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 (
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1 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 f (z) f(z) f 2 (z) = +z z = Re z> f (z) Taylor z < f(z) f 2 (z) f (z) f(z) 35
2 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 (z) D D z D C D z f f 2 (z) z Taylor C D C f (z) f 2 (z) C C z D D f (z) =f 2 (z)
3 3. 37 D f(z) D f (z) f (z) D 2 f 2 (z) D 2 D f 2 (z) =f(z) f(z) =z + z 2 + z z 2n + z < z = 3.2 D D 2 C f (z) D f 2 (z) D 2 C C f (z) =f 2 (z) f (z) f 2 (z) D = D D 2 C D D 2 D f(z) f (z) z D f(z) = f 2 (z) z D 2 D C f(z) C f(z)dz = f (z)dz + f 2 (z)dz = C C C 2 C C D C L C 2 C D 2 C L L C C 2 f (z) f 2 (z) Morera f(z) D D D 2
4 Schwarz D f(z) D x f(x) D z f(z) =f(z) (3.) (3.) f(x) f(z) =u(x, y)+iv(x, y) (3.) f(z) =u(x, y) iv(x, y) f(z) =f(z) z = x + i u(x, ) iv(x, ) = u(x, ) + iv(x, ) v(x, ) = f(x) f(z) D f(z) F (z) =f(z) =U(x, y)+iv (x, y) f(z) =u(x, y) iv(x, y) U(x, y) =u(x, t) V (x, y) = v(x, t) (t = y ) f(x + it) =u(x, t) +iv(x, t) x + it u(x, t) v(x, t) D Cauchy-Riemann u(x, t) =U(x, y), v(x, t) = V (x, y) U x = u x = v t = V y U y = u t = v x = V x U(x, y) V (x, y) Cauchy-Riemann U, V u, v D F (z) =f(z) D
5 3. 39 f(x) =u(x, ) + iv(x, ) v(x, ) = f(x) =u(x, ) F (x) =U(x, ) + iv (x, ) = u(x, ) iv(x, ) = u(x, ) = f(x) D z F (z) =f(z) =f(z) D F (z) =f(z) F (z) =f(z) =f(z) D F (z) =f(z) f(z) =f(z) R C P P 2 C P, P 2 O r, r 2 r r 2 = R 2 C C C Q OQ C C C C C f(z) C AB AB AB ω = f(z) ω C A B z C ω C z = az + b cz + d ω = a ω + b c ω + d f (z ) ω = f (z ) L L z ω = f(z) f(z) AB C C
6 Riemann Riemann Riemann f(z) Riemann z f(z) log z Riemann ω = log z = log z + i arg z log z z 2π R 2 z 4π arg z< 2π R z 2π arg z< R z R z R 2 z arg z<2π 2π arg z<4π 4π arg z<6π R k log z log z = log z + i arg z (2kπ arg z<2(k +)π ),R 2,R,R,R,R 2, ω, 4π v< 2π, 2π v<, v<2π, 2π v<4π, 4π v<6π, R k k =, ±, ±2, R k R k+ R k k =, ±, ±2, R
7 3.2 Riemann 4 log z R ω = log z z ω R log z Riemann 2 z /2 Riemann ω = z /2 ( z = r e iθ ) z /2 = r e iθ/2 arg ω = θ 2 θ<2π arg ω<πz ω 2π θ<4π π arg ω<2π z ω z ω R R R R R R R R z z = r R R R z z z ω = ω = z /2 R R R ω =(z z ) /2 Riemann z ω = z /2 z 3 (z 2 ) /2 Riemann () ω =(z 2 ) /2 f(z) =(z 2 ) /2 f(z) =(z ) /2 (z +) /2 D (z ) /2 f (z) D 2 (z +) /2 f 2 (z) f(z) =f (z)f 2 (z) D D 2 (z 2 ) /2 f (z) = r e iθ /2 ( r = z >, <θ = arg (z ) < 2π ) f 2 (z) = r 2 e iθ 2/2 ( r 2 = z + >, <θ 2 = arg (z +)< 2π )
