Esercizi svolti di analisi reale e complessa - Dipartimento di ...

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Kn ⊂ Kn+1 n > 1 n Fp n > 1 Kn z α dx dy dz = = = 1 dz 1 n 1 1 n 1 1 n x 2 +y 2 ≤z 2p z α dx dy = z α Area (B 2 zp(0)) dz = z α πz 2p dz = 1 = π z α+2p dz = = ⎧ ⎨ ⎩ 1 n π [log z] 1 1 π n α+2p+1 z α+2p+1 1 1 n se α + 2p = −1 se α + 2p = −1 π log n se α + 2p = −1 = π 1 α+2p+1 1 − nα+2p+1 se α + 2p = −1 . n → +∞ z Kn α ⎧ ⎨ +∞ se α + 2p = −1 n→∞ dx dy dz −→ +∞ se α + 2p < −1 ⎩ π α+2p+1 se α + 2p > −1 . zα Fp a > −1 − 2p π α+2p+1 R n Γ ⎧ ⎨ Φ(t) = ⎩ x(t) = t y(t) = t 2 z(t) = t 3 t ∈ I = (1, 2). Γ = Φ(I) Φ ∈ C 1 (I) ˙ Φ(t) = (1, 2t, 3t 2 ) = (0, 0, 0) ∀ t ∈ I Γ fdσ1 = Γ 2 1 log t3 √ 1 + 4t2 + 9t4 dt = 3(log 4 − 1). 1 + 4t2 + 9t4
S = Φ(A) A = (a, b) × (0, α) Φ(t, θ) = (u(t) cos θ, u(t) sin θ, v(t)) ∈ C 1 (A) Γ (0, ∞) × R γ(t) = (u(t), v(t)) ∂Φ ∂t A(S) = dσ2 = ∂Φ ∂t ∧ ∂Φ ∂θ dt dθ = S A ∂Φ ∧ ∂θ = u(t) ˙u(t) 2 + ˙v(t) 2 u(t) > 0 b = α u(t) a ˙u(t) 2 + ˙v(t) 2 dt. A(S) = α Γ x dσ1. Γ l(γ) A(S) = α l(γ) 1 x dσ1 = α l(γ) xb l(γ) Γ xb Γ α Γ Γ γ x(θ) = ρ(θ) cos θ = a (1 + cos θ) cos θ θ ∈ (0, 2π) . y(θ) = ρ(θ) sin θ = a (1 + cos θ) sin θ γ 2π lungh (γ) = ds := x γ 0 ′2 + y ′2 dθ . ′ ′ ′ ′ x (θ) = ρ (θ) cos θ − ρ(θ) sin θy (θ) = ρ (θ) sin θ + ρ(θ) cos θ ,
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Rn Rn Rn
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Rn Rn Rn
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ℓ 1 ℓ ∞ x = {xn}n ∈ R N
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P ≡ {(x, y) ∈ R 2 y = x 2 (x
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∂ 5 f (1, 1, .., 1) n > 5 ∂x1
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z = z(x, y) (1, 1) z 3 −
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K ⊂ R2 \ y = 1 2 0 < δ < β
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(i) ∞ n=0 n 3 2 n (ii) ∞ n=1 n
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v ∈ R n 1 f(x) ≡ x +
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X ⊂ R n o X = ∅ X Qx ,
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E = {(x, y) ∈ R 2 : x ≥ 0, (x 2
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R 3 Γ ≡ {(u(t), v(t)) : t ∈ (
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F F (x, y, z) = (x, y, z) ;
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Im {(1 + i) n + (1 − i) n } Re
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x dz σ 1 + i σ x dz
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f Imz ≥ 0 ∃ lim f(z) <
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∞ n=2 1 − 1 n2 = 1 2 . a
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Ω ∂
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a) b) c) d) e) f) g) h) i) 2π 0
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Λ = {λ ∈ R n : λ = λ(t), a
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x, y ∈ E k > N0 |(fk(x) −
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x ∈ ℓ 1 k > N0 x1 ≤ x
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X
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limk→+∞ x (k) = limk→+∞ y (
Page 50 and 51: (⇐=) i = 1, . . . , m fi
Page 52 and 53: |f(x) − f(y)| ≤ 2 1 1
Page 54 and 55: L 4m3 m→+∞ −→ +∞ 3
Page 56 and 57: F ′ (t) = ∂ f ∂ x (g(t), 1
Page 58 and 59: ∂f ∂y = ∂f ∂x = ∂f ∂t =
Page 60 and 61: • M1 = ( 2 √ 3 , 1 √ 3 ) M
Page 62 and 63: s(x, y) ≡ x2s(x) c(x, y) ≡ y2s
Page 64 and 65: f ∂xf(x, y) = 2x − y 2 = 0
Page 66 and 67: 2 3 4 4 i=1 xi i x2 4 i=1 xii x3 4
Page 68 and 69: dist lim dist (x(t), y(t)), y = t
Page 70 and 71: • y(t, β) = e βt .
