(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)
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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
2 (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) (iii), V 0. V 0. (2) (iv), y x, x. (3) x, y V, x y x y = x + ( y). x y x y. 2
3 (4) 0 x = 0. 0 C 0, 0 V.. (5) ( 1)x = x H., H, H x, y, (x, y), (i) (iv) : x, y, z H, α C, (i) (x, y) 0,, (x, y) = 0 z = 0. (ii) (x, y) = (y, x). (iii) (x + y, z) = (x, z) + (y, z). (iv) (αx, x) = α(x, y). (x, y) x y H, (1), (2) : x, y, z H, α C, (1) (x, y + z) = (x, y) + (x, z). (2) (x, αy) = α(x, y) H, x H, x = (x, x), x H.,. 3
4 1.1.1 H, x, (1) (3) : x, y H, α C, (1) x 0., x = 0, x = 0. (2) x + y x + y. (3) αx = α x., H x. H x, x = 1. H 0 x, e = x x x H., (1) (3) : x, y H, (1) (x, y) x y. ( ). (2) (x, y) = 1 4 ( x + y 2 x y 2 + i( x + iy 2 x iy 2 )). (3) x + y 2 + x y 2 = 2( x 2 + y 2 ). ( ) H, H x y d(x, y) = y x., (1) (3) : (1) d(x, ; y) 0., x = y, d(x, y) = 0. 4
5 (2) d(x, y) = d(y, x). (3) d(x, z) d(x, y) + d(y, z). ( ) 1.1.3, H. H, H x H x. H {x n } H x, lim x n x = 0 n. lim x n = x, x n x, (n ) n., x n x., {x n }. {x n },, x n x m x n x + x x m lim x n x m = 0 m, n.., {x n },., H, H, H, d(x, y) = y x. 5
6 1.1.3, H, H {x n }, x H x n x H, (x, y) x y., x n x, y n y, lim (x n, y n ) = (x, y) n., x n x,. lim x n = x n 1.1.5( ) H 1., H, H 1 H.. H H {xn }., H {x n } {y n } x, y, x + y α αx x + y = {x n + y n }, αx = {αx n }, H., x y (x, y) = lim n (x n, y n )., H., x, y H, x y = lim n x n y n = 0 6
7 . x y., H., x, y, z, (1) (3) : (1) x x. (2) x y, y x. (3) x y, y z, x z. H H = H/, H, H 1 H. H H 1.,,.., H. H K, x, y K 0 α 1 α, αx + (1 α)y K. 7
8 1.1.6 H, K H, α = inf x K x., K {x n } lim x n = α n., {x n } {x n } K H, H., L 2 (a, b) R (a, b), 2, L 2 (a, b)., f, g L 2 (a, b), (f, g) = b a f(x)g(x)dx., L 2 (R) = L 2 (, ) L 2 (a, b), {f n (x)} f(x), {f nk (x)}, (a, b) x f nk (x) f(x) d 1, Ω R d., L 2 (Ω) Ω, 8
9 2. L 2 (Ω)., f, g L 2 (Ω), (f, g), (f, g) = f(x)g(x)dx Ω., L 2 (R d ) l 2 x = {ξ n }, ξ n 2 <., l 2 x = {ξ n } y = (η n ) (x, y) = ξ n η n, l ,,.,. H., H x y, (x, y) = 0., x y. 0 H x. 9
10 H 0 {f 1, f 2,, f n, }, (f m, f n ) = 0, (m, n = 1, 2, )., {f n }, f n = 1, (n = 1, 2, 3, )., (f m, f n ) = δ mn, (mn = 1, 2, 3, )., δ mn, 1, (m = n), δ mn = 0, (m n). H {f 1, f 2,, f n } 1, 1, n α j f j = 0, (α j C, (1 j n)) j=1, α 1 = α 2 = = α n = 0. {f 1, f 2,, f n } 1, 1., H 0 {f 1, f 2,, f n } 1. 10
11 1.2.1( ) H 1 {f 1, f 2, },., e 1 = f 1 f 1, e 2 = f 2 (f 2, e 1 )e 1 f 2 (f 2, e 1 )e 1, e n = n 1 f n (f n, φ j )e j j=1, (n 2) n 1 f n (f n, e j )e j j=1, {e 1, e 2, }., e n f 1, f 2,, f n 1,, f n e 1, e 2, ;, e n L 2 ( π, ; π), (1) (3) : (1) {cos nx; n = 0, 1, 2, }. (2) {sin nx; n = 1, 2, }. (3) {cos nx; n = 0, 1, 2, } {sin nx; n = 1, 2, } L 2 ( π, ; π), (1) (3) : (1) { 1 2π, 1 π cos nx; n = 1, 2, }. (2) { 1 π sin nx; n = 1, 2, }. 11
12 (3) { 1 2π, 1 π cos nx, 1 π sin nx; n = 1, 2, }. H, {e n }., H x, x = α 1 e 1 + α 2 e α n e n +., {α n }, α n = (x, e n ), (n = 1, 2, ). H x, {α n } x {e n }, x., ( ) H, {e n } H., {α n } α n 2 <, x = α n e n, (x, e n ) = α n, (n 1) x H ( ) H {e n } , x H, (x, e n ) 2 x 2 12
