cos θ + i sin θ) 1.2 (, ). 1. < 2. > 3. ±,, 1.3 ( ). A
y = f(x) (x, y : ) w = f(z) (w, z : ) df(x) df(z), f(x)dx dx dz f(z)dz : e iωt = cos(ωt) + i sin(ωt) [ ] : y = f(t) f(ω) = 1 2π f(t)e iωt d
![y = f(x) (x, y : ) w = f(z) (w, z : ) df(x) df(z), f(x)dx dx dz f(z)dz : e iωt = cos(ωt) + i sin(ωt) [ ] : y = f(t) f(ω) = 1 2π f(t)e iωt d](/thumbs/104/162030620.jpg "y = f(x) (x, y : ) w = f(z) (w, z : ) df(x) df(z), f(x)dx dx dz f(z)dz : e iωt = cos(ωt) + i sin(ωt) [ ] : y = f(t) f(ω) = 1 2π f(t)e iωt d")
8. y = f(x) (x, y : ) w = f(z) (w, z : ) df(x) df(z), f(x)dx dx dz f(z)dz : e iωt = cos(ωt) + i sin(ωt) [ ] : y = f(t) f(ω) = π f(t)e iωt dt. : ϕ(x, y) x + ϕ(x, y) y = ( ).. . : 3. (z p, e z, sin z, sinh
(1) D = [0, 1] [1, 2], (2x y)dxdy = D = = (2) D = [1, 2] [2, 3], (x 2 y + y 2 )dxdy = D = = (3) D = [0, 1] [ 1, 2], 1 {
![(1) D = [0, 1] [1, 2], (2x y)dxdy = D = = (2) D = [1, 2] [2, 3], (x 2 y + y 2 )dxdy = D = = (3) D = [0, 1] [ 1, 2], 1 {](/thumbs/94/118399854.jpg "(1) D = [0, 1] [1, 2], (2x y)dxdy = D = = (2) D = [1, 2] [2, 3], (x 2 y + y 2 )dxdy = D = = (3) D = [0, 1] [ 1, 2], 1 {")
7 4.., ], ], ydy, ], 3], y + y dy 3, ], ], + y + ydy 4, ], ], y ydy ydy y y ] 3 3 ] 3 y + y dy y + 3 y3 5 + 9 3 ] 3 + y + ydy 5 6 3 + 9 ] 3 73 6 y + y + y ] 3 + 3 + 3 3 + 3 + 3 ] 4 y y dy y ] 3 y3 83 3
grad φ(p ) φ P grad φ(p ) p P p φ P p l t φ l t = 0 g (0) g (0) (31) grad φ(p ) p grad φ φ (P, φ(p )) xy (x, y) = (ξ(t), η(t)) ( )
2 9 2 5 2.2.3 grad φ(p ) φ P grad φ(p ) p P p φ P p l t φ l t = g () g () (3) grad φ(p ) p grad φ φ (P, φ(p )) y (, y) = (ξ(t), η(t)) ( ) ξ (t) (t) := η (t) grad f(ξ(t), η(t)) (t) g(t) := f(ξ(t), η(t))
29
9 .,,, 3 () C k k C k C + C + C + + C 8 + C 9 + C k C + C + C + C 3 + C 4 + C 5 + + 45 + + + 5 + + 9 + 4 + 4 + 5 4 C k k k ( + ) 4 C k k ( k) 3 n( ) n n n ( ) n ( ) n 3 ( ) 3 3 3 n 4 ( ) 4 4 4 ( ) n n
[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s
![[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s](/thumbs/94/118579915.jpg "[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s")
[ ]. lim e 3 IC ) s49). y = e + ) ) y = / + ).3 d 4 ) e sin d 3) sin d ) s49) s493).4 z = y z z y s494).5 + y = 4 =.6 s495) dy = 3e ) d dy d = y s496).7 lim ) lim e s49).8 y = e sin ) y = sin e 3) y =
() Remrk I = [0, ] [x i, x i ]. (x : ) f(x) = 0 (x : ) ξ i, (f) = f(ξ i )(x i x i ) = (x i x i ) = ξ i, (f) = f(ξ i )(x i x i ) = 0 (f) 0.
