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

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(⇐=) i = 1, . . . , m fi x0 ∀ ε > 0 ∃ δi = δi(ε) > 0 : |x − x0| < δ =⇒ |fi(x) − fi(x0)| < ε √ m . δ = δ(ε) = min{δ1(ε), . . . , δm(ε)} ; |x − x0| ≤ δ |f(x) − f(x0| ≤ √ m max i=1,...,m {|fi(x) − fi(x0)|} ≤ ≤ √ m ε √ m = ε . | · | Rn x, y ∈ B1(0) f(x) − f(y) = 1 1 − 2 − |x| 2 − |y| = 1 1 − 2 − |x| 2 − |y| , ≤ sin n i=1 xi − sin n i=1 yi . 2 − |y| − 2 + |x| (2 − |x|)(2 − |y|)) = | |x| − |y| | (2 − |x|)(2 − |y|) ≤ |x − y| ≤ |x − y| . (2 − |x|)(2 − |y|) m N xm = ym = | |x| − |y| | ≤ |x − y| x, y ∈ R n 2 − |x| ≥ 1 x ∈ B1(0) 1 − 1 , 0, . . . , 0 2m 1 − 1 , 0, . . . , 0 m
1 2 − |xm| − 1 2 − |ym| = = = = = = 1 2 − 1 + 1 2m 1 1 + 1 2m 2m (1 + 2m) − − − 1 1 + 1 m 1 2 − 1 + 1 = m (1 + m) = m (2m + 1)(m + 1) = 2m2 1 (2m + 1)(m + 1) 2m = m = 2m 2 (2m + 1)(m + 1) |xm − ym| . Lm xm ym Lm := L ≥ 1 sin n i=1 xi − sin n i=1 yi 2m2 m→+∞ −→ 1 (2m + 1)(m + 1) ≤ sin n n xi − sin yi i=1 i=1 ≤ n n xi − yi i=1 i=1 ≤ ≤ x1 n n n n xi − x1 yi + x1 yi − y1 yi i=2 i=2 i=2 i=2 ≤ n n ≤ |x1| xi − yi i=2 i=2 + n yi i=2 |x1 − y1| ≤ n n ≤ |x1 − y1| + xi − yi ≤ ≤ . . . ≤ | |x| − |y| | ≤ |x − y| x, y ∈ R n | sin t − sin s| ≤ |t − s| t, s ∈ R x1 ≤ √ n|x| x ∈ R n i=2 i=2 ≤ |x1 − y1| + . . . + |xn − yn| = = x − y1 ≤ ≤ √ n |x − y| .
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 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 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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