13 ., x H {e n }., x H {e n }.,. H {e n }, x H, (x, e n ) = 0, (n 1), x = 0. {e n }, e n,., H {x α }, α, (x, x α ) = 0, x = 0., {x α } H,.,., H, {e n }, (1) (5) : (1) {e n }. (2) x, y H, (x, e n ) = (y, e n ), (n 1), x = y. 13
14 (3) x H, x = α n e n, α n = (x, e n ), (n 1). (4) x, y H, α n = (x, e n ), β n = (y, e n ), (n 1),. (x, y) = α n β n (5) x H,. (x, e n ) 2 = x 2 H., H {e n } H, , H {f n } H. H {e n }, {e n } H. H H, {e n } H., {e n }, x H ε > 0, {e n } 1 y = N α n e n 14
15 , x y < ε H, {e n } H., x H x = α n e n, α n = (x, e n ), (n 1), α = (α n ) l 2., x H α = (α n ) l 2.,. H = l l 2, e n = (δ ni, i 1), (n 1), {e 1, e 2, } { 1 2π, 1 π cos nx, 1 sin nx; n = 1, 2, } π L 2 ( π, π) L 2 ( π, π) f, : f(x) = 1 a (a n cos nx + b n sin nx), 2π π 15
16 a n = 1 π π π b n = 1 π π π f(x) cos nxdx, (n = 0, 1, 2, ), f(x) sin nxdx, (n = 1, 2, ). L { 1 2π e inx ; n = 0, ±1, ±2, } L 2 ( π, π) L 2 ( π, π) f, : f(x) = 1 a n e inx, 2π a n = 1 2π π n= π f(x)e inx dx, (n = 0, ±1, ±2, ). L { 1 π, 2 } cos nx; n = 1, 2, π L 2 (0, π) { 2 } sin nx; n = 1, 2, π L 2 (0, π) < a < b <., (a, b) { 1 b a e 2πinx/(b a) ; n = 0, ±1, ±2, } 16
17 L 2 (a, b) d n P n (x) = 1 2 n n! dx n (x2 1) n, (n = 0, 1, 2, )., { 2n + 1 P n (x); n = 0, 1, 2, } 2 L 2 ( 1, 1) dn H n (x) = ( 1) n e x2 dx n e x2, (n = 0, 1, 2, )., { 1 2 n n! π H n(x)e x2 /2 ; n = 0, 1, 2, } L 2 (, ) L n (x) = e x dn dx n (xn e x ), (n = 0, 1, 2, )., { 1 n! L n(x)e x/2 ; n = 0, 1, 2, } L 2 (0, ). 17
18 1.3,. H M, H., M x, y α C, (i), (ii) : (i) x + y M. (ii) αx M. H, H M x, y H (x, y) M M., H M H M H , H, H, M H. H M N, (x, y) = 0, (x M, y N)., M N., M N = {0}., H x M, (x, y) = 0, (y M) 18
19 ., x M. H A, A A = {u H; v A, (u, v) = 0}., : 1.3.2, A H. H M, M M., M M., M M = {0} H. M H, M H, M {0}., H x 0, x M H M, x H,. x = y + z, y M, z M ,. (M ) = M H S, S 1 H L(S) L(S) = M S 19
20 ., M = L(S) H.,. H., H x. x x., x, y H α C, (1) (4) : (1) x = x. (2) x + y = x + y. (3) αx = α x. (4) (x, y) = (x, y)., x H, x = x., x H,,, x 1 = x + x, x 2 = x x 2 2i x 1 = x 1, x 2 = x 2 x = x 1 + ix 2, x = x 1 ix 2. x 1 = Re (x), x 2 = Im (x), x 1 x, x 2 x., H 1 = {x H; x = x} 20
21 , H 1,. H = H 1 ih 1 21
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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)
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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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A S1-20 http://www2.mth.kyushu-u.c.jp/ hr/lectures/lectures-j.html 1 1 1.1 ϵ-n 1 ϵ-n lim n n = α n n α 2 lim n = 0 1 n k n n k=1 0 1.1.7 ϵ-n 1.1.1 n α n n α lim n n = α ϵ N(ϵ) n > N(ϵ) n α < ϵ (1.1.1)
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
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6 2 6.1 2 2, 2 5.2 R 2, 2 (R 2, B, µ)., R 2,,., 1, 2, 3,., 1, 2, 3,,. () : = 1 + 2 + 3 + (6.1.1).,,, 1 ,,,,., = (),, (1) (4) :,,,, (1),. (2),, =. (3),,. (4),,,,.. (1) (3), (4).,,., () : = 1 + 2 + 3 +,