![() Remrk I = [0, ] [x i, x i ]. (x : ) f(x) = 0 (x : ) ξ i, (f) = f(ξ i )(x i x i ) = (x i x i ) = ξ i, (f) = f(ξ i )(x i x i ) = 0 (f) 0.](/thumbs/89/98384840.jpg "() Remrk I = [0, ] [x i, x i ]. (x : ) f(x) = 0 (x : ) ξ i, (f) = f(ξ i )(x i x i ) = (x i x i ) = ξ i, (f) = f(ξ i )(x i x i ) = 0 (f) 0.")
() 6 f(x) [, b] 6. Riemnn [, b] f(x) S f(x) [, b] (Riemnn) = x 0 < x < x < < x n = b. I = [, b] = {x,, x n } mx(x i x i ) =. i [x i, x i ] ξ i n (f) = f(ξ i )(x i x i ) i=. (ξ i ) (f) 0( ), ξ i, S, ε >
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KENZOU 2008 8 9 5 1 2 3 4 2 5 6 2 6.1......................................... 2 6.2......................................... 2 6.3......................................... 4 7 5 8 6 8.1.................................................
But nothing s unconditional, The Bravery R R >0 = (0, ) ( ) R >0 = (0, ) f, g R >0 f (0, R), R >
2 2 http://www.ozawa.phys.waseda.ac.jp/inde2.html But nothing s unconditional, The Bravery > (, ( > (, f, g > f (,, > sup f( ( M f(g( (i > g [, lim g( (ii g > (, ( g ( < ( > f(g( (i g < < f(g( g(
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1, R f : R R,.,, b R < b, f(x) [, b] f(x)dx,, [, b] f(x) x ( ) ( 1 ). y y f(x) f(x)dx b x 1: f(x)dx, [, b] f(x) x ( ).,,,,,., f(x)dx,,,, f(x)dx. 1.1 Riemnn,, [, b] f(x) x., x 0 < x 1 < x 2 < < x n 1
1 : f(z = re iθ ) = u(r, θ) + iv(r, θ). (re iθ ) 2 = r 2 e 2iθ = r 2 cos 2θ + ir 2 sin 2θ r f(z = x + iy) = u(x, y) + iv(x, y). (x + iy) 2 = x 2 y 2 +
1.3 1.4. (pp.14-27) 1 1 : f(z = re iθ ) = u(r, θ) + iv(r, θ). (re iθ ) 2 = r 2 e 2iθ = r 2 cos 2θ + ir 2 sin 2θ r f(z = x + iy) = u(x, y) + iv(x, y). (x + iy) 2 = x 2 y 2 + i2xy x = 1 y (1 + iy) 2 = 1
c y /2 ddy = = 2π sin θ /2 dθd /2 [ ] 2π cos θ d = log 2 + a 2 d = log 2 + a 2 = log 2 + a a 2 d d + 2 = l
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c 28. 2, y 2, θ = cos θ y = sin θ 2 3, y, 3, θ, ϕ = sin θ cos ϕ 3 y = sin θ sin ϕ 4 = cos θ 5.2 2 e, e y 2 e, e θ e = cos θ e sin θ e θ 6 e y = sin θ e + cos θ e θ 7.3 sgn sgn = = { = + > 2 < 8.4 a b 2
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
f(x) = f(x ) + α(x)(x x ) α(x) x = x. x = f (y), x = f (y ) y = f f (y) = f f (y ) + α(f (y))(f (y) f (y )) f (y) = f (y ) + α(f (y)) (y y ) ( (2) ) f
