E Riemann-Stieltjes 65 teaching.html (epsilon-delta )

23 4 4 2 5 3 9 4 6 5 22 6 24 7 3 8 34 9 4 42 48 2 49 3 53 A Goursat 56 B 56 6 D 62

E Riemann-Stieltjes 65 http://www.math.nagoya-u.ac.jp/~yamagami/teaching/ teaching.html (epsilon-delta ) http://sss.sci.ibaraki.ac.jp/teaching/set/set25.pdf, http://sss.sci.ibaraki.ac.jp/teaching/set/real.pdf [] Richard A. Silverman, Introductory omplex Analysis, 985, Dover. [2] S. Lang, omplex Analysis, 3rd edition, 998, Springer. [3], (23). [4] S. G. Krantz and H. R. Parks, The Implicit Function Theorem, 22, Birkhäuser. (**) (*) 2

θ e iθ = cos θ + i sin θ * (Euler s formula) ( ) θ 2 + 3 4 +... {, /2, /3, /4,... } (generating function) f(t) f(t) = t 2 t2 + 3 t3 4 t4 +... f (t) = t + t 2 t 3 + = t + x = f(x) = x dt = log(x + ). t + 2 + 3 4 + = log 2 x = i f(i) = (i) 2 ( ) + 3 ( i) 4 () + 5 (i) 6 ( ) + 7 ( i) +... = i ( 3 + 5 7 ) +... + ( 2 2 + 3 4 ) +.... + i = 2e πi/4 f(i) = log(i + ) = log 2 + log e πi/4 = 2 log 2 + π 4 i 3 + 5 7 + = π 4 * http://en.wikipedia.org/wiki/euler s_formula 3

(*). 2 ( ) = = ( )( ) = = i 2 =. a + ib i 2 i 2 = a + ib (a, b) (a, b) (a, b) + (a, b ) = (a + a, b + b ), (a, b)(a, b ) = (aa bb, ab + a b) (a, ) a (, ) (, ) (a, b) (a, b ) (a, b) + (a, b ) = (a, b ) + (a, b) = (, ) (a, b) (a, b) = ( a, b) (a, b) (x, y) x, y (a, b)(x, y) = (x, y)(a, b) = (, ) ax by =, bx + ay = a 2 + b 2 (a, b) (a, b) (a,b) ( ) a (a, b) = a 2 + b 2, b a 2 + b 2 (a, ) a (, ) = i (a, b) = a + ib (, ) 2 = (, ) i 2 = 4

2. 3. (x, y)(x, y) = (, ) (x, y) 4 (*). i 2 = αi + β (α, β ) n z n = z z z n z n if n >, z n = if n =, (/z) n if n < m, n z z m z n = z m+n, (z m ) n = z mn 5. a (x, y) (ax, ay) π π/2 i x + iy ( ) x y y x 2 z = x + iy x = Re z, y = Im z, z (real part, imaginary part) z = x 2 + y 2 (modulus) z = x iy z (complex conjugate) z + w = z + w, zw = z w, z = z, z + w z + w, z 2 = zz, zw = z w Remark. i j i z ± 5

i i i i z = x + iy 6. zw = z w zw = z w z + w z + w 7. z = x + iy, w = u + iv zw 2 = z 2 w 2 8. a, a,..., a n a + a z + + a n z n = z = x + iy z z = x + iy (x, y) *2 (complex plane) (polar coordinates) (r, θ) z = r(cos θ + i sin θ) (polar form) z = re iθ. r z z θ z (argument *3 ) θ = arg z 2π 2 e iθ e iθ = e i(θ+θ ) zz = z z, arg(zz ) = arg z + arg z mod 2πZ e iθ θ 9. (*). z + z + z 2 + + z n = zn z *2 2 *3 argument variable parameter argument argument phase phase 6

z = e iθ. a + ib (a, b R) z 2 = a + ib z a n n z n = a a = a e iα z = re iθ z n = a z = a /n (cos θ k + i sin θ k ), r n = a, nθ = α + 2πk, k Z. k n θ k = α + 2πk, k =,,..., n n 2.. n 2 n z n = z = e 2πik/n ( k n) n n n 2 (*). z 3 = i 2.2. z, z 2, z 3, z 4 z 4 z z 2 z z 2 z 3 z 4 z 3 = z 4 z z 2 z 3 z 2 z z 4 z 3. 3. 3 a = + i, b = 2 i, c = x + iy c x, y cos θ = eiθ + e iθ 2, sin θ = eiθ e iθ. 2i cos(nθ) t = cos θ z = e iθ n = 2, 3 2(cos θ) 2 = 2 ( z + ) 2 = (z 2 + z ) z 2 2 + 2 = cos(2θ) +. (z 2 + z ) 2 2 = 2t 2 cos(2θ) = 2 cos 2 θ. z + z z 3 + z 3 = 4t 3 3t 2 cos(3θ) = 4 cos 3 θ 3 cos θ. 7

t = cos θ n (hebyshev polynomial) T n (t) 2tT n (t) = (z + z ) zn + z n = T n+ (t) + T n (t), T =, T (t) = t 2 cos(nθ) = T n (cos θ) *4 4. sin(nθ) 5 (**)., z 3 + az 2 + bz + c = ζ = z + d d ζ 3 + 3aζ + 2b = 2, 3 (Niccolò Fontana Tartaglia) ζ = u + v u 3 + v 3 + 2b + 3(uv + a)(u + v) = ζ u, v u, v uv + a = ζ u, v u 3 + v 3 = 2b, uv = a u 3 v 3 = a 3 u 3, v 3 t 2 + 2bt a 3 = u, v λ ± = b ± b 2 + a 3 uv = a λ ± µ ± (µ + µ ) 3 = λ + λ = a 3 µ + µ = aω k (k =,, 2), ω 2 + ω + = *4 http://en.wikipedia.org/wiki/hebyshev_polynomials 8

u, v (u, v) = (µ + ω k, µ ), (µ + ω k, µ ω 2 ), (µ + ω 2 k, µ ω) ζ = µ + ω k + µ, µ + ω k + µ ω 2, µ + ω 2 k + µ ω a, b b 2 + a 3 λ ± µ ± µ + µ = a ζ = µ + + µ, µ + ω + µ ω 2, µ + ω 2 + µ ω. µ + = µ b 2 +a 3 = µ + +µ = 2b /3, µ + = b /3 ζ 3 + 3aζ + 2b = (ζ + 2b /3 )(ζ b /3 ) 2 b 2 + a 3 < λ ± µ ± a < µ + µ = a µ = µ + ζ = µ + µ, µω + µω, µω 2 + µω 2 µ, µω, µω 2 µ 3 = λ + Rafael Bombelli Joseph Louis Lagrange *5 3 (topology) {z n = x n + iy n } n z = x + iy lim z n z = n lim x n = x, lim y n = y n n *5 J.L. Lagrange, Réflexions sur la résolution algébrique des équations, 77. 9

z {z n } n z = lim n z n 6. x + y 2 z x + y 3.. {a n }, {b n } a, b lim a nb n = ab. n a b n lim = b n a n a. Proof. a n b n ab a n a b n + a b n b a n a ( b n b + b ) + a b n b a n a = a n a a n a a a n a ( a a n a ) 7. z z < + z + z 2 + + z n = zn z ( lim + z + z 2 + + z n ) n n n z n + a z n + + a n = 3.2 ( ). f(z) = z n + a z n + + a n (i) f(ζ) = ζ (ii) c,..., c n f(z) = (z c ) (z c n ) (iii) f(z) f(z)

