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

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ℓ 1 ℓ ∞ x = {xn}n ∈ R N x1 ≡ n∈N R N |xn| e x∞ ≡ sup |xn| . n∈N ℓ 1 ≡ {x ∈ R N : x1 < ∞} ℓ ∞ ≡ {x ∈ R N : x∞ < ∞} . 1 (ℓ 1 , · 1) (ℓ ∞ , · ∞) 2 ℓ 1 ⊂ ℓ ∞ (ℓ ∞ , · ∞) (ℓ 1 , · ∞) 3 ∗∗ ℓ 1 · ∞ (ℓ 1 , · 1) Ω := {x ∈ ℓ 1 : x1 ≤ 1} D := {x ∈ ℓ 1 : |xk| ≤ 1 ∀ k, xk = 0 ∀ k > 10} . R n R m . . . f(x) = xi x ɛ > 0 δ |f(x) − f(x0)| < ɛ |x − x0| < δ f = |x| α , α > 0 x ∈ R 4 , x0 = (0, 1, 1, 2) x0 = (0, .., 0) f = sin 1 x1x2 , x ∈ R 3 3 , x0 = (−1, 0, −1)
f = log[cos ( n i=1 xi)] , x ∈ R n , x0 = (0, .., 0) f = +∞ k=0 e−k|x|2 , x ∈ R n , x0 = (1, .., 1) f = tanh |x|1 , x ∈ R n , x0 = (1, .., 1) f = ( |x| 3 2 , tanh |x|1 ) , x ∈ R 4 , x0 = (0, 1, 1, 2) S 2 ≡ { x ∈ R 3 : |x| = 1 } , x ≡ (2, 0, 0) , x0 = (1, 0, 0) , f ≡ x1x2(sin |x − x|) −1 . δ |f(x) − f(x0)| < ɛ x ∈ S 2 |x − x0| < δ f : R 4 −→ R 2 x ↦−→ 1 (f1(x), f2(x)) ≡ , sin(x1x4) . 1 + |x| x0 = (0, 0, 0, 0) f : E ⊂ R n −→ R m i = 1, . . . , m fi : E ⊂ R n −→ R L > 0 |f(x) − f(y)| ≤ L|x − y| ∀ x, y ∈ Ω x ∈ Rn f | · | (i) 1 f(x) = 2 − |x| , sin n xi , Ω = B1(0) ; (ii) f(x) = 1 2 − |x| 1 2 i=1 , Ω = B1(x0), x0 = (2, . . . , 2) oppure x0 = (0, . . . , 0) (per il primo dominio dobbiamo supporre che n = 3, 4, 5, 6 ); (iii) f(x) = e |x|2 x , Ω = Br(0), r > 0 .
Page 2 and 3: Rn Rn Rn
Page 4 and 5: Rn Rn Rn
Page 8 and 9: P ≡ {(x, y) ∈ R 2 y = x 2 (x
Page 10 and 11: ∂ 5 f (1, 1, .., 1) n > 5 ∂x1
Page 12 and 13: z = z(x, y) (1, 1) z 3 −
Page 14 and 15: K ⊂ R2 \ y = 1 2 0 < δ < β
Page 16 and 17: (i) ∞ n=0 n 3 2 n (ii) ∞ n=1 n
Page 18 and 19: v ∈ R n 1 f(x) ≡ x +
Page 20 and 21: 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
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 |
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• [0, +∞) • [0, +∞)
Page 78 and 79:
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 .
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g(x, y) = e x2 +y 2 − x 2 − 2y
Page 86 and 87:
R n R n • q = m
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∂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
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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
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Kn ⊂ Kn+1 n > 1 n Fp
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x ′2 + y ′2 = (ρ ′ (θ) cos
Page 104 and 105:
(2, 0) A f dx dy = = = A A π =
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T R3 ·1 (
Page 108 and 109:
xy z ∈ [0, 1] z C C(z) ≡
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(ρ, ϕɛ, θ) P = (x, y,
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Area (∂E2) = = = = = = ∂E2 π
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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 =
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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 = {
Page 130 and 131:
2z + 3 z + 1 1 = 2 = ((z − 1) + 5
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1 = |z − a| 2 1 ρ 2 − |a|
Page 134 and 135:
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
Page 142 and 143:
Res √ i 2 = i√2 Res √ i 3 =
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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)
Page 152 and 153:
1 n −n ∞ sin πz = π
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g = f ◦ S−1 f(z0
Page 156 and 157:
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,
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
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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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