22 A 3,4 No.3 () (2) (3) (4), (5) (6) (7) (8) () n x = (x,, x n ), = (,, n ), x = ( (x i i ) 2 ) /2 f(x) R n f(x) = f() + i α i (x ) i + o( x ) α,, α n g(x) = o( x )) lim x g(x) x = y = f() + i α i(x )
, 1 ( f n (x))dx d dx ( f n (x)) 1 f n (x)dx d dx f n(x) lim f n (x) = [, 1] x f n (x) = n x x 1 f n (x) = x f n (x) = x 1 x n n f n(x) = [, 1] f n (x
![, 1 ( f n (x))dx d dx ( f n (x)) 1 f n (x)dx d dx f n(x) lim f n (x) = [, 1] x f n (x) = n x x 1 f n (x) = x f n (x) = x 1 x n n f n(x) = [, 1] f n (x](/thumbs/93/113428934.jpg ", 1 ( f n (x))dx d dx ( f n (x)) 1 f n (x)dx d dx f n(x) lim f n (x) = [, 1] x f n (x) = n x x 1 f n (x) = x f n (x) = x 1 x n n f n(x) = [, 1] f n (x")
1 1.1 4n 2 x, x 1 2n f n (x) = 4n 2 ( 1 x), 1 x 1 n 2n n, 1 x n n 1 1 f n (x)dx = 1, n = 1, 2,.. 1 lim 1 lim 1 f n (x)dx = 1 lim f n(x) = ( lim f n (x))dx = f n (x)dx 1 ( lim f n (x))dx d dx ( lim f d
= M + M + M + M M + =.,. f = < ρ, > ρ ρ. ρ f. = ρ = = ± = log 4 = = = ± f = k k ρ. k
7 b f n f} d = b f n f d,. 5,. [ ] ɛ >, n ɛ + + n < ɛ. m. n m log + < n m. n lim sin kπ sin kπ } k π sin = n n n. k= 4 f, y = r + s, y = rs f rs = f + r + sf y + rsf yy + f y. f = f =, f = sin. 5 f f =.
lim lim lim lim 0 0 d lim 5. d 0 d d d d d d 0 0 lim lim 0 d
lim 5. 0 A B 5-5- A B lim 0 A B A 5. 5- 0 5-5- 0 0 lim lim 0 0 0 lim lim 0 0 d lim 5. d 0 d d d d d d 0 0 lim lim 0 d 0 0 5- 5-3 0 5-3 5-3b 5-3c lim lim d 0 0 5-3b 5-3c lim lim lim d 0 0 0 3 3 3 3 3 3
(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t
![(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t](/thumbs/67/56612994.jpg "(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t")
6 6.1 6.1 (1 Z ( X = e Z, Y = Im Z ( Z = X + iy, i = 1 (2 Z E[ e Z ] < E[ Im Z ] < Z E[Z] = E[e Z] + ie[im Z] 6.2 Z E[Z] E[ Z ] : E[ Z ] < e Z Z, Im Z Z E[Z] α = E[Z], Z = Z Z 1 {Z } E[Z] = α = α [ α ]
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I 9-9 (3) 9 9, x, x (t)+a(t)x (t)+b(t)x(t) = f(t) (9), a(t), b(t), f(t),,, f(t),, a(t), b(t),,, x (t)+ax (t)+bx(t) = (9),, x (t)+ax (t)+bx(t) = f(t) (93), b(t),, b(t) 9 x (t), x (t), x (t)+a(t)x (t)+b(t)x(t)
Email: kawahiraamath.titech.ac.jp (A=@) 27 2 25 . 2. 3. 4. 5. 3 4 5 ii 5 50 () : (2) R: (3) Q: (4) Z: (5) N: (6) : () α: (2) β: (3) γ, Γ: (4) δ, : (5) ϵ: (6) ζ: (7) η: (8) θ, Θ: (9) ι: (0) κ: () λ, Λ:
W u = u(x, t) u tt = a 2 u xx, a > 0 (1) D := {(x, t) : 0 x l, t 0} u (0, t) = 0, u (l, t) = 0, t 0 (2)