Remark. Proof. z n + c z n + + c n = c n = z = c n r > z = r iθ f(r iθ ) = r n e inθ + c r n e i(n )θ + + c n r, θ r > θ 2π r c n r f(r iθ ) = r n e inθ ( + c e iθ r + + c n e inθ ) r n c e iθ r e inθ + + c n r n r r n (r ) r ( = max{ c k }) r n e inθ ( θ 2π) z = r n n < r < f(r iθ ) = re iθ 8. (ii), (iii) (i) 9 (**). f(z) f(ζ) = ζ (i) lim z f(z) = f(z) z ζ (ii) f(z) z ζ f(z) = f + f l (z ζ) l + f l+ (z ζ) l+ + + f n (z ζ) n, f l (iii) f z ζ z ζ f(z) < f (iv) f = f(ζ) = 2. x P (x) P (x) (x R) x Q(x) P (x) = Q(x)Q(x) Q(x) = j c jx j Q(x) = j c jx j ( e x = lim + x ) n n n

2. log( + t) lim ( n log + x ) = x n n z = x + iy ( e x (cos y + i sin y) = lim + z ) n n n *6 e z + z n = r n(cos θ n + i sin θ n ), r 2 n = ( + x ) 2 y 2 + n n 2, tan θ n = y n + x ( + z n) n = r n n (cos(nθ n ) + i sin(nθ n )) n log r n = n ( 2 log + 2x n + x2 + y 2 ) n 2 = n 2 ( 2x n + y2 x 2 ) n 2 + x lim n rn n = e x tan θ θ n (n ) lim = θ θ lim nθ n = lim n tan θ ny n = lim n n n n + x = y 3.3. e z e w = e z+w {e z ; z } = = \ {} re iθ (r > ) {z ; e z = re iθ } = log r + iθ + 2πiZ = {log r + iθ + 2πin; n Z} *7 t z(t) z(t) t z(t) = x(t) + iy(t) x(t), y(t) 22 (*). < r < z(t) = e it + re 2it *6 http://en.wikipedia.org/wiki/euler s_formula *7 2

z (t) = lim h z(t + h) z(t) h z (t) = x (t) + iy (t) d dt (z(t)w(t)) = z (t)w(t) + z(t)w (t). 23. w(t) = u(t) + iv(t) z(t)w(t) z(t) t x(t), y(t) t b a z(t) dt = lim + j= z(τ j )(t j t j ), = max{ t j t j ; j n}, τ j [t j, t j ] b a z(t) dt = b a x(t) dt + i b a y(t) dt b a z (t) dt = z(b) z(a) 3.4. c d dt ect = ce ct e ct dt = c ect. Proof. a, b c = a + ib e ct = e at (cos(bt) + i sin(bt)) d dt ect = ae at (cos(bt) + i sin(bt)) + e at ( b sin(bt) + ib cos(bt)) = ce ct. 3.5. e ct dt = c ect = e at cos(bt) dt = eat a 2 (a cos(bt) + b sin(bt)), + b2 a cos(bt) + b sin(bt) + ia sin(bt) ib cos(bt) a 2 + b 2 e at e at sin(bt) dt = eat a 2 (a sin(bt) b cos(bt)). + b2 3

24 (*). te ct te at sin(bt) dt z (t) a < t < b lim t a+ z (t), lim t b z (t) z(t) (a t b) * 8 b a z (t) dt = b a (x (t)) 2 + (y (t)) 2 dt 25. z(t) = e a t(cos(bt) + i sin(bt)) ( t ) b b z(t) dt z(t) dt, a b a a λ b z(t) dt = b a a z(τ j )(t j t j ) j= λz(t) dt, λ z(τ j ) (t j t j ) j= λ z(τ j )(t j t j ) = λz(τ j )(t j t j ) j= j= 26. b b z(t) dt = e iθ z(t) dt a a * 9 Re(e iθ z(t)) = x(t) cos θ + y(t) sin θ x(t) 2 + y(t) 2 27 (**). ( 2 ( 2 b b x(t) dt) + y(t) dt) b a a a x(t)2 + y(t) 2 dt *8 *9 4

* D f(x, y) dxdy D f(x, y) dxdy f(x, y) f() f() = f (t) dt f (t) dt 28. f(t) ( t ) f() f() f (t) ( < t < ) 29 (**). f(t) (a < t < b) f (t) φ(s, t) (a < s, t < b) φ(s, t) = φ { f(s) f(t) s t if s t, f (t) if s = t z (t) z (t) f (t) d = A + dt. z n (t). z n (t). f n (t) a a 2... a n a 2 a 22... a 2n A =...... a n a n2... a nn f j (t) z j (t) f j (t) = ( j n) z j (t) z (t). = e λt z n (t) ζ.. ζ n λ ζ. = A ζ n ζ.. ζ n * 5

λ A ζ A n v,..., v n A A v j = λ j vj f (t) z (t), f (t) z (t) = n j= c j (t)e λ jt vj, f (t) = g j (t) v j j c j(t)e λ jt vj = j j g j (t) v j c j (t) = t g j (s)e λjs ds + c j () 3.6. A = ( ) ω ω A v ± = ±iω v ±, ( ) v ± = ±i ( ) f (t) = g f 2 (t) + (t) v + + g (t) v g ± (t) ( ) ( t ) ( t ) z (t) = g z 2 (t) + (s)e iω(t s) ds + c + e iωt + g (s)e iω(t s) ds + c e iωt. 4 D * z z * { z c < r} D D 6

4.. (i) f(z) = q(z), D = {z ; p(z) }. p(z), q(z) z p(z) p(z) = p (ii) f(z) = e z, D =. cos z = eiz + e iz, sin z = eiz e iz 2 2i z R (iii) z = re iθ (r >, π < θ < π) Logz = log r + iθ, D = \ (, ] (iv) α z α = e αlogz, D = \ (, ] 3. z (, ] e Logz = z, Loge z = z ( π < Imz < π) 3. e z = e z, cos z = cos z, sin z = sin z Logz = Logz 32. z = re iθ ( r < ) Log( + z) 33. (z α ) β = z αβ 34 (*). z z z z = x + iy D f(z) = u(x, y) + iv(x, y) f u(x, y), v(x, y) 4.2. (i) f(z) = z 2 u(x, y) = x 2 y 2, v(x, y) = 2xy. (ii) f(z) = z u(x, y) = x, v(x, y) = y. (iii) f(z) = z 2 u(x, y) = x 2 + y 2, v(x, y) =. (iv) f(z) = /z (D = \ {}) u(x, y) = x y x 2, v(x, y) = + y2 x 2 + y 2. f z D w w = f(z) 4.3. c φ f(z) = e iφ z + c (D = ) 35. φ f(z) = e iφ z 36 (**). z z < w = + z + z 2 D z z 2 + z + w = z < w z = w 7