1 yousuke.itoh/lecture-notes.html [0, π) f(x) = x π 2. [0, π) f(x) = x 2π 3. [0, π) f(x) = x 2π 1.2. Euler α
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III ϵ-n ϵ-n lim n a n = α n a n α 1 lim a n = 0 1 n a k n n k= ϵ-n 1.1
III http://www2.mth.kyushu-u.c.jp/~hr/lectures/lectures-j.html 1 1 1.1 ϵ-n ϵ-n lim n = α n n α 1 lim n = 0 1 n k n k=1 0 1.1.7 ϵ-n 1.1.1 n α n n α lim n = α ϵ Nϵ n > Nϵ n α < ϵ 1.1.1 ϵ n > Nϵ n α < ϵ 1.1.2
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i 1 1 2 3 6 6 7 8 10 10 11 12 12 12 13 2 15 15 16 17 17 18 19 20 20 21 ii CONTENTS 25 26 26 28 28 29 30 30 31 32 35 35 35 36 37 40 42 44 44 45 46 49 50 50 51 iii 52 52 52 53 55 56 56 57 58 58 60 60 iv
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i 1 1 2 2 3 3 4 4 4 5 7 8 8 9 9 10 11 13 14 15 16 17 19 ii CONTENTS 2 21 21 22 25 26 32 37 38 39 39 41 41 43 43 43 44 45 46 47 47 49 52 54 56 56 iii 57 59 62 64 64 66 67 68 71 72 72 73 74 74 77 79 81 84
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i 1 1 1 2 2 2 3 4 4 5 6 7 7 9 10 11 12 13 14 15 17 ii CONTENTS 2 19 19 20 23 24 25 25 26 29 29 31 31 33 35 36 36 39 39 41 44 45 46 48 iii 50 50 52 54 55 57 57 59 61 63 64 66 66 67 70 70 73 74 74 77 77
, 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
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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. http://www.morikita.co.jp/books/mid/009192 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. i 2 t 1. 2. 3 2 3. 6 4. 7 5. n 2 ν 6. 2 7. 2003 ii 2 2013 10 iii 1987
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
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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
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1 2 3 4 5 6 7 8 9 10 I II III 11 IV 12 V 13 VI VII 14 VIII. 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 _ 33 _ 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 VII 51 52 53 54 55 56 57 58 59
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211 kotaro@math.titech.ac.jp 1 R *1 n n R n *2 R n = {(x 1,..., x n ) x 1,..., x n R}. R R 2 R 3 R n R n R n D D R n *3 ) (x 1,..., x n ) f(x 1,..., x n ) f D *4 n 2 n = 1 ( ) 1 f D R n f : D R 1.1. (x,
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1 1 1 + 1 4 + + 1 n 2 + π2 6 x [10 n x] x = lim n 10 n n 10 k x 1.1. a 1, a 2,, a n, (a n ) n=1 {a n } n=1 1.2 ( ). {a n } n=1 Q ε > 0 N N m, n N a m a n < ε 1 1. ε = 10 1 N m, n N a m a n < ε = 10 1 N
ii iii iv CON T E N T S iii iv v Chapter1 Chapter2 Chapter 1 002 1.1 004 1.2 004 1.2.1 007 1.2.2 009 1.3 009 1.3.1 010 1.3.2 012 1.4 012 1.4.1 014 1.4.2 015 1.5 Chapter3 Chapter4 Chapter5 Chapter6 Chapter7
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[ ] 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 ),
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
春期講座 ~ 極限 1 1, 1 2, 1 3, 1 4,, 1 n, n n {a n } n a n α {a n } α {a n } α lim n an = α n a n α α {a n } {a n } {a n } 1. a n = 2 n {a n } 2, 4, 8, 16,
