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Trasformata di Laplace
Linearit`a ( a, b 2 IR ) Lax(t) + by(t) = aX(s) + bY (s) s > max{s x , s y }
Traslazione ( a 2 IR ) Lx(t a)u(t a) = e
as X(s) s > sx
Modulazione ( a 2 IR ) L[e
at x(t)](s) = X(s a) s > a + sx
Riscalamento ( a > 0 ) L [x(at)] (s) =
1
a
X
⇣ s
a
⌘
s > as x
Derivazione rispetto a s L [t
n x(t)] (s) = ( 1)
n X
(n) (s) s > s x
Derivazione rispetto a t L[x
0 (t)](s) = sX(s) x(
) s > max{s x , s x 0 }
L[x
00 (t)](s) = s
2 X(s) sx(
) x
0 (
) s > max{s x , s x 0 , s x 00 }
Integrale della trasformata
Z
s
X(r)dr = L
x(t)
t
(s) s > sx
Convoluzione L[(x ⇤ y)(t)] = X(s) Y (s) s > max{s x , s y }
Trasformata dell’integrale L
Z t
0
x(r)dr
(s) =
X(s)
s
s > max{ 0 , s x }
x(t) X(s) = Lx(t) sx
u(t)
1
s
0
e
at (a 2 IR)
1
s a
a
t
n (n 2 IN )
n!
s
n+
0
sin(at) (a 2 IR)
a
s
2
2
0
cos(at) (a 2 IR)
s
s 2
0
sinh(at) (a 2 IR)
a
s
2 a
2
|a|
cosh(at) (a 2 IR)
s
s 2 a 2
|a|
e
at sin(bt) (a, b 2 IR)
b
(s a)
2
2
a
e
at cos(bt) (a, b 2 IR)
s a
(s a)
2
2
a
sin t
t
arctan
✓
1
s
◆
0
Sviluppi in serie di MacLaurin
Funzione Sviluppo di McLaurin Intervallo di convergenza
e
x
n=
x
n
n!
R = + 1
1
1 x
n=
x
n ( 1 , 1)
log(1 x)
n=
x
n
n
( 1 , 1]
arctan x
n=
( 1)
n
2 n + 1
x
2 n+ [ 1 , 1]
sin x
n=
( 1)
n
(2n + 1)!
x
2 n+ R = + 1
cos x
n=
( 1)
n
(2n)!
x
2 n R = + 1
sinh x
n=
x
2 n+
(2n + 1)!
R = + 1
cosh x
n=
x
2 n
(2n)!
R = + 1
(1 + x)
↵
n=
✓
↵
n
◆
x
n R = 1
Serie di Fourier
f (x) ⇡ a 0 +
1 X
k=
ak cos
✓
k
2 ⇡
T
x
◆
✓
k
2 ⇡
T
x
◆
,
a 0 =
1
T
Z T
0
f (x) dx, ak =
2
T
Z T
0
f (x) cos
✓
k
2 ⇡
T
x
◆
dx k 1 , bk =
2
T
Z T
0
f (x) sin
✓
k
2 ⇡
T
x
◆
dx k 1.