Page 72 and 73: α > 0 • [0, 1) • [0, a
Page 74 and 75: N ≥ N0 |(1+x) α − N−1 k=0 |
Page 76 and 77: • [0, +∞) • [0, +∞)
Page 78 and 79: k 2k + 1 2n = (−1)k (2k + 1)! n=
Page 80 and 81: I − T f ′ (y)∞,∞ = max {|
Page 82 and 83: g Br(0) r ≡ ρ ρ = 2 In 2 .
Page 84 and 85: g(x, y) = e x2 +y 2 − x 2 − 2y
Page 86 and 87: R n R n • q = m
Page 88 and 89: ∂A ∂A
Page 90 and 91: Q = {0} × R y x = 0 Q0 = R
Page 92 and 93: D xy dx dy = = = = 1 dx √ 1−x
Page 94 and 95: x ′ xi i = 1−|x4| i = 1, 2, 3
Page 96 and 97: Area(E) = = = 1 2 = 1 4 E π 4
Page 98 and 99: (∆) = 2π dx dy dz = dθ ρ d
Page 102 and 103: x ′2 + y ′2 = (ρ ′ (θ) cos
Page 104 and 105: (2, 0) A f dx dy = = = A A π =
Page 106 and 107: T R3 ·1 (
Page 108 and 109: xy z ∈ [0, 1] z C C(z) ≡
Page 110 and 111: (ρ, ϕɛ, θ) P = (x, y,
Page 112 and 113: Area (∂E2) = = = = = = ∂E2 π
Page 114 and 115: Φ + F (∂E1) Ψ 1 u
Page 116 and 117: F Φ + F (∂E) = = = Vol (E) =
Page 118 and 119: 0 ≤ α < 2π 2π ω = γα =
Page 120 and 121: ω 1 R 2 \ {0} f(x, y) ω =
Page 122 and 123: • • • ∞ n=1 1 n 2 = ∞ n=1
Page 124 and 125: X ′′ (x) X(x) = −µ X(0) = X(
Page 126 and 127: {sin nx}n ∆v = 0 ⇐⇒ c ′
Page 128 and 129: • inf | sin z| = 0 4√ i( i = {
Page 130 and 131: 2z + 3 z + 1 1 = 2 = ((z − 1) + 5
Page 132 and 133: 1 = |z − a| 2 1 ρ 2 − |a|
Page 134 and 135: f R = {aω1 + bω2 : 0
Page 136 and 137: •
Page 138 and 139: c > 1 c = 1 Rez = 1 c < 1
Page 140 and 141: Resπ = −1 . Reskπ = (−1) k
Page 142 and 143: Res √ i 2 = i√2 Res √ i 3 =
Page 144 and 145: sin 2 z (0, π 2 ( π 2 ) ,
Page 146 and 147: R → ∞ z 2 − z + 2 CR
Page 148 and 149: π 2 i 2 ∞ = lim f(z) dz =
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f(0) = 0 g(z) = f(z) − f(0)
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1 n −n ∞ sin πz = π
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g = f ◦ S−1 f(z0
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un [a, b] un (a, b)
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f(x) = sin 2 (xy) x 2 +y 2 (x, y)
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2 y(x) (0, 1 2 ) {(x,
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Ac 1 + 3x2 dxdy + y2 Ac ≡ {(x, y
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ϕ : R ↦→ R 1 1 ϕ f(x)d
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P x P A, B, C, D ∈
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f : R 3 → R µf (x, r) = 1 4
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D = {(x, y, z) : x 2 + y 2 + z 2
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az + b T z = . T (R∪∞)
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cos z f(z) = z f(z) = e 1 z f(z)
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