8 42 3 f(z) = r r 2 e i(θ +θ 2 )/2 ( r k >, <θ k < 2π : k =, 2) y z r 2 r P 2 θ 2 - P θ x 3.2: ω =(z 2 ) /2 f(z) x x r 2 θ 2 = x< θ π, θ 2 π, θ + θ 2 = π (z 2 ) /2 f(z) x F (z) = r r 2 e i(θ +θ 2 )/2 ( r k >, θ k < 2π : k =, 2) F (z) G(z) = r r 2 e i(θ +θ 2 )/2 ( r k >, π <θ k <π : k =, 2) G(z) G(z) F (z) z P P 2 F (z) i r r 2 ( θ = π, θ 2 =) i r r 2 ( θ = π, θ 2 =2π ) F (z) P P 2 θ θ 2 π ω arg ω =(θ +θ 2 )/2 π z ω θ θ 2 π 2π arg ω =(θ + θ 2 )/2 π 2π z ω P P 2 ω v (2) (z 2 ) /2 Riemann
9 3.2 Riemann 43 z P P 2 P P 2 θ θ 2 2π arg ω =(θ + θ 2 )/2 2π (z 2 ) /2 z z = z = θ θ 2 4π (z 2 ) /2 z z = z = θ θ 2 2π (z 2 ) /2 z z = z = (z 2 ) /2 P P 2 P P 2 R, R P P 2 R P P 2 R P P 2 R P P 2 R P P 2 R R (z 2 ) /2 Riemann R R ω R ω =(z 2 ) /2 z = ±
10 Γ B ζ 3.3. Γ t z = x + iy Γ (z) Γ (z) = e t t z dt (Rez> ) (3.2) t z = t x e t t z dt e t t x dt (3.3) t = x =Rez> Re z< (3.2) z [ ] e t t z dt = z e t t z + e t t z dt (3.4) z Re (z +) > z = Re z> Γ (z) = Γ (z + ) (3.5) z Γ Γ (z) = Γ (z + n +) z(z +) (z + n) (3.6) z = n n =,, 2, lim (z + n) Γ (z) =( )n z n n! z /Γ (z) z < Γ (z) = z n= ( + ) z n + z n (3.7) (3.8) Γ (z) = z e γz n= [( + z ) ] e z/n n (3.9) Γ (z) = lim n (n )! n z z(z + )(z +2) (z + n ) (3.)
11 3.3 Γ B ζ 45 γ Euler γ = lim ( n ) log n = (3.) n n =, 2, Γ () =, Γ(n +)=n! (3.2) Γ ( )= )!! π, Γ(n + )=(2n 2 2 π, Γ( n + 2 n )= ( )n 2 n 2 π (3.3) (2n )!! π Γ (iy) = y (3.4) y sinh πy Γ (z +)=zγ(z) (3.5) π Γ (z) Γ ( z) = Γ (z) Γ ( z) = π z sin πz sin πz (3.6) ( Γ z + ) ( ) Γ 2 2 z = π cos πz (3.7) Γ (2z) = (z 22z 2 π Γ (z) Γ + ) 2 (3.8) z Γ (z) 2π e z z z (/2) [ + 2 z z z z z z ] z 7 + z = n + Stirling (3.9) n! 2πn n n e n (3.2) B t p, q B(p, q) B(p, q) = t p ( t) q dt (Rep>, Re q> ) (3.2)
12 46 3 B(p, q) = t p π/2 ( + t) p+q dt =2 cos 2p θ sin 2q θ dθ (Rep>, Re q>) (3.22) B(p, q) = Γ (p) Γ (q) Γ (p + q) = B(q, p) (3.23) B(p, p) = π B(p, q) B(p + q, r) =B(q, r) B(q + r, p) (3.24) sin πp ( ) ( ) n + m n + m B(n, m) = m = n ( n, m =, 2, ) (3.25) n m z γ(z,p) = Γ (z,p) = p p e t t z dt (Rez> ) (3.26) e t t z dt (Rez> ) (3.27) Γ (z,p)+γ(z,p) =Γ (z) (3.28) γ(z + n, p) = Γ (z + n) Γ (z) [ n γ(z,p) e p p z r= ] p r z(z +) (z + r) (3.29) ζ z ζ(z) ζ(z) = n= n z (Rez> ) (3.3)
13 3.3 Γ B ζ 47 z = z z = 2, 4, 6, < Re z< ( ) z ζ(z) Γ 2 B n Bernoulli ( ) z = π z (/2) ζ( z) Γ 2 ζ( z) = 2 z π z ζ(z) Γ (z) cos πz 2 ζ() = 2, π2 π4 ζ(2) =, ζ(4) = 6 9 ζ(2n) = 22n π 2n B n, ζ( 2n) = ( )n B n (2n)! 2n (3.3) (3.32) (3.33) (3.34) x e x + x 2 = ( ) n B n (2n)! x2n (3.35) n B = 6, B 2 = 3, B 3 = 42, B 4 = 3, B 5 = 5 66 B 6 = , B 7 = 7 6, B 8 = 367 5, B 9 = , B = ζ(z) = 2 z n= n= (3.36) (2n ) z (Rez> ) (3.37) ζ(z) = p ζ(z) = ( p z ) 2 z ( ) n n= n z (Rez> ) (3.38) (Rez>) p (3.39) ζ(z) = ζ(z) = ζ(z) = t z Γ (z) e t dt (Rez>) (3.4) 2 z e t t z (2 z )Γ (z) e 2t dt (Rez>) (3.4) t z ( 2 z )Γ (z) e t dt (Rez>) (3.42) +