4.4. D f(z) c D lim f(z) = f(z) z c 4.5. D f(z) c D f (c) = lim z c f(z) f(c) z c f (z) D f * 2 (holomorphic * 3 function) 4.6. n (z n ) = nz n. n < D = {z } 37. 4.7. x, y e x+iy lim =. (x,y) (,) x + iy Proof. * 4 e a+ib = a + ib e x+iy x + iy e t(a+ib) dt a + ib e ta dt a + ib e a = (e t(x+iy) ) dt e t(x+iy) dt x + iy 4.8. (i) (e z ) = e z (z ). (ii) (Logz) = z (z (, ]). te t x dt x + iy e x 2 *2 ( Goursat ) *3 holos = whole morphe = shape *4 e x = + x + O(x 2 ), cos y = + O(y 2 ), sin y = y + O(y 3 ) 8

Proof. c (, ] z = ce x+iy z c (x, y) (, ) Log(c x+iy ) = Log c + x + iy Logz Logc z c = x + iy c(e x+iy ) c. 4.9. f(z) = 2x = z + z, f(z) = x iy = z (x = Rez, y = Imz) z 4.. f(z), g(z) f(z)g(z), /f(z), f(g(z)) (f(z)g(z)) = f (z)g(z) + f(z)g (z), ( ) = f (z) f(z) f(z) 2, (g(f(z))) = g (f(z))f (z). Proof. w = f(z), b = f(a) g(f(z)) g(f(a)) z a = g(w) g(b) f(z) f(a) w b z a z a w b F (z) = { f(z) f(a) z a z a, f (a) otherwise f(z) f(a) = (z a)f (z) lim z a F (z) = f (a) G(w) g (f(a)) (w f(a)) g(f(z)) g(f(a)) z a 4. ( ). α = F (z)g(f(z)) f (a)g (f(a)) (z α ) = αz α, z (, ]. x = z + z 2, y = z z 2i z = x z x + y z z = x z x + y z y = ( 2 y = 2 x i ), y ( x + i ). y 9

4.2. f(z) 38. z z = z z =, d f f(z(t)) = dt z (z(t))z (t) + f z (z(t))z (t). z z = z z =, z z = z z = ( ) 2 4 x 2 + 2 y 2. 4.3 (auchy-riemann * 5 ). u(x, y), v(x, y) u(x, y) + iv(x, y) f(x + iy) = f(x, y) f f z = u x = v y, u y = v x Proof. chain rule f(z) f(c) = = = f z d f(tz + ( t)c) dt dt f (tz + ( t)c)(z c) dt + z f (c)(z c) + (c)(z c) + (z c) + (z c) z ( f h f (tz + ( t)c) z f (tz + ( t)c)(z c) dt z ) z (c) ) ( f f (tz + ( t)c) z z (c) (h(tz + ( t)c) h(c)) dt max{ h(w) h(c) ; w c z c } ( z c ) (z c)/(z c) = e 2i arg(z c) Remark. f = f z z z dt dt 39 (*). f(x + iy) = e x (a cos y + i sin y) a *5 auchy Riemann (d Alembert, 77 783) 752 d Alembert 2

4. f(x + iy) = u(x, y) + iv(x, y) u(x, y), v(x, y) ( ) ( ) 2 x 2 + 2 2 y 2 u = x 2 + 2 y 2 v = 4. f (z) f(z) 42. f(z) f(z) auchy-riemann f(z) = u(x, y) + iv(x, y) ( x φ : y) φ ( ) ux u y = v x v y ( ) u(x, y) v(x, y) ( ) ux u y (conformal mapping, conformal transformation) * 6 u y ( ) ( ) a b a b, b a b a det(φ ) = f f = = f(z) f(z) z z f(z) c = a + ib f (a + ib) = f (c) c = a + ib u, v u = f (c) (X 2 Y 2 )/2, v = f (c) XY X, Y (x a, y b) (a, b) u, v 3 π/4 ( X Y z n, z + /z ) ( cos θ sin θ sin θ cos θ u x ) ( x a y b ). *6 conformal 2

5 z(t) [ ] (smooth curve) z(t) (a t b) * 7 (i) z(t) (a, b) (ii) z (t) (a < t < b) z (a) lim t a+ z (t), z (b) lim t b z (t) (iii) t (piece-wise smooth curve) z(t) (a t b) [a, b] a = c < c < < c n = b z(t) (c j t c j ) (closed curve) (simple curve) Remark. { t 2 + it 4 t, z(t) = t 2 it 4 t w(t) = { t + it 2 t, t it 2 t 43., a, ib (a >, b > ) w(t) z(t) D f(z) D : z(t) (a t b) b a f(z(t)) dz dt dt *7 22

f(z k )(z k z k ), z k = z(t k ), a = t < t < < t n = b k= * 8 * 9 f(z) dz f(z) (line integral) (path) (contour integral) f(z) dz b a dz dt dt z k z k k= 2 n f(z) dz = f(z)dz + + f(z) dz. n : z(ta + ( t)b) ( t ) f(z)dz = f(z)dz. f(z)dz f(z) dz max{ f(z) ; z }. 44. 45 (**). z(t) z (t) tk z(t k ) z(t k ) = z (t k )(t k t k ) + (t k s)z (s) ds t k *8 z (t) Riemann-Stieltjes *9 23

5.. r > : z(t) = re it ( t 2π) { z n 2πi if n =, dz = otherwise. /z 46. a b f(z) f (z)dz = f(b) f(a) D = \ {} f (z) = /z f(z) 47 (*)., + i 2, + i 3 f(z) = z 2, f(z) = e z, f(z) = x + y f(z)dz + f(z)dz, 2 3 f(z)dz 48 (*). z 2 + = i ( 2 z + i ) z i z 2 + dz \ {±i} Remark. f(z) = u(x, y) + iv(x, y) (z = x + iy) : z(t) = x(t) + iy(t) f(z) dz = b a (ux (t) vy (t)) dt + i (u(x, y)dx v(x, y)dy), b a (uy (t) + vx (t)) dt (u(x, y)dy v(x, y)dx) (u(x, y), v(x, y)), (v(x, y), u(x, y)) (x(t), y(t)) (a t b) 6 (i) Green (ii) (homotopy invariance) * 2 *2 24