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)
.5 z = a + b + c n.6 = a sin t y = b cos t dy d a e e b e + e c e e e + e 3 s36 3 a + y = a, b > b 3 s363.7 y = + 3 y = + 3 s364.8 cos a 3 s365.9 y =,
[ ] IC. r, θ r, θ π, y y = 3 3 = r cos θ r sin θ D D = {, y ; y }, y D r, θ ep y yddy D D 9 s96. d y dt + 3dy + y = cos t dt t = y = e π + e π +. t = π y =.9 s6.3 d y d + dy d + y = y =, dy d = 3 a, b
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
49 2 I II 2.1 3 e e = 1.602 10 19 A s (2.1 50 2 I SI MKSA 2.1.1 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 = 3 10 8 m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq F = k r
Tips KENZOU PC no problem 2 1 w = f(z) z 1 w w z w = (z z 0 ) b b w = log (z z 0 ) z = z 0 2π 2 z = z 0 w = z 1/2 z = re iθ θ (z = 0) 0 2π 0
Tips KENZOU 28 7 6 P no problem 2 w = f(z) z w w z w = (z z ) b b w = log (z z ) z = z 2π 2 z = z w = z /2 z = re iθ θ (z = ) 2π 4π 2 θ θ 2π 4π z r re iθ re i2π = r re i4π = r w r re iθ/2 re iπ = r re
( : December 27, 2015) CONTENTS I. 1 II. 2 III. 2 IV. 3 V. 5 VI. 6 VII. 7 VIII. 9 I. 1 f(x) f (x) y = f(x) x ϕ(r) (gradient) ϕ(r) (gradϕ(r) ) ( ) ϕ(r)
( : December 27, 215 CONTENTS I. 1 II. 2 III. 2 IV. 3 V. 5 VI. 6 VII. 7 VIII. 9 I. 1 f(x f (x y f(x x ϕ(r (gradient ϕ(r (gradϕ(r ( ϕ(r r ϕ r xi + yj + zk ϕ(r ϕ(r x i + ϕ(r y j + ϕ(r z k (1.1 ϕ(r ϕ(r i
36 3 D f(z) D z f(z) z Taylor z D C f(z) z C C f (z) C f(z) f (z) f(z) D C D D z C C 3.: f(z) 3. f (z) f 2 (z) D D D D D f (z) f 2 (z) D D f (z) f 2 (
3 3. D f(z) D D D D D D D D f(z) D f (z) f (z) f(z) D (i) (ii) (iii) f(z) = ( ) n z n = z + z 2 z 3 + n= z < z < z > f (z) = e t(+z) dt Re z> Re z> [ ] f (z) = e t(+z) = (Rez> ) +z +z t= z < f(z) Taylor
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12 -- 1 1 2009 1 17 1-1 1-2 1-3 1-4 2 2 2 1-5 1 1-6 1 1-7 1-1 1-2 1-3 1-4 1-5 1-6 1-7 c 2011 1/(21) 12 -- 1 -- 1 1--1 1--1--1 1 2009 1 n n α { n } α α { n } lim n = α, n α n n ε n > N n α < ε N {1, 1,
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)
( ) sin 1 x, cos 1 x, tan 1 x sin x, cos x, tan x, arcsin x, arccos x, arctan x. π 2 sin 1 x π 2, 0 cos 1 x π, π 2 < tan 1 x < π 2 1 (1) (
6 20 ( ) sin, cos, tan sin, cos, tan, arcsin, arccos, arctan. π 2 sin π 2, 0 cos π, π 2 < tan < π 2 () ( 2 2 lim 2 ( 2 ) ) 2 = 3 sin (2) lim 5 0 = 2 2 0 0 2 2 3 3 4 5 5 2 5 6 3 5 7 4 5 8 4 9 3 4 a 3 b
