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

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1 = |z − a| 2 1 ρ 2 − |a| 2 2πia • |a| < ρ |ρ2−|a| 2 | 2πiρ • |a| > ρ 2 a|ρ2−|a| 2 | a ρ2 + z − a ρ2 − az • n ≤ 0 |z|=1 • n > 0 n • n > 0 n ≡ 1 4 • n > 0 n ≡ 3 4 • n ≥ 0 m ≥ 0 • n ≥ 0m < 0 • n < 0m ≥ 0 • n < 0 m < 0 |z|=2 z n (1−z) m dz = 2πi sin z zn dz = 0 sin z |z|=1 zn dz = 0 sin z |z|=1 zn dz = 2πi (n−1)! sin z |z|=1 zn dz = − 2πi (n−1)! |z|=2 zn (1 − z) m dz = 0 |z|=2 zn (1−z) m dz = 2πi(−1) m |z|=2 zn (1−z) m dz = 2πi(−1) n+1 |m| + |n| − 2 |n| − 1 n |m| − 1 m |n| − 1 |m| + |n| − 2 + |m| − 1 z |z| ≤ ρ δ = R − ρ > 0 Bδ(z) ⊂ BR(0) f f f n (γ) = n! 2πi ∂Bδ(z) f(ζ) (ζ − z) n+1 x = 1 2
|γ − z| < δ |f (n) (z)| ≤ n!M 2π = ∂Bδ(z) 1 δ n+1 n!M 2δ n+1 π |∂Bδ(z)| = n!Mδ −n = n!M(R − ρ) −n sup |f Bρ(0) (n) (z)| ≤ n!M(R − ρ) −n . δ Bδ(z) ⊂ Ω Ω f n!n n ≤ |f (n) (z)| sup |f(ζ)|n!δ Bδ(z) −n , sup |f(ζ)| ≥ n Bδ(z) n δ n n→+∞ −→ +∞ |f| Bδ(z) f z ≥ 0 z ∈ C g(z) = e −f(z) . g C |g(z)| = e −f(z) ≤ 1 f • f(z) = sin 1 1−z • f g |z| ≤ 1 g ≡ 0 {|z| ≤ 1} g f
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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
Page 10 and 11:
∂ 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 ,
Page 22 and 23:
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
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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
Page 42 and 43:
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
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|f(x) − f(y)| ≤ 2 1 1
Page 54 and 55:
L 4m3 m→+∞ −→ +∞ 3
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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
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s(x, y) ≡ x2s(x) c(x, y) ≡ y2s
Page 64 and 65:
f ∂xf(x, y) = 2x − y 2 = 0
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2 3 4 4 i=1 xi i x2 4 i=1 xii x3 4
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dist lim dist (x(t), y(t)), y = t
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• y(t, β) = e βt .
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α > 0 • [0, 1) • [0, a
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N ≥ N0 |(1+x) α − N−1 k=0 |
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• [0, +∞) • [0, +∞)
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k 2k + 1 2n = (−1)k (2k + 1)! n=
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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 ′
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 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 =
Page 150 and 151: f(0) = 0 g(z) = f(z) − f(0)
Page 152 and 153: 1 n −n ∞ sin πz = π
Page 154 and 155: g = f ◦ S−1 f(z0
Page 156 and 157: un [a, b] un (a, b)
Page 158 and 159: f(x) = sin 2 (xy) x 2 +y 2 (x, y)
Page 160 and 161: 2 y(x) (0, 1 2 ) {(x,
Page 162 and 163: Ac 1 + 3x2 dxdy + y2 Ac ≡ {(x, y
Page 164 and 165: ϕ : R ↦→ R 1 1 ϕ f(x)d
Page 166 and 167: P x P A, B, C, D ∈
Page 168 and 169: f : R 3 → R µf (x, r) = 1 4
Page 170 and 171: D = {(x, y, z) : x 2 + y 2 + z 2
Page 172 and 173: az + b T z = . T (R∪∞)
Page 174 and 175: cos z f(z) = z f(z) = e 1 z f(z)
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