Z T
0
|f (x)|
2 dx = T a
2
0
T
2
1 X
k=
(a
2
k
2
k ) ( identit`a di Parseval )
424 Tables and Formulas
F u n d a m e n t a l limits
lim x^ = +00, lim x" = 0 , a > 0
lim x^ = 0, lim x^ = +(X), a < 0
^.^ a^x" +... + a i X + ao ^ a , ^.^ ^ , _ ^
x ^ ± o o bmX^ +... + biX + 6 o ^m a:->±oo
lim a^ = +CX), lim a^ = 0 , a > 1
a;—)-+oo X—)• —oo
lim a^ = 0, lim a^ = + o c , a < 1
rr—>+oo x—> —oo
lim log^ X — +0 0 , lim log^ x = —oo , a > I
X—>-+oo X—>-0+
lim log^ X = —oo , lim log^ x = +oo , a < 1
X—>-+oo X—>-0+
lim sin x , lim cos x , lim tan x do not exist
X—>-zbcx) X—^±0 0 x ^ i b o o
TT
lim tanx = +oo , VA: G Z , lim arctanx = ± —
x -. ( f + / c 7 r ) ^ ^ - ± o o 2
TT
lim arcsina: = ±— = arcsin(±l)
x-.±i 2 ^ ^
lim arccosx = 0 = arccos 1 , lim arccosx = TT = arccos(—1)
X—>- + l X—> — 1
, sinx ^ ,. 1 — cos a; 1
lim = 1, lim ^ = -
x-> 0 X x-^0 X"^ 2
lim fl + - ) ^ e " " , a G R , lim(l + x ) - = e
lim — , a > 0 ; m particular, lim = 1
x->o X log a x-^^ X
c^ — 1 e*^ — 1
lim = log a, a > 0 ; in particular, lim = 1
x-^O X x - ^ 0 X
Tables and Formulas 425
Derivatives of elementary functions
/(^)
x^
sinx
cosx
tanx
a r f c i r i T
arrrrm T
arctan x
a^
loga 1^ 1
sinh X
cosh X
n^)
ax"^-^ , ^aeM.
cosx
— sin X
9 1
± 1 Laii X — „
COS^ X
v ^ r = ^
v^n^^
l + x 2
(log a) a^
(log a) X
cosh X
sinh X
Differentiation rules
(a/(x)+/35(x))'-a/'(x)+/35'(x)
(/(^)5(x))'
/^/(^)V
(gifix)))-
= f'{x)g{x) + f{x)g'{x)
r{x)9{x) - f{x)g'{x)
(fl(^))'
= g'{f{x))nx)
Tables and Formulas 427
Integrals of e l e m e n t a ry functions
/ ( ^ )
x^
X
sinx
cosx
e^
sinh X
cosh X
1 + ^ 2
x/^2Tri
J/(x)dx
— — - + c , ay^-
a + 1
log|x| + c
— cos X -- c
sin x + c
e^ + c
cosh X -{- c
sinh X + c
arctan x - h c
arcsin x + c
log(x + V x^ + 1 ) + c = sett sinh x + c
log(x - h v x^ — 1 ) + c = sett cosh x + c
I n t e g r a t i o n rules
j(af{x) +
j f{x)9\x)
J ^{x)
(3g{x) j dx = a / /(x) dx + / 3 / ^(x) dx
dx = f{x)g{x) - / /(x)5f(x)dx
= log|(^(x)|+c
j f{(p{x))Lp\x) dx = / f{y) dy where y = : (P(X)
Basic definitions and formulas
Sequences and series
Geometric sequence (p. 3):
lim
n→∞
q
n
0 if |q| < 1,
1 if q = 1,
+∞ if q > 1,
does not exist if q ≤ − 1
The number e (p. 3):
e = lim
n→∞
n
n
∞ ∑
k=
n!