6.. * 2 f(s, t) d ds f(s, t)dt = Proof. f(s, t)dt x ( ) f (s, t)dt ds = s = = f (s, t)dt. s ( x f s [ f(s, t) ) (s, t)ds dt ] s=x f(x, t)dt dt s= f(, t)dt f (s, t)dt s {(s, t); s, t } D z(s, t) < s, t < z(s, t) = s z t (u, t) du + z(, t) = s z (s, u) du + z(s, ) t z(t, ), z(, t), z( t, ), z(, t) ( t ) 6.2. D f(z) f(z) dz =. Proof. s z(s, t) ( t ) s d f(z)dz = d ds s ds = = = [ = f(z(s, t)) z (s, t)dt t ( f(z(s, t)) z ) (s, t) dt s t ( f (z(s, t)) z (s, t) z s t (s, t) + f(z(s, t)) 2 z s t ( f(z(s, t)) z ) (s, t) dt t s f(z(s, t)) z (s, t) s ] t= t= = f(z(s, )) z (s, ) f(z(s, )) z (s, ). s s s ) (s, t) dt 49 (*). D -homotopy *2 f(s, t) g(s, t) s 25

6.3. (i) a, b, c D [a,b] f(z) dz + f(z) dz + f(z) dz =. [b,c] [c,a] (ii) D : z(t) ( t ) c D sz(t) + ( s)c D ( s, t ) f(z) dz = L z() z() (iii) D L f(z) dz =. f(z) dz. 6.4 (auchy ). D D D D f(z) D f(z)dz =. D D Proof. D D = + + n j L j L j f(z) D f(z) dz = j= j f(z) dz = j= L j f(z) dz =. 6.5. {z ; Im z } {z ; z R} f(z) M(r) = max{ f(z) ; z = r} lim M(r) = t > r lim e itz f(z)dz = r r r : z = re iθ ( θ π). Proof. π e itz f(z) dz rm(r) e rt sin θ dθ 2rM(r) r π/2 e 2rtθ/π dθ = π t M(r)( e rt ). 6.6. 26

(i) r z R e iz z dz = R sin x lim R x dx = π 2 (ii) r R, θ π/4 e z2 dz = R lim R e ix2 dx = π 2 eπi/4 (Fresnel ) 5 (*). 6.7 (auchy ). auchy 6.4 z D f(z) = f(ζ) 2πi ζ z dζ. Proof. z r > r r f(ζ) ζ z dζ = r f(ζ) ζ z dζ. r f(ζ) ζ z dζ = = r r f(ζ) f(z) dζ + f(z) ζ z r ζ z dζ f(ζ) f(z) dζ + 2πif(z) ζ z f(ζ) dζ 2πif(z) ζ z = r f(ζ) f(z) ζ z dζ ( f (z) + ϵ) r r Remark. D f(z) D 6.8. c r > D f(z) c D z F (z) D F (z) F (z) = f(z) 27

Proof. c = z = x + iy F (z) = x f(t) dt + i y f(x + it) dt = i y f(it) dt + x F (z) x, y f(t + iy) dt F y = if(z), F x = f(z). F (z) f(z) F z = f(z) + iif(z) = Remark. c 6.9. f (n) (z) = n! f(ζ) dζ 2πi (ζ z) n+ Proof. a b B D B f(ζ) dz dζ = (ζ z) n+ = = n dζf(ζ) [ f(ζ) B (ζ z) n(ζ z) n ( f(ζ) dz n+ ] z=b z=a dζ (ζ b) n (ζ a) n ) dζ B f(ζ) (ζ z) n dζ z D d dz f(ζ) (ζ z) n dζ = n f(ζ) dζ (ζ z) n+ f(z) z = a ( z a R) 6.7 f(z) z a < R, ζ a = R f(z) = f(ζ) 2πi ζ a =R ζ z dζ. 28

z a / ζ a < f(z) = n ζ z = n (z a) n (ζ a) n+ c n (z a) n, z a < R, c n = f(ζ) dζ 2πi ζ a =R (ζ a) n+ n f(z) c k (z a) k = (z a)k f(ζ) dζ k= 2πi ζ a =R (ζ a) k+ k n z a k f(ζ) dζ 2π ζ a k+ ( z a < R ) ζ a =R k n z a k M R k k n M z a n = R z a R n 6. (Taylor expansion). D f(z) f a D D D r > f(z) = n= n! f (n) (a)(z a) n, z a < r z a z a ( ) f(z) (Taylor series) 6.. * 22 (i) z e z = cos z = sin z = n= k= k= n! zn = + z + 2 z2 +, ( ) k (2k)! z2k = 2 z2 4! z4 +, ( ) k (2k + )! z2k+ = z 3! z3 +. *22 29

(ii) z < α ( ) n+ Log( + z) = z n = z n 2 z2 + 3 z3 4 z4 +, n= ( ) α ( + z) α = z n α(α ) = + αz + z 2 +. n 2 n= Remark. 7 c k = lim k= c k n k= ( ) {a i } i I A [, ] * 23 A = a i {a i } i I i I (i) {a i } i F a i A i F (ii) A I I = I n n a i = ( i I n i I n a i ). {a i } i I, {b i } i I a i b i (i I) a i b i. i I i I 5 (**). ) {a i } i I i I a i < ϵ > {i I; a i ϵ} {i I; a i > } *23 A = sup{ i F a i; F I } 3

+ n dx < + α >, xα < α >. nα n= n =. n + x dx n k= n+ k x dx + 2 lim + + n n log n =, log n k k+ k k+ x dx = x k k+ k kx dx k kx dx k 2 γ = lim ( + 2 + + n ) log n n 52. a lim n ( a + + a + 2 + + a + n log a + n ) a + 53. a n 2 n(log n) a 54 (*). (k 2 + l 2 ) a k= l= a > 7.. c n n= c n < n 3

(converge absolutely) * 24 7.2 ( ). c n c n n n Proof. c n n= c n c n = a n b n, a n, b n, c n = a n + b n a n b n n= c n = n a n n b n {c n } c n = x n + iy n (x n, y n R) x n c n, n n y n c n n n n x n, n y n c n = lim k + iy k ) = n n= k=(x x n + i n n c k c k k= Remark. (summable) 7.3. z = x + iy (x, y R) k= e z = e x (cos y + i sin y) = n= n! zn *24 (absolute value) y n 32

( 5.) n n! z n = e z n= zn /n! ( 55 (*). cos π ) n n 56. f(t) z < f(n)z n {c i } i I c i < c i n i I c i c i. i I i I i I 7.4. a n < +, n b n < + n c i < + i I c i = lim i I I = {i = (m, n); m, n }, n a k n k= l= b l = ( lim n k= ) ( a k lim n c i = a m b n ) b l = m l= a m b n. n m,n a m b n 57. z, w m z m m! n w n n! = k (z + w) k k! 2 + 3 4 +... + p q n 33

