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

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k 2k + 1 2n = (−1)k (2k + 1)! n=0 2k + 1 2k + 1 = 2n 2k + 1 − 2n = (−1)k 1 (2k + 1)! 2 2k+1 n=0 2k + 1 n = 1 (2k + 1)! = (−1) k 2 22k+1 e iπ = cos π + i sin π = cos 2 π π π π π + i sin 2π = cos2 − sin2 + 2i cos sin 2 2 2 2 2 2 = π cos π 2 = 0 2 π π = − sin = cos2 − 1 = −1 2 2 φɛ R f ∈ L 1 (R) C ∞ f ∈ C ∞ 0 +∞ φɛdx = −∞ ɛ γ 0 γ(x) ≡ φɛdx ≤ φɛ∞ɛ < ∞ . x −∞ φɛdx +∞ . φɛdx −∞ φɛ C ∞ φɛ ∀k ≥ 1 γ (k) (x) = φ(k−1) ɛ (x) +∞ −∞ φɛdx 2k+1 2k + 1 n=0 = (1 + 1) n 2k+1 = 22k+1 φɛ [0, ɛ]
γ φɛ ≥ 0 x −∞ x ≤ 0 γ(x) = φɛdx +∞ −∞ φɛdx = (−∞, x) ∩ φɛ = ∅ 0 +∞ φɛdx −∞ = 0 x −∞ x ≥ ɛ γ(x) = φɛdx +∞ −∞ +∞ = φɛdx −∞ φɛdx +∞ = 1 φɛdx −∞ (−∞, x] ⊇ φɛ x −∞ φɛdx = +∞ • x = 0 γ(0) = 0 ∀ k ≥ 1 γ (k) (0) = 0 φɛ C∞ −∞ φɛdx [0, ɛ] x = 0 • x = ɛ γ(ɛ) = 1 ∀ k ≥ 1 γ (k) (0) = 0 φɛ C ∞ [0, ɛ] x = ɛ ≡ 0 ≡ 1 (⇒) : M(t) ∀ i, j Mij(t) ∀ ɛ > 0 , ∃ δij |Mij(t) − Mij(t0)| < ɛ , ∀ |t − t0| < δij δ = mini,j δij (⇐) : ∀ i, j |Mij(t) − Mij(t0)| < M(t) − M(t0) f ∈ C ∞ ({y0}, R 2 ) f ′ (y0) = 1 + 2y1 cos y2 −y 2 1 sin y2 2y1 1 |y0=(0,0) = 1 0 0 1 det f ′ (y0) = 1 = 0 ⇒ ∃! f C ∞ ({f(y0)}, R 2 )
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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
Page 24 and 25:
R 3 Γ ≡ {(u(t), v(t)) : t ∈ (
Page 26 and 27:
F F (x, y, z) = (x, y, z) ;
Page 28 and 29: Im {(1 + i) n + (1 − i) n } Re
Page 30 and 31: x dz σ 1 + i σ x dz
Page 32 and 33: f Imz ≥ 0 ∃ lim f(z) <
Page 34 and 35: ∞ n=2 1 − 1 n2 = 1 2 . a
Page 36 and 37: Ω ∂
Page 38 and 39: a) b) c) d) e) f) g) h) i) 2π 0
Page 40 and 41: Λ = {λ ∈ R n : λ = λ(t), a
Page 42 and 43: x, y ∈ E k > N0 |(fk(x) −
Page 44 and 45: x ∈ ℓ 1 k > N0 x1 ≤ x
Page 46 and 47: X
Page 48 and 49: 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 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 100 and 101: Kn ⊂ Kn+1 n > 1 n Fp
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 ′
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• inf | sin z| = 0 4√ i( i = {
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2z + 3 z + 1 1 = 2 = ((z − 1) + 5
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1 = |z − a| 2 1 ρ 2 − |a|
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f R = {aω1 + bω2 : 0
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•
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c > 1 c = 1 Rez = 1 c < 1
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Resπ = −1 . Reskπ = (−1) k
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Res √ i 2 = i√2 Res √ i 3 =
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sin 2 z (0, π 2 ( π 2 ) ,
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R → ∞ z 2 − z + 2 CR
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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
Page 172 and 173:
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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