z z x = y = /x lim y = + x + lim y = x (x a ) a (x a+) lim z z f(z) = A, lim z z g(z) = B () lim z z {f(z) ± g(z)} = A ± B (2) lim {f(z) g(z)} = AB z
Tips KENZOU 28 6 29 sin 2 x + cos 2 x = cos 2 z + sin 2 z = OK... z < z z < R w = f(z) z z w w f(z) w lim z z f(z) = w x x 2 2 f(x) x = a lim f(x) = lim f(x) x a+ x a z z x = y = /x lim y = + x + lim y
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 ),
66 σ σ (8.1) σ = 0 0 σd = 0 (8.2) (8.2) (8.1) E ρ d = 0... d = 0 (8.3) d 1 NN K K 8.1 d σd σd M = σd = E 2 d (8.4) ρ 2 d = I M = EI ρ 1 ρ = M EI ρ EI
65 8. K 8 8 7 8 K 6 7 8 K 6 M Q σ (6.4) M O ρ dθ D N d N 1 P Q B C (1 + ε)d M N N h 2 h 1 ( ) B (+) M 8.1: σ = E ρ (E, 1/ρ ) (8.1) 66 σ σ (8.1) σ = 0 0 σd = 0 (8.2) (8.2) (8.1) E ρ d = 0... d = 0 (8.3)
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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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5. [. ] z = f(, y) () z = 3 4 y + y + 3y () z = y (3) z = sin( y) (4) z = cos y (5) z = 4y (6) z = tan y (7) z = log( + y ) (8) z = tan y + + y ( ) () z = 3 8y + y z y = 4 + + 6y () z = y z y = (3) z =
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第1章 微分方程式と近似解法
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2000年度『数学展望 I』講義録
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1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1
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1 I 1.1 ± e = = - =1.602 10 19 C C MKA [m], [Kg] [s] [A] 1C 1A 1 MKA 1C 1C +q q +q q 1 1.1 r 1,2 q 1, q 2 r 12 2 q 1, q 2 2 F 12 = k q 1q 2 r 12 2 (1.1) k 2 k 2 ( r 1 r 2 ) ( r 2 r 1 ) q 1 q 2 (q 1 q 2
p = mv p x > h/4π λ = h p m v Ψ 2 Ψ
II p = mv p x > h/4π λ = h p m v Ψ 2 Ψ Ψ Ψ 2 0 x P'(x) m d 2 x = mω 2 x = kx = F(x) dt 2 x = cos(ωt + φ) mω 2 = k ω = m k v = dx = -ωsin(ωt + φ) dt = d 2 x dt 2 0 y v θ P(x,y) θ = ωt + φ ν = ω [Hz] 2π
Gmech08.dvi
145 13 13.1 13.1.1 0 m mg S 13.1 F 13.1 F /m S F F 13.1 F mg S F F mg 13.1: m d2 r 2 = F + F = 0 (13.1) 146 13 F = F (13.2) S S S S S P r S P r r = r 0 + r (13.3) r 0 S S m d2 r 2 = F (13.4) (13.3) d 2
2019 1 5 0 3 1 4 1.1.................... 4 1.1.1......................... 4 1.1.2........................ 5 1.1.3................... 5 1.1.4........................ 6 1.1.5......................... 6 1.2..........................