+ 3 + + 2p 2 4 2q +... + 2(n )p + + + 2np 2(n )q + 2 2nq () + ( 3 + + 2np 2 + 4 + + ) 2nq + 2 + + 2np 2 4 2np 2 ( + 2 + + ) qn γ n = + 2 + + n log n log(2pn) + γ 2pn 2 (log(pn) + γ pn) 2 (log(qn) + γ qn) = log(2p) 2 log p 2 log q + γ 2pn 2 γ pn 2 γ qn n log(2p) 2 log p 2 log q = log(2 p/q) 7.5. 2 + 3 + = log 2, 4 + 3 2 + 5 + 7 4 + = 3 log 2. 2 58 (**). B. Riemann 2 + 3 4 + 8 {c n } n z c n z n n= 34

z (power series) * 25 (domain of convergence) D = {z ; lim n k= c k z k } 8.. 6. e z = + z + 2 z2 + 3! z3 +, cos z = 2 z2 + 4! z4 6! z6 +, sin z = z 3! z3 + 5! z5 7! z7 +. 8.2. {z ; z < }. z = + z + z2 +... 8.3. (i) r > c k r k < = {z ; z r} D. k (ii) z D r < z c k r k <. k D ρ { z < ρ} D { z ρ} ρ c k z k k (radius of convergence) 59 (*). n= n zn S n = + z + + z n z n = S n S n (Abel transformation) *25 série entière entier naturel entier relatif quatre-vingt 35

Bachmann-Landau {c n } n { n } n M > N c n M n, n N c n = O( n ) 6. n = c n = O( n ) c n 6 (**). k c kz k ρ = sup{r > ; c n = O(/r n )} n log n, n a, r n, n! * 26 8.4. a > r > log n n a lim n n a = lim n r n = lim r n n n! =. n log n, n a, r n, n! 62. r n << n! 8.5. n= nz n2 z = ρ < r < nr n2 /2 (n ) nr n2 = nr n2 /2 r n2 /2 n= r n2/2 k= n= nr n2 r k/2 < < r ρ r < ρ ρ =. 63. n c nz n, n c nz n 64. a > n n a zn *26 http://www.math.nagoya-u.ac.jp/~yamagami/teaching/calculus/cal22haru.pdf 36

65. n n!zn 8.6. c n z n, n nc n z n n Proof. ρ, ρ n= n c n r r < ρ = r < ρ ρ ρ c + r n c n r n < r < ρ r( + ϵ) < ρ ϵ > N n ( + ϵ) n (n N) r n N n c n r n n N n= c n r n ( + ϵ) n < r ρ Remark. ϵ-δ 8.7. n 2 n(n )c nz n 2 66 (*). p(n) n p n c nz n, n p(n)c nz n 8.8. f(z) = n c nz n f (z) = nc n z n n= Proof. g(z) = n nc nz n f(z) ρ > g(z) w < ρ f(z) z = w f (w) = g(w) w < R < ρ R z < R (z w) z k w k z w z k + z k 2 w + + w k kr k z w, k 37

f(z) f(w) g(w) z w c n (z n w n ) + (z n 2 w w n ) + + (zw n 2 w n ) n=2 z w n=2 = z w n 2 ( ) c n + 2 + + (n ) R n 2 n(n ) c n R n 2 2 f(z) f(w) lim = g(w) z w z w f(z) z < ρ f(z) g(z) f (z) = g(z) 67. n Z c n φ(θ) = c n e inθ n Z θ R c n = 2π 2π φ(θ)e inθ dθ 6. D f(z) a D D r f(z) = c n (z a) n, n= z a < r ρ r ρ z a < r D {z n } ( z n a < r) lim f(z n) n ρ = r 8.9. lim log( + ( + /n)) = n log( + z) = z 2 z2 + 3 z3 4 z4 + 38

68. /( + z + z 2 ) z = 8. (Ratio Test). f(z) = n c nz n f(z) lim n c n c n+ Proof. ρ = lim n c n / c n+ ϵ > k ρ ϵ c k c k+ ρ + ϵ k = N, k = N +,..., k = n (ρ ϵ) n N c N c n (ρ + ϵ)n N c N (ρ + ϵ) n N c c N n (ρ ϵ) n N, n > N n c nz n z < ρ ϵ z > ρ + ϵ ϵ ρ 8.. α ( α ) ( n α n+ ) = n + (n ) α n ( + z) α = n= ( ) α z n n Taylor Newton (, ) + αx + α(α ) x 2 + 2 α(α )(α 2) x 3 +... 3! f(x) f(x) ( + x)f (x) = αf(x) ( + x) α ( ) f(x) ( + x) α = f(x) = ( + x) α ( 4) x = = 39

69. e x, sin x, cos x, log( + x) cos x, sin x cos x + i sin x Remark. lim sup c n /n n (auchy-hadamard ) 9 9.. D f(z) 3 (i) f f (z) D (ii) f D D f(z)dz =. (iii) f c D r > {f n } n f(z) = f n (z c) n for z c < r n 9.2. (iii) (analytic function) Remark. f n = f (n) (c)/n! 9.3. f(z) z = c (i) g(z) z = c f(z)g(z) z = c (ii) g(w) w = f(c) g(f(z)) z = c g(z) = /z f(c) /f(z) z = c 4

(ring) (formal power series) f(z) = m f mz m g(z) = n g nz n f(z)g(z) = (f g n + f g n + + f n g )z n n 7. 7. f(z), g(z) r, s f(z)g(z) min{r, s} f(z) = n f nz n f(z)g(z) = g(z) f f(z)g(z) = g(z) = f(z) 72. /f(z) 9.4 (). z e z = z n B n n!, n= = ( + 2 z + 3! z2 +... )(B + B z + 2 B 2z 2 +... ) B =, B = 2, B 2 = 6, B 4 = 3, B 6 = 42, B 8 = 3, B = 5 66. 73. z e z + z 2 = z e z/2 + e z/2 2 e z/2 e z/2 B 2k+ = (k ) 74. z 4 e z + z,, tan z. + z + z2 (composition of power series) f(z) = n f nz n {fl k} k,l (f(z)) k = fl k z l, fl k = l f l... f lk l + +l k =l w = f(z) g(w) = k g kw k ( ) g(f(z)) = g k fl k z l (2) l= k= 4

k f = fl k = (k > l) 75. e log(+z) = + z f(z) ρ f, g(w) ρ g f < ρ g f(z) f r = n f n r n r r < ρ f R = sup{r > ; f r ρ g } z < R f(z) n f n z n < ρ g (2) g k fl k z l < k,l 76. f(z) = z + z 2 g(z) = log( + z) log( + z + z 2 ) 9.5. + z + z 2 = (z + z2 ) + (z + z 2 ) 2 (z + z 2 ) 3 + = z + z 3 z 4 +. 77. ω = ( + 3i)/2 /( + z + z 2 ) f(z) z a z = a (annulus) {r < z a < R} r < z a < R auchy f(z) = 2πi ζ a =R f(ζ) ζ z dζ 2πi ζ a =r f(ζ) ζ z dζ. ζ a = R ζ a = r ζ z = n ζ z = n (z a) n (ζ a) n+, (ζ a) n (z a) n+ f(z) = n Z c n (z a) n, c n = 2πi c n = 2πi ζ a =R ζ a =r f(ζ) dζ, n (ζ a) n+ f(ζ)(ζ a) n, n 42

