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

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Λ = {λ ∈ R n : λ = λ(t), a ≤ t ≤ b } Λ = Γ ∪ Φ P = (rP , θP ) , Q = (rQ, θQ) ∈ A γ : t ↦→ (x = (trQ+(1−t)rP ) cos(tθQ+(1−t)θP ) , y = (trQ+(1−t)rP ) sin(tθQ+(1−t)θp)). · Lip : (E, R m ) −→ R • fLip ≥ 0 f ∈ (E, R m ) fLip = 0 ⇐⇒ sup |f(x)| = 0 ⇐⇒ x∈E ⇐⇒ f(x) = 0 ∀ x ∈ E. • a ∈ R afLip = |af(x) − af(y)| sup x,y∈E |x − y| x=y |f(x) − f(y)| = |a| sup x,y∈E |x − y| x=y = |a| fLip . + sup |af(x)| = x∈E + |a| sup |f(x)| = x∈E • f g (E, R m ) |(f(x) + g(x)) − (f(y) + g(y))| |x − y| ≤ + |g(x) − g(y)| |f(x) − f(y)| + |x − y| . |x − y|
f + gLip = |(f(x) + g(x)) − (f(y) + g(y)| sup + x,y∈E |x − y| x=y + sup |f(x) + g(x)| ≤ x∈E ≤ sup x,y∈E x=y |f(x) − f(y)| |x − y| |g(x) − g(y)| + sup x,y∈E |x − y| x=y = fLip + gLip . + sup |f(x)| + x∈E + sup |g(x)| = x∈E ((E, Rm ), fLip) {fk}k ε > 0 N0 = N0(ε) > 0 k, h > N0 fk − fhLip = |(fk(x) − fh(x)) − (fk(y) − fh(y))| sup + x,y∈E |x − y| x=y + sup x∈E |fk(x) − fh(x)| ≤ ε . {fk}k (C(E, R m ), · ∞,E) f ∈ C(E, R m ) N1 = N1(ε) > 0 k > N1 fk − f∞,E ≤ ε . x, y ∈ E x = y k, h > N0 |(fk(x) − fh(x)) − (fk(y) − fh(y))| |x − y| h → +∞ |(fk(x) − f(x)) − (fk(y) − f(y))| |x − y| ≤ ε ≤ ε . · ∞,E f ∈ C(E, R m ) f∞,E ≡ sup |f(x)| . x∈E
Page 2 and 3: Rn Rn Rn
Page 4 and 5: Rn Rn Rn
Page 6 and 7: ℓ 1 ℓ ∞ x = {xn}n ∈ R N
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 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 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 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 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 =
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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