Geometric series (p. 6):
∞ ∑
k=
q
k
converges to
1 − q
if |q| < 1 ,
diverges to + ∞ if q ≥ 1 ,
is indeterminate if q ≤ − 1
Mengoli’s series (p. 7):
∞ ∑
k=
(k − 1)k
Generalised harmonic series (p. 15):
∞ ∑
k=
k
α
converges if α > 1 ,
diverges if α ≤ 1
C. Canuto, A. Tabacco: Mathematical Analysis II, 2nd Ed.,
UNITEXT – La Matematica per il 3+2 85, DOI 10.1007/978-3-319-12757-6,
© Springer International Publishing Switzerland 2015
Definitions and formulas 535
Fourier series
Fourier coefficients of a map f (p. 82):
a 0
2 π
2 π
0
f (x) dx
a k
π
2 π
0
f (x) cos kx dx , k ≥ 1
b k
π
2 π
0
f (x) sin kx dx , k ≥ 1
c k
2 π
2 π
0
f (x)e
−ikx dx , k ∈ ZZ
Fourier series of a map f ∈
C
2 π (p. 85):
f ≈ a 0
∞ ∑
k=
(a k cos kx + b k sin kx) ≈
+∞ ∑
k=−∞
c k e
ikx
Parseval’s formula (p. 91):
2 π
0
|f (x)|
2 dx = 2πa
2
0
+∞ ∑
k=
(a
2
k
2
k
) = 2π
+∞ ∑
k=−∞
|c k
2
Square wave (p. 85):
f (x) =
− 1 if −π < x < 0 ,
0 if x = 0, ±π ,
1 if 0 < x < π ,
f ≈
π
∞ ∑
m=
2 m + 1
sin(2m + 1)x
Rectified wave (p. 87):
f (x) = | sin x| , f ≈
π
π
∞ ∑
m=
4 m
2 − 1
cos 2mx
Sawtooth wave (p. 87):
f (x) = x , x ∈ (−π,π ) , f ≈
∞ ∑
k=
k
k+ sin kx
536 Definitions and formulas
Real-valued functions
Partial derivative (p. 157):
∂f
∂x i
(x 0 ) = lim
∆x→ 0
f (x 0
∆x
Gradient (p. 157):
∇f (x 0 ) = grad f (x 0
∂f
∂x i
(x 0
1 ≤i≤n
Differential (p. 162):
df x 0
(∆x) = ∇f (x 0 ) · ∆x
Directional derivative (p. 163):
∂f
∂v
(x 0 ) = ∇f (x 0 ) · v =
∂f
∂x 1
(x 0 ) v 1
∂f
∂x n
(x 0 ) v n
Second partial derivative (p. 168):
2 f
∂x j ∂x i
(x 0
∂x j
∂f
∂x i
(x 0
Hessian matrix (p. 169):
Hf (x 0 ) = (h ij
1 ≤i,j≤n
with h ij
2 f
∂x j ∂x i
(x 0
Taylor expansion with Peano’s remainder (p. 172):
f (x) = f (x 0 ) + ∇f (x 0 ) · (x − x 0 ) +
1
2
(x − x 0 ) · Hf (x 0 )(x − x 0 ) + o(∥x − x 0 ∥
2 )
Curl in dimension 2 (p. 206):
curl f =
∂f
∂x 2
i −
∂f
∂x 1
j
Fundamental identity (p. 209):
curl grad f = ∇ ∧ (∇f ) = 0
Laplace operator (p. 211):
∆f = div grad f = ∇ · ∇f =
n ∑
j=
2 f
∂x
2
j
538 Definitions and formulas
Polar coordinates
From polar to Cartesian coordinates (p. 230):
Φ : [0, +∞) × R → R
2 , (r,θ ) .→ (x, y) = (r cos θ, r sin θ)
Jacobian matrix and determinant (p. 230):
JΦ(r,θ ) =
cos θ −r sin θ
sin θ r cos θ
, det JΦ(r,θ ) = r
Partial derivatives in polar coordinates (p. 231):
∂f
∂r
∂f
∂x
cos θ +
∂f
∂y
sin θ ,
∂f
∂θ
∂f
∂x
r sin θ +
∂f
∂y
r cos θ
∂f
∂x
∂f
∂r