f(z) z = a (Laurent expansion) a f(z) (isolated singularity) (pole) (z a) c f(z) z = a (residue) Res a (f) a f Res a (f) = Res a (f) = 78... (i) n a (ii) z 2 + ez z = a (z a) n e a k (z a)k n k! z = ±i 2 z = ±i i 2 z i + 4 + i (z i) +..., 8 i 2 z + i + 4 i (z + i) +... 8 (iii) 79. e /z = n n! z n. ez sin z.2 (). f(z) n a,..., a n f(z)dz = 2πi Res aj (f) j=.3. 2π dθ ( < a < ) + 2a sin θ + a2 43

Proof. t = tan θ 2t, sin θ = 2 + t 2, dθ = + t 2 dt 2π z = e iθ, sin θ = z z, dθ = 2i iz dz + 2a sin θ + a 2 dθ = z = (az + i)(z + ia) dz z = ia z = ia w = z + ia (az + i)(z + ia) = w(a(w ia) + i) = w i ia 2 + aw = ( ) w i ia 2 +... z = Res z= ia = i a 2. (az + i)(z + ia) dz = 2πiRes z= ia = 2π a 2..4. x 4 + dx Proof. x 4 + : [ R, R] 2 z 4 + = (z ζ)(z ζ 3 )(z ζ 5 )(z ζ 7 ) (ζ = e πi/4 ) z 4 + dz = 2πiRes z=ζ + 2πiRes z=ζ 3. Res z=ζ = Res z=ζ 3 = (ζ ζ 3 )(ζ ζ 5 )(ζ ζ 7 ) = 4ζ 3 = ζ 4, (ζ 3 ζ)(ζ 3 ζ 5 )(ζ 3 ζ 7 ) = 4ζ πi 2 (ζ ζ) = π 2. 2 z 4 + dz 44

z = Re iθ z 4 + z 4 = R 4 z 4 + dz 2 R 4 = π R R 4 as R + 2 R lim R R x 4 + dx = π 2 8. (i) (ii) + 2π x 2 dx =. + x + 3 x n + dx.5. n 2 + a 2 = π e πa + e πa 2a e πa e πa 2a 2. n= Proof. π cot πz z = n (n Z) f(z) [ N, N] N 2i f(n) = f(z) cot πzdz. n= N f(z) lim z zf(z) = R: x + y + x y = N + 2 (N ) N R cot πz 2 f(n) = π (f(z) f(z) cot πz ) n= n= n 2 + a 2 = 2 n= n 2 + a 2 2a 2 f(z) = /(z 2 + a 2 ) z = ±(N + 2 ) + iy cot(πz) = eπy e πy e πy, + e πy z = x ± ir (R > ) cot(πz) eπr + e πr e πr e πr. 45

.6. e itx x 2 + a 2 dx = π a e t a, t R, a > f(z) (i) f R (ii) f(z) = O(/ z ) (z ) t > + e itx f(x)dx = 2πi Iz> Res z (e itz f(z)) f(z) = f n (z a) n f f(a) f(z) a f f(z) = f k (z a) k, f l, l k=l l a (multiplicity) r > f(z) z a = r f(z) z a l ( f l f l+ z a... ) = r l ( f l f l+ r f l+2 r 2... ) > z a =r f (z) f(z) dz = 2πiRes a(f /f) = 2πl D D f(z) f(z) (z D) f(z) D D D D f (z) dz = 2πi {z D; f(z) = } f(z).7 ( ). f(z) f (a) w = f(a) g(w) g(f(z)) = z, f(g(w)) = w g f Proof. f(z) z = a f (a) f(z) f(a) (< z a r) r > D = {z ; z a < r} D f (z) dz = 2πi f(z) f(a) 46

f(z) w z D, w δ > z D, w f(a) < δ = f(z) w w D f (z) f(z) w dz w f(a) < δ 2πi f(z) = w z D g(w) ( w f(a) < δ) g(w) = zf (z) 2πi D f(z) w dz g(w) w w f(a) < δ { f(z) w dw = 2πi if surrounds f( D), otherwise dw dz zf (z) D f(z) w = 2πi g(w) D zf (z) dz = f(g(w)) = w g (f(a)) g(w) g(h(z)) = z, h(a) = f(a) z = a h(z) z a w = h(z) f(a) g(f(z)) = z f(z) = f(g(h(z))) = f(g(w)) = w = h(z) f(z) = n f nz n f g(z) = n g nz n z = f(g(z)) = f n g m z m n m n.8. arctan z = n= ( ) n 2n + z2n+, z < tan z 8. f(z) = 2z + z 2 g(w) + w 47

f(z) ( {a n } n a f f(z) z = a f(z) = c + c (z a) + c 2 (z a) 2 +... f(a f {a n } c = f(a) = lim n f(a n )c = lim n) c n a n a f(a n ) c c (a n a) c m (a n a) m c m = lim n (a n a) m D, D 2 f (z), f 2 (z) a D D 2 f f 2 D D 2 D D 2 f(z) f(z) = { f (z) if z D, f 2 (z) if z D 2 (analytic continuation).. n z n /( z) \ {} 82. f(z) = n 2 n(n ) zn 83. f(x) (auchy transform) F (z) = f(t) t z dt (i) F (z) z \ R (ii) Stieltjes 2πif(x) = lim y (F (x + iy) F (x iy)) (iii) F (z) z = t x = t f(x) = 48

2 log x x = log x = (x ) 2 (x )2 + 3 (x )3... z < z log z = (z ) 2 (z )2 + 3 (z )3... < x < 2 {x R; x } D Logz = z D D z D ζ D z 2.. D f(z) D f(z) z ζ dζ f(z)dz = Proof. D c z D g(z) g(z) = z c f(ζ)dζ g (z) = f(z) g(z) f(z) = g (z) z 2.2. Logz z z < z = re iθ (r >, π < θ < π) Logz = log r + iθ Proof. < x < 2 49