14 e t Ei( z) = dt z t ( z ) ( ) (3.43) x Erf x = e t2 dt = ( ) 2 γ 2,x2 (3.44) Erfc x = e t2 dt = ( ) 2 Γ 2,x2 ( Gauss ) (3.45) x Φ(x) = 2π x e t2 /2 dt = 2 + ( ) x Erf π 2 ( ) (3.46) sin t si x = dt x t ( ) (3.47) cos t ci x = dt x t ( ) (3.48) Im 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
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
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 =
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
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
(ii) (iii) z a = z a =2 z a =6 sin z z a dz. cosh z z a dz. e z dz. (, a b > 6.) (z a)(z b) 52.. (a) dz, ( a = /6.), (b) z =6 az (c) z a =2 53. f n (z
B 4 24 7 9 ( ) :,..,,.,. 4 4. f(z): D C: D a C, 2πi C f(z) dz = f(a). z a a C, ( ). (ii), a D, a U a,r D f. f(z) = A n (z a) n, z U a,r, n= A n := 2πi C f(ζ) dζ, n =,,..., (ζ a) n+, C a D. (iii) U a,r
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
, 3, 6 = 3, 3,,,, 3,, 9, 3, 9, 3, 3, 4, 43, 4, 3, 9, 6, 6,, 0 p, p, p 3,..., p n N = p p p 3 p n + N p n N p p p, p 3,..., p n p, p,..., p n N, 3,,,,
6,,3,4,, 3 4 8 6 6................................. 6.................................. , 3, 6 = 3, 3,,,, 3,, 9, 3, 9, 3, 3, 4, 43, 4, 3, 9, 6, 6,, 0 p, p, p 3,..., p n N = p p p 3 p n + N p n N p p p,
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
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
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
1 1 1 1 1 1 2 f z 2 C 1, C 2 f 2 C 1, C 2 f(c 2 ) C 2 f(c 1 ) z C 1 f f(z) xy uv ( u v ) = ( a b c d ) ( x y ) + ( p q ) (p + b, q + d) 1 (p + a, q + c) 1 (p, q) 1 1 (b, d) (a, c) 2 3 2 3 a = d, c = b
, ( ) 2 (312), 3 (402) Cardano
214 9 21, 215 4 21 ( ) 2 (312), 3 (42) 5.1.......... 5.2......................................... 6.2.1 Cardano................................... 6.2.2 Bombelli................................... 6.2.3
(iii) 0 V, x V, x + 0 = x. 0. (iv) x V, y V, x + y = 0., y x, y = x. (v) 1x = x. (vii) (α + β)x = αx + βx. (viii) (αβ)x = α(βx)., V, C.,,., (1)
1. 1.1...,. 1.1.1 V, V x, y, x y x + y x + y V,, V x α, αx αx V,, (i) (viii) : x, y, z V, α, β C, (i) x + y = y + x. (ii) (x + y) + z = x + (y + z). 1 (iii) 0 V, x V, x + 0 = x. 0. (iv) x V, y V, x + y
(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 :.