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No. No. 4 No f(z) z = z z n n sin x x dx = π, π n sin(mπ/n) x m + x n dx = m, n m < n e z, sin z, cos z, log z, z α 4 4 9
4 4 No. pdf pdf II Fourier No. No. 4 No. 4 4 38 f(z) z = z z n n sin x x dx = π, π n sin(mπ/n) x m + x n dx = m, n m < n e z, sin z, cos z, log z, z α 4 4 9 4 9 i = imaginary unit z = x + iy x, y R x real
sec13.dvi
13 13.1 O r F R = m d 2 r dt 2 m r m = F = m r M M d2 R dt 2 = m d 2 r dt 2 = F = F (13.1) F O L = r p = m r ṙ dl dt = m ṙ ṙ + m r r = r (m r ) = r F N. (13.2) N N = R F 13.2 O ˆn ω L O r u u = ω r 1 1:
SFGÇÃÉXÉyÉNÉgÉãå`.pdf
SFG 1 SFG SFG I SFG (ω) χ SFG (ω). SFG χ χ SFG (ω) = χ NR e iϕ +. ω ω + iγ SFG φ = ±π/, χ φ = ±π 3 χ SFG χ SFG = χ NR + χ (ω ω ) + Γ + χ NR χ (ω ω ) (ω ω ) + Γ cosϕ χ NR χ Γ (ω ω ) + Γ sinϕ. 3 (θ) 180
phs.dvi
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量子力学 問題
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Radiation from moving charges#1 Liénard-Wiechert potential Yuji Chinone 1 Maxwell Maxwell MKS E (x, t) + B (x, t) t = 0 (1) B (x, t) = 0 (2) B (x, t)
Radiation from moving harges# Liénard-Wiehert potential Yuji Chinone Maxwell Maxwell MKS E x, t + B x, t = B x, t = B x, t E x, t = µ j x, t 3 E x, t = ε ρ x, t 4 ε µ ε µ = E B ρ j A x, t φ x, t A x, t
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mugensho.dvi
1 1 f (t) lim t a f (t) = 0 f (t) t a 1.1 (1) lim(t 1) 2 = 0 t 1 (t 1) 2 t 1 (2) lim(t 1) 3 = 0 t 1 (t 1) 3 t 1 2 f (t), g(t) t a lim t a f (t) g(t) g(t) f (t) = o(g(t)) (t a) = 0 f (t) (t 1) 3 1.2 lim
1.2 y + P (x)y + Q(x)y = 0 (1) y 1 (x), y 2 (x) y 1 (x), y 2 (x) (1) y(x) c 1, c 2 y(x) = c 1 y 1 (x) + c 2 y 2 (x) 3 y 1 (x) y 1 (x) e R P (x)dx y 2
1 1.1 R(x) = 0 y + P (x)y + Q(x)y = R(x)...(1) y + P (x)y + Q(x)y = 0...(2) 1 2 u(x) v(x) c 1 u(x)+ c 2 v(x) = 0 c 1 = c 2 = 0 c 1 = c 2 = 0 2 0 2 u(x) v(x) u(x) u (x) W (u, v)(x) = v(x) v (x) 0 1 1.2
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Gmech08.dvi
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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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
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Microsoft Word - 信号処理3.doc
Junji OHTSUBO 2012 FFT FFT SN sin cos x v ψ(x,t) = f (x vt) (1.1) t=0 (1.1) ψ(x,t) = A 0 cos{k(x vt) + φ} = A 0 cos(kx ωt + φ) (1.2) A 0 v=ω/k φ ω k 1.3 (1.2) (1.2) (1.2) (1.1) 1.1 c c = a + ib, a = Re[c],
chap9.dvi
9 AR (i) (ii) MA (iii) (iv) (v) 9.1 2 1 AR 1 9.1.1 S S y j = (α i + β i j) D ij + η j, η j = ρ S η j S + ε j (j =1,,T) (1) i=1 {ε j } i.i.d(,σ 2 ) η j (j ) D ij j i S 1 S =1 D ij =1 S>1 S =4 (1) y j =
A 2 3. m S m = {x R m+1 x = 1} U + k = {x S m x k > 0}, U k = {x S m x k < 0}, ϕ ± k (x) = (x 0,..., ˆx k,... x m ) 1. {(U ± k, ϕ± k ) 0 k m} S m 1.2.
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[1] 1.1 x(t) t x(t + n ) = x(t) (n = 1,, 3, ) { x(t) : : 1 [ /, /] 1 x(t) = a + a 1 cos πt + a cos 4πt + + a n cos nπt + + b 1 sin πt + b sin 4πt = a
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