Logz log z = : z z z 2πiZ ζ dζ (multivalued function) 2.3. z = {ζ ; ζ =, Iζ } log z z = 2.4. r > log z = πi + z + t(z + ) dt πi (z + ) 2 (z + )2 3 (z + )3... r r r r x + i dx x + i dx = z + i dz i r 2 + r r 2θ tan θ = r dz = 2iθ = 2i arctan r z + i r r dx = 2i arctan r x + i ζ \ (, ] ζ + + ( ) m ζ m = + ( )m ζ m+ z 2 z2 + + ( ) m m + zm+ = 5 z + ζ + ( ) m ζ m+ dζ + ζ

m = 2n, z = i ( i ) 3 + + ( )n + ( 2n + 2 2 + + ) i + ζ 2n+ ( )n+ = dζ. n + ζ it ( t ) n i + ζ dζ = log( + i) = 2 log 2 + π 4 i 2.5. + x a 2πi F (x)dx = e 2πia Res cj (z a F (z)) j < a <, F (z) [, ) c,..., c n F (z) = O(/ z 2 )(z = ), zf (z) = O()(z = ) 2 Proof. r R (, ) ϵ > D D D z a = e a log z ( t) a = e πai f(z) = z a F (z) f(z)dz = 2πi Res cj (f) R f(ρe iϵ )e iϵ dρ + 2π ϵ D j= f(re iθ )ire iθ dθ r ϵ r ϵ R f(ρe i(2π ϵ) )e i(2π ϵ) dρ 2π ϵ f(re iθ )ire iθ dθ F (z) = O(/ z 2 ) (z ), zf (z) = O() (z ) r +, R + + f(ρe iϵ )e iϵ dρ + f(ρe i(2π ϵ) )e i(2π ϵ) dρ f(ρe iβ ) = ρ a e iaβ F (ρe iβ ) ϵ + ( e 2πia ) + ρ a F (ρ)dρ 5

2.6. F (z) = z(z+) z = (n =, c = z a /(z + ) z = e (a ) log z z= = e πi(a ) = e πia + x a e πia dx = 2πi x + e 2πia = π sin πa. (Riemann surface) z w w = z {z z ; z }, {w w ; w } z w = = { } 2.7. \ {} /z n \ {} w - w n 2.8. log z meromorphic function 2.9. 84. 2.. 85. 2.. {z ; z < } 52

3 f(z) = 2πi f(ζ) ζ z dζ = {ζ ; ζ z = r} ζ = z + re iθ ( θ 2π) f(z) = 2π f(z) 2π 2π 2π f(z + re iθ )dθ f(z + re iθ ) dθ f(z) f(ζ) ( ζ z r) f(z) 2π 2π f(z + re iθ ) = f(z), f(z + re iθ ) dθ f(z) θ R f(z) {f(z + r iθ ); θ < 2π} f(z) = f(z + re iθ ) ( θ 2π) r > f(ζ) f(z) f(z) 3. (maximum modulus principle). 3.2. Proof. f(z) /f(z) lim z f(z) = 3.3. { z < r; Rz > } lim f(z) = z iy for y ( r, r), Proof. h(z) = { f(z) if Rz >, otherwise 53

h(z) h(z)dz = h(z) 3.4 ( (three line theorem)). D = {z ; Rz f(z) D M x = sup{ f(x + iy) ; y R}, x M x M x M x, x Proof. M = M = f(z) M, M F (z) = f(z)m z M z F (z) = f(z) M Rz M Rz F (iy) = f(iy) M, F ( + iy) = f( + iy) M lim z F (z) = F (z) f(z) M Rz M Rz n F n (z) = F (z)e (z2 )/n F n (iy) = F (iy) e (y2 +)/n F (iy), F n ( + iy) = F ( + iy) e y2 /n F ( + iy) F n (z) = F (z) e (y2 + x 2 )/n as y with x F n (z) n F (z) = lim n F n(z) 3.5. n a,..., a n, b,..., b n, c,..., c n x j= a x j b x j c j b j c j j= x a j c i j= x 54

b j = c j = (j =,..., n) a x j n x j= j= a j x Proof. z f(z) = j= c j a z j b z j z = x + iy f(z) j= c j a x+iy j b x iy j = f(z) Rz j= ( ) x aj b j c j = f(x) b j f(x + iy) f() x f() x, x, y R y = Remark. x a x j, x a j j= j= a = (a,..., a n ) a + + a n = a x j j= a j a x j a j 3.6 (Hölder s inequality). p, q p + q = n z,..., z n, w,..., w n z j w j z j p j= j= /p w j q Proof. 6.5 a j = z j p, b j = w j q x = /p a j, b j 86. j= /q 55

A Goursat Goursat A. Pringsheim * 27 D I = f(z) dz k f(z) dz I k 4 k c g(z) = f(z) f(c) z c f (c) g(z) D L f(z) dz = g(z)(z c) dz k max{ g(z) ; z k} L L k 2 k+ 2 k I L2 2 max{ g(z) ; z k}. k L 2 g(c) /2 = I B f(z) /f(z) F (z) = F n z n f(z) (majorant) f n F n (n ) f(z) ρ > f n = O(R n ) ( < R < ρ) f n M/R n (n ) M > F (z) = f + n Mzn /R n f(z) majorant g(z) g(z) majorant G(z) *27 E. Hille, Analytic Function Theory, volume I, Ginn and ompany, 959, 7.2 56

g(z) majorant f(z) f(z)/f f = g n f g n + + f n g + f n g(z) majorant G(z) = n G nz n G =, G n = k= k= M R k G n k, n 87. g n G n n M R k g n k, g = f = G(z) G(z) = = l l= k= k= M R k zk M R k G l kz l = G j z j = j= k= l=k Mz R z G(z) G(z) G(z) = M R k G l kz l = R z R (M + )z k= j= M R K G jz j+k G(z) G(z) z < R/(M + ) ( z < R G(z) g(z) z < R/(M + ) 88. R/(M + ) 6/5 (R = 3, M = 3/2) 2π f(z) f g(z) h l = k g k f k l h(z) = l h lz l R > f f n M/R n (n ) M > F (z) = f + n Mzn /R n f(z) z < R f(z) f n z n f + n= n= M z n R n = f + M z R z 57

g(z) ρ fl k z l l l F k l z l = ( z < R + M ) (3) ρ f f + M R n z n n k = ( f + M z ) k R z g k fl k z l ( g k f + M z ) k < R z k,l k l h l = k g k f k l h l z l ( g k f + M z ) k < R z l k (3) g k fl k z l k,l h l z l = g k f(z) k l= k= 89. φ(z) = n φ nz n φ = g(f(z)), g(z) = φ(z) + z, f(z) = φ(z) φ φ 9. f(z) = z + z 2 g(z) = log( + z) log( + z + z 2 ) g(z) g(z) k = l k g k l z l, g k l = f(z) = z + l + +l k =l 58 f k z k k=2 g l g lk, g l = g l, g l l = g l =

f(g(z)) l g l = f k gl k (l 2). k=2 g k l (k 2) g 2,..., g l G k l (g 2,..., g l ) ( ) l 2 G l l =, G l l = (l )g 2, G l 2 l = (l 2)g 3 + g 2 g 3, 2 G 2 l = 2g l + g 2 g l 2 + + g l 2 g 2. f k F k (k 2) G l (l 2) l G l = F k G k l (G 2,..., G l ) k=2 g l G l F k = M/R k (k 2) G(z) z = G l z l = l=2 = = = l l=2 k=2 k=2 l=k k=2 M R k Gk l (G 2,..., G l )z l M R k Gk l (G 2,..., G l )z l M z R k + k G l z l l 2 k=2 = M R M R k G(z)k G(z) 2 R G(z). G(z) = z + l 2 G lz l (M + R)G(z) 2 R(z + R)G(z) + R 2 z = G(z) = R(z + R) R2 2( + 2M/R)(z/R) + (z/r) 2 2(M + R) G(z) D 2(R + 2M)z z 2 < R 2 z R + 2M D z < R 2 R + 2M G(z) R 2 /(R + 2M) 59