Z[i] Z[i] π 4,1 (x) π 4,3 (x) 1 x (x ) 2 log x π m,a (x) 1 x ϕ(m) log x 1.1 ( ). π(x) x (a, m) = 1 π m,a (x) x modm a 1 π m,a (x) 1 ϕ(m) π(x)
![Z[i] Z[i] π 4,1 (x) π 4,3 (x) 1 x (x ) 2 log x π m,a (x) 1 x ϕ(m) log x 1.1 ( ). π(x) x (a, m) = 1 π m,a (x) x modm a 1 π m,a (x) 1 ϕ(m) π(x)](/thumbs/94/118373946.jpg "Z[i] Z[i] π 4,1 (x) π 4,3 (x) 1 x (x ) 2 log x π m,a (x) 1 x ϕ(m) log x 1.1 ( ). π(x) x (a, m) = 1 π m,a (x) x modm a 1 π m,a (x) 1 ϕ(m) π(x)")
3 3 22 Z[i] Z[i] π 4, (x) π 4,3 (x) x (x ) 2 log x π m,a (x) x ϕ(m) log x. ( ). π(x) x (a, m) = π m,a (x) x modm a π m,a (x) ϕ(m) π(x) ϕ(m) x log x ϕ(m) m f(x) g(x) (x α) lim f(x)/g(x) = x α mod m (a,
D = [a, b] [c, d] D ij P ij (ξ ij, η ij ) f S(f,, {P ij }) S(f,, {P ij }) = = k m i=1 j=1 m n f(ξ ij, η ij )(x i x i 1 )(y j y j 1 ) = i=1 j
![D = [a, b] [c, d] D ij P ij (ξ ij, η ij ) f S(f,, {P ij }) S(f,, {P ij }) = = k m i=1 j=1 m n f(ξ ij, η ij )(x i x i 1 )(y j y j 1 ) = i=1 j](/thumbs/103/158001152.jpg "D = [a, b] [c, d] D ij P ij (ξ ij, η ij ) f S(f,, {P ij }) S(f,, {P ij }) = = k m i=1 j=1 m n f(ξ ij, η ij )(x i x i 1 )(y j y j 1 ) = i=1 j")
6 6.. [, b] [, d] ij P ij ξ ij, η ij f Sf,, {P ij } Sf,, {P ij } k m i j m fξ ij, η ij i i j j i j i m i j k i i j j m i i j j k i i j j kb d {P ij } lim Sf,, {P ij} kb d f, k [, b] [, d] f, d kb d 6..
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
Z: Q: R: C:
0 Z: Q: R: C: 3 4 4 4................................ 4 4.................................. 7 5 3 5...................... 3 5......................... 40 5.3 snz) z)........................... 4 6 46 x
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 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
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
I ( ) ( ) (1) C z = a ρ. f(z) dz = C = = (z a) n dz C n= p 2π (ρe iθ ) n ρie iθ dθ 0 n= p { 2πiA 1 n = 1 0 n 1 (2) C f(z) n.. n f(z)dz = 2πi Re
I ( ). ( ) () a ρ. f() d ( a) n d n p π (ρe iθ ) n ρie iθ dθ n p { πia n n () f() n.. n f()d πi es f( k ) k n n. f()d n k k f()d. n f()d πi esf( k ). k I ( ). ( ) () f() p g() f() g()( ) p. f(). f() A
Z: Q: R: C: 3. Green Cauchy
7 Z: Q: R: C: 3. Green.............................. 3.............................. 5.3................................. 6.4 Cauchy..................... 6.5 Taylor..........................6...............................
2 2 L 5 2. L L L L k.....
L 528 206 2 9 2 2 L 5 2. L........................... 5 2.2 L................................... 7 2............................... 9. L..................2 L k........................ 2 4 I 5 4. I...................................
平成 29 年度 ( 第 39 回 ) 数学入門公開講座テキスト ( 京都大学数理解析研究所, 平成 29 ~8 年月 73 月日開催 31 日 Riemann Riemann ( ). π(x) := #{p : p x} x log x (x ) Hadamard de
Riemann Riemann 07 7 3 8 4 ). π) : #{p : p } log ) Hadamard de la Vallée Poussin 896 )., f) g) ) lim f) g).. π) Chebychev. 4 3 Riemann. 6 4 Chebychev Riemann. 9 5 Riemann Res). A :. 5 B : Poisson Riemann-Lebesgue
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
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7 f(z) H f(z)dz = f(z), f (z)dz = 9 3 4 7 7.. 7. 7.. 3 f(z) D D z 7. f(z) = f () d (7.) z f(z) - =( z) = z = z Res(z) =lim!z ( z) z = f(z): (7.) 7. (7.) f(z) 7. 5 7.. f(z) z = a jz aj = r z = a r Ref(a)
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RIMS Kokyuroku, vol.084, (999), 45 59. Euler Fourier Euler Fourier S = ( ) n f(n) = e in f(n) (.) I = 0 e ix f(x) dx (.2) Euler Fourier Fourier Euler Euler Fourier Euler Euler Fourier Fourier [5], [6]
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医系の統計入門第 2 版 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, 第 2 版 1 刷発行時のものです.