春期講座 ~ 極限 1 1, 1 2, 1 3, 1 4,, 1 n, n n {a n } n a n α {a n } α {a n } α lim an = α n a n α α {a n } {a n } {a n } 1. a n = 2 n {a n } 2, 4, 8, 16, 32, n a n {a n } {a n } 2. a n = 10n + 1 {a n } lim an
1
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活用ガイド (ソフトウェア編)
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1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2
2005 9/8-11 2 2.2 ( 2-5) γ ( ) γ cos θ 2πr πρhr 2 g h = 2γ cos θ ρgr (2.1) γ = ρgrh (2.2) 2 cos θ θ cos θ = 1 (2.2) γ = 1 ρgrh (2.) 2 2. p p ρgh p ( ) p p = p ρgh (2.) h p p = 2γ r 1 1 (Berry,1975) 2-6
() 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 ()
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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
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1 I 3 1 1.1 R x, y R x + y R x y R x, y, z, a, b R (1.1) (x + y) + z = x + (y + z) (1.2) x + y = y + x (1.3) 0 R : 0 + x = x x R (1.4) x R, 1 ( x) R : x + ( x) = 0 (1.5) (x y) z = x (y z) (1.6) x y =
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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 )
2010 II / y = e x y = log x = log e x 2. ( e x ) = e x 3. ( ) log x = 1 x 1.2 Warming Up 1 u = log a M a u = M a 0
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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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u = u(t, x 1,..., x d ) : R R d C λ i = 1 := x 2 1 x 2 d d Euclid Laplace Schrödinger N := {1, 2, 3,... } Z := {..., 3, 2, 1,, 1, 2, 3
2 2 1 5 5 Schrödinger i u t + u = λ u 2 u. u = u(t, x 1,..., x d ) : R R d C λ i = 1 := 2 + + 2 x 2 1 x 2 d d Euclid Laplace Schrödinger 3 1 1.1 N := {1, 2, 3,... } Z := {..., 3, 2, 1,, 1, 2, 3,... } Q
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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活用ガイド (ソフトウェア編)
ii iii iv NEC Corporation 1998 v vi PA RT 1 vii PA RT 2 viii PA RT 3 PA RT 4 ix P A R T 1 2 3 1 4 5 1 1 2 1 2 3 4 6 1 2 3 4 5 7 1 6 7 8 1 9 1 10 1 2 3 4 5 6 7 8 9 10 11 11 1 12 12 1 13 1 1 14 2 3 4 5 1
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 {
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パソコン機能ガイド
PART12 ii iii iv v 1 2 3 4 5 vi vii viii ix P A R T 1 x P A R T 2 xi P A R T 3 xii xiii P A R T 1 2 3 1 4 5 1 6 1 1 2 7 1 2 8 1 9 10 1 11 12 1 13 1 2 3 4 14 1 15 1 2 3 16 4 1 1 2 3 17 18 1 19 20 1 1
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2000年度『数学展望 I』講義録
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6 FIR FIR FIR FIR 6.1 FIR 6.1.1 H(e jω ) H(e jω )= H(e jω ) e jθ(ω) = H(e jω ) (cos θ(ω)+jsin θ(ω)) (6.1) H(e jω ) θ(ω) θ(ω) = KωT, K > 0 (6.2) 6.1.2 6.1 6.1 FIR 123 6.1 H(e jω 1, ω
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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
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 =
基礎数学I
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数学概論I
{a n } M >0 s.t. a n 5 M for n =1, 2,... lim n a n = α ε =1 N s.t. a n α < 1 for n > N. n > N a n 5 a n α + α < 1+ α. M := max{ a 1,..., a N, 1+ α } a n 5 M ( n) 1 α α 1+ α t a 1 a N+1 a N+2 a 2 1 a n
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