9. G(z) 92. tan z = sin z cos z R2 /(R + 2M) (curve) (path= ) R n (motion) [a, b] (a < b) R n ϕ R n ( ) ϕ [ϕ] [ϕ] ϕ[a, b] = [ϕ] = [ϕ] ϕ (simple) ϕ(s) = ϕ(t) (a s < t b) s = a, t = b ϕ(a) = ϕ(b) t = h(τ) (α τ β) * 28 ϕ (total variation) l sup{ ϕ(t j ) ϕ(t j ) ; a = t < t < < t l = b} j= x = (x,..., x n ) R n x = (x ) 2 + + (x n ) 2 x ϕ = [ϕ] = [ϕ].. ϕ = b a dϕ dt dt..2. f(x) = x 2 sin x ϕ(t) = (t, f(t)) (t R) *28 h h (τ) > (α τ β) 6

x r > ϕ(t) = (r + e t )(cos(at + b), sin(at + b)) r = s = e t/2 φ(s) = s 2 (cos( 2a log s + b), sin( 2a log s + b)), s > s = s (cos b, sin b) b b ( b < 2π) π * 29 N {,,..., N } N a = (a n ), b = (b n ) a n = b n n N [, ] [a] N = N k a k k= 3 {(2a n ); (a n ) 2 } (antor set) t antor [, ] (antor function) c = [c c 2... ] 3 f(c) c n c j for j n and c n+ = [ c f(c) = 2 c 2 2... c n 2 ]2 = j= c j 2 j+ + 2 n+ c c j for j [ c f(c) = 2 c 2 2... ] 2 = j= c j 2 j+ (i) f (ii) [, ] \ f (x) = (the devil s staircase) 2 *29 URL 6

2 2 = 2 2 [, ] [.] [, ] \ [, ] [ϕ] [, ] [, ] (Peano curve) * 3 D * 3 D D p D p z p < r D D r >, z D, z \ D, z p < r, z p < r. D D (boundary) D p D (i) p p x D (ii) p p p p p p D x D D D D *3 http://en.wikipedia.org/wiki/space-filling_curve *3 V.J. Katz, The History of Stokes Theorem, Mathematics Magazine, 52(979) Georg Green Michael Ostrogradsky Stokes Kelvin 62

F (x, y)i + G(x, y)j D (F (x, y)dx + G(x, y)dy) = D ( ) G F (x, y) (x, y) dxdy. x y F i + Gj D D (x(t), y(t)) (a t b) D (F (x, y)dx + G(x, y)dy) = b a ( F (x(t), y(t)) dx ) + G(x(t), y(t))dy dt dt dt D D D.. D D D = D D Proof. D * 32 ϕ : [a, b] R 2 ϕ[a, b] R,, R n R + + R n M = max{ ϕ (t) ; a t b} : a = t < < t n = b ϕ(t) ϕ(t j ) M t t j, t [t j, t j ] R j ϕ(t j )±M(t j t j )(, ) ϕ[t j, t j ] R j R j 2 = 4M 2 (t j t j ) 2 4M 2 (t j t j ) = 4M 2 (b a) j= j= j= = max{t j t j ; j n} b F (x, ψ(x)) dx b F (x, φ(x)) dx = b ψ(x) F y (x, y) dydx = a a a φ(x) a x b,φ(x) y ψ(x) F y (x, y) dxdy. D x = y 2 sin(a/y) R = [, ] [, ] φ : (s, t) z(s, t) 2 z s t, 2 z t s * 33 s s z(s, t) ( t ) *32 *33 2 z s t = 2 z t s 63

d f(z) dz = d ds s ds = = = f(z(s, t)) z t dt = f z z z s t dt + f z z z s t dt + f (z, z) z (s, t) f z z z s ( f(z(s, t)) z ) s t t dt + f z z z s t dt + dt + f(z(s, )) z s s f(z) dz = φ( R) dt f(z(s, t)) 2 z s t dt ( f(z(s, t)) z ) t s (s, ) f(z(s, )) z (s, ) s R f (z, z) z (s, t) dsdt. dt f(z(s, t)) z t s dt φ R D φ R D f f(z) dz = dzdz = 2i z D D D D f z dxdy. ( u (udx vdy) = D y + v ) dxdy x D φ : z(t) ( t ) c φ : (s, t) sz(t) + ( s)c (z, z) (s, t) = s(z(t) c)dz s(z(t) c)dz dt dt f(z) dz L f(z) dz = R f (z, z) z (s, t) dsdt f z z(t) c dz φ(r) dt dt. L [z(), c], [c, z()] f z = max{ f (φ(s, t)) φ(r) z ; s, t } D D z(t) ( t ) D z = z(), z = z(t ),..., z n = z(t n ) c j ( j n) z j c j ir, z j c j R z n = z 64

z j c j z j z j, z j c j z j z j D = + + n j L j L = L + + L n f(z) dz f(z) dz n f(z) dz f(z) dz L j= j L j f tj z z(t) c j dz j= t j dt dt f z D. = max{ z(t) c j ; j n, t j t t j } z(t) L D L lim D L f z dxdy = f D z dxdy L = D L * 34 Green 93. = lim L E Riemann-Stieltjes f(t) (a t b) δ > H δ = max{ f(s) f(t) ; s t δ} lim δ H δ = * 35 (uniform continuity) *34 Lang IV, 3 *35 http://sss.sci.ibaraki.ac.jp/teaching/set/real.pdf 65

z(t) (a t b) : a = t < < t n = b z k = z(t k ), = max{t k t k ; k n} tk z k z k z (t k )(t k t k ) = (z (t) z (t k )) dt t k ( f(z k ) z k z k z (t k )(t k t k )) k= lim k= (4) f(z k )(z k z k ) = lim k= tk t k z (t) z (t k ) dt H (t k t k ) (4) f(z k ) H (t k t k ) (b a) f H k= f(z(t k ))z (t k )(t k t k ) = b a f(z(t))z (t) dt. z (t k ) (t k t k ) H (t k t k ) z k z k z (t k ) (t k t k ) + H (t k t k ) k z (t k ) (t k t k ) (b a)h k= lim k= z k z k k= z k z k = lim k= z (t k ) (t k t k ) + (b a)h k= z (t k ) (t k t k ) = b a z (t) dt 66