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数学の基礎訓練I
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= M + M + M + M M + =.,. f = < ρ, > ρ ρ. ρ f. = ρ = = ± = log 4 = = = ± f = k k ρ. k
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ma22-9 u ( v w) = u v w sin θê = v w sin θ u cos φ = = 2.3 ( a b) ( c d) = ( a c)( b d) ( a d)( b c) ( a b) ( c d) = (a 2 b 3 a 3 b 2 )(c 2 d 3 c 3 d
A 2. x F (t) =f sin ωt x(0) = ẋ(0) = 0 ω θ sin θ θ 3! θ3 v = f mω cos ωt x = f mω (t sin ωt) ω t 0 = f ( cos ωt) mω x ma2-2 t ω x f (t mω ω (ωt ) 6 (ωt)3 = f 6m ωt3 2.2 u ( v w) = v ( w u) = w ( u v) ma22-9
I A A441 : April 15, 2013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida )
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p1 i 2 = 1 i 2 x, y x + iy 2 (x + iy) + (γ + iδ) = (x + γ) + i(y + δ) (x + iy)(γ + iδ) = (xγ yδ) + i(xδ + yγ) i 2 = 1 γ + iδ 0 x + iy γ + iδ xγ + yδ xδ = γ 2 + iyγ + δ2 γ 2 + δ 2 p7 = x 2 +y 2 z z p13
「産業上利用することができる発明」の審査の運用指針（案）
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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. 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)
II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re
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II 29 7 29-7-27 ( ) (7/31) II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I Euler Navier
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KENZOU 28 8 9 9/6 4 1 2 3 4 2 2 5 2 2 5.1............................................. 2 5.2......................................... 3 5.2.1........................................ 3 5.2.2...............................
CIII CIII : October 4, 2013 Version : 1.1 A A441 Kawahira, Tomoki TA (Takahiro, Wakasa 3 )
III203 00- III : October 4, 203 Version :. Kawahira, Tomoki TA (Takahiro, Wakasa 3 ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/3w-kansuron.html pdf 0 4 0 0 8 2 0 25 8 5 22 2 6 2 3 2 20 0 7 24 3
2.1 H f 3, SL(2, Z) Γ k (1) f H (2) γ Γ f k γ = f (3) f Γ \H cusp γ SL(2, Z) f k γ Fourier f k γ = a γ (n)e 2πinz/N n=0 (3) γ SL(2, Z) a γ (0) = 0 f c
GL 2 1 Lie SL(2, R) GL(2, A) Gelbart [Ge] 1 3 [Ge] Jacquet-Langlands [JL] Bump [Bu] Borel([Bo]) ([Ko]) ([Mo]) [Mo] 2 2.1 H = {z C Im(z) > 0} Γ SL(2, Z) Γ N N Γ (N) = {γ SL(2, Z) γ = 1 2 mod N} g SL(2,
1 12 ( )150 ( ( ) ) x M x 0 1 M 2 5x 2 + 4x + 3 x 2 1 M x M 2 1 M x (x + 1) 2 (1) x 2 + x + 1 M (2) 1 3 M (3) x 4 +
( )5 ( ( ) ) 4 6 7 9 M M 5 + 4 + M + M M + ( + ) () + + M () M () 4 + + M a b y = a + b a > () a b () y V a () V a b V n f() = n k= k k () < f() = log( ) t dt log () n+ (i) dt t (n + ) (ii) < t dt n+ n
II 1 II 2012 II Gauss-Bonnet II
II 1 II 212 II Gauss-Bonnet II 1 1 1.1......................................... 1 1.2............................................ 2 1.3.................................. 3 1.4.............................................
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
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I A A441 : April 21, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) Google
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[ ] 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
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)
() 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.
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() 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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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...
arctan 1 arctan arctan arctan π = = ( ) π = 4 = π = π = π = =
arctan arctan arctan arctan 2 2000 π = 3 + 8 = 3.25 ( ) 2 8 650 π = 4 = 3.6049 9 550 π = 3 3 30 π = 3.622 264 π = 3.459 3 + 0 7 = 3.4085 < π < 3 + 7 = 3.4286 380 π = 3 + 77 250 = 3.46 5 3.45926 < π < 3.45927