cos θ −
∂g
∂θ
sin θ
r
∂f
∂y
∂g
∂r
sin θ +
∂f
∂θ
cos θ
r
Variable change in double integrals (p. 320):
Ω
f (x, y) dx dy =
Ω ′
f (r cos θ, r sin θ) r dr dθ
Cylindrical coordinates
From cylindrical to Cartesian coordinates (p. 233):
Φ : [0, +∞) × R
2 → R
3 , (r, θ, t) .→ (x, y, z) = (r cos θ, r sin θ, t)
Jacobian matrix and determinant (p. 233):
JΦ(r, θ, t) =
cos θ −r sin θ 0
sin θ r cos θ 0
, det JΦ(r, θ, t) = r
Partial derivatives in cylindrical coordinates (p. 233):
∂f
∂r
∂f
∂x
cos θ +
∂f
∂y
sin θ ,
∂f
∂θ
∂f
∂x
r sin θ +
∂f
∂y
r cos θ ,
∂f
∂t
∂f
∂z
∂f
∂x
∂f
∂r
cos θ −
∂f
∂θ
sin θ
r
∂f
∂y
∂f
∂r
sin θ +
∂f
∂θ
cos θ
r
∂f
∂z
∂f
∂t
Variable change in triple integrals (p. 328):
Ω
f (x, y, z) dx dy dz =
Ω
′
f (r cos θ, r sin θ, t) r dr dθ dt
Definitions and formulas 539
Spherical coordinates
From spherical to Cartesian coordinates (p. 234):
Φ : [0, +∞) × R
2 → R
3 ,
(r, ϕ,θ ) .→ (x, y, z) = (r sin ϕ cos θ, r sin ϕ sin θ, r cos ϕ)
Jacobian matrix (p. 234):
JΦ(r, ϕ,θ ) =
sin ϕ cos θ r cos ϕ cos θ −r sin ϕ sin θ
sin ϕ sin θ r cos ϕ sin θ r sin ϕ cos θ
cos ϕ −r sin ϕ 0
Jacobian determinant (p. 234):
det JΦ(r, ϕ,θ ) = r
2 sin ϕ
Partial derivatives in spherical coordinates (p. 233):
∂f
∂r
∂f
∂x
sin ϕ cos θ +
∂f
∂y
sin ϕ sin θ +
∂f
∂z
cos ϕ
∂f
∂ϕ
∂f
∂x
r cos ϕ cos θ +
∂f
∂y
r cos ϕ sin θ −
∂f
∂z
r sin ϕ
∂f
∂θ
∂f
∂x
r sin ϕ sin θ +
∂f
∂y
r sin ϕ cos θ
∂f
∂x
∂f
∂r
sin ϕ cos θ +
∂f
∂ϕ
cos ϕ cos θ
r
∂f
∂θ
sin θ
r sin ϕ
∂f
∂y
∂f
∂r
sin ϕ sin θ +
∂f
∂ϕ
cos ϕ sin θ
r
∂f
∂θ
cos θ
r sin ϕ
∂f
∂z
∂f
∂r
cos ϕ −
∂f
∂ϕ
sin ϕ
r
Variable change in triple integrals (p. 329):
Ω
f (x, y, z) dx dy dz =
Ω
′
f (r sin ϕ cos θ, r sin ϕ sin θ, r cos ϕ) r
2 sin ϕ dr dϕ dθ
Definitions and formulas 541
Integrals on curves and surfaces
Integral along a curve (p. 368):
γ
f =
b
a
f
γ(t)
∥γ
′ (t)∥ dt
Path integral (p. 375):
γ
f · τ =
γ
f τ
b
a
f
γ(t)
· γ
′ (t) dt
Integral on a surface (p. 378):
σ
f =
R
f
σ(u, v)
∥ν(u, v)∥ du dv
Flux integral (p. 384):
σ
f · n =
σ
f n
R
f
σ(u, v)
· ν(u, v) du dv.
Divergence Theorem (p. 391):
Ω
div f =
∂Ω
f · n
Green’s Theorem (p. 394):
Ω
∂f 2
∂x
∂f 1
∂y
dx dy =
∂Ω
f · τ
Stokes’ Theorem (p. 396):
Σ
(curl f ) · n =
∂Σ
f · τ
542 Definitions and formulas
Examples of quadrics
x
y
Ellipsoid
x
2
a
2
y
2
b
2
z
2
c
2
= 1
z
x
y
Hyperbolic paraboloid
z
c
= −
x
2
a
2
y
2
b
2
z
x
y
Elliptic paraboloid
z
c
=
x
2
a
2
y
2
b
2
z
