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ZAKŁAD ANALIZY I TEORII OSOBLIWOŚCI
PRZEKSZTAŁCENIE L
WZÓR CAŁKOWY FOURIERA
+∞
L [ f (t )] = ∫ e − st f (t )dt
+∞
df
f (t ) = ∫ [a(ω ) cos ωt + b(ω ) sin ωt ]dω
a (ω ) =
π
+∞
∫ f (τ ) cosωτdτ
1
b(ω ) =
π
−∞
f (t )
+∞
∫ f (τ ) sin ωτdτ
−∞
F [ f (t )] = ∫ e
df
− j ωt
( = F ( jω ) )
f (t )dt
1
F [ F ( jω )] =
2π
∞
∫e
j ωt
F ( j ω ) dω
−∞
e −αt , α ∈ C
( = f (t ))
− π ≤ θ (ω ) ≤ +π
a (ω )
a 2 (ω ) + b 2 (ω )
sin θ (ω ) =
− b(ω )
a 2 (ω ) + b 2 (ω )
JeŜeli cos θ (ω ) = −1 i sin θ (ω ) = 0 , to θ (ω ) = π sgn ω dla ω ≠ 0
θ - funkcja nieparzysta na Dθ lub Dθ − {0}
s +ω2
s
2
s +ω2
cos ωt , ω ∈ R
2
e −αt ⋅ sin ωt ,
α ∈ C, ω ∈ R
ω
(s + α )2 + ω 2
chβt , β ∈ R
s
s −β2
2
(n)
(t )] = s n ⋅ f ( s ) − ∑ s n− k ⋅ f ( k −1) (0+ )
L [ f ' (t )] = s f ( s ) − f (0+ ) ,
L [ f ' ' (t )] = s 2 f ( s ) − sf (0+ ) − f ' (0+ )
1 s
L ( f (at )) = ⋅ f , gdy a > 0
a a
L [ f (t − t 0 )] = e − st0 ⋅ f ( s ), t 0 ≥ 0
[
2
,
s −β
2
ω
2
k =1
F ( jω ) = π a (ω ) + b (ω ) ( funkcja parzysta )
cos θ (ω ) =
sin ωt , ω ∈ R
f (s )
n
WIDMO AMPLITUDOWE
WIDMO FAZOWE
1
s
n!
ozn
F ( jω ) = π [a (ω ) − jb(ω )]
2
f (t )
β
shβt , β ∈ R
L[ f
F ( jω ) = F ( jω ) e jθ (ω ) ,
; f – oryginał
ozn
f (s )
s n +1
1
s +α
ozn
−∞
−1
1(t )
tn, n∈N
PRZEKSZTAŁCENIE F
+∞
( = f (s))
(CAŁKA FOURIERA)
0
1
0
]
L e −αt ⋅ f (t ) = f ( s + α ) , α ∈ C
n
t
f (s)
n
n d f (s)
L ∫ f (τ )dτ =
, L [t f (t )] = (−1)
,
0
s
ds n
T
Dla oryginału okresowego f :
Dla f ( s ) =
∫
f ( s) =
L( s )
(funkcja wymierna) :
M (s)
0
f (t ) ⋅ e − st dt
1 − e − sT
n∈N
(T – okres)
f (t ) = ∑ res si [ f ( s ) ⋅ e st ]
i
SPLOT ORYGINAŁÓW: F (t ) = f 1 (t ) ∗ f 2 (t )
t
F (t ) = ∫ f 1 (τ ) f 2 (t − τ )dτ ⋅ 1(t ) ,
df 0
L [ F (t )] = f 1 ( s ) ⋅ f 2 ( s )
sin 2α = 2 sin α cos α
cos 2a = 2 cos 2 α − 1 = cos 2 α − sin 2 α = 1 − 2 sin 2 α
PRZEKSZTAŁCENIE Z
∞
Z [( x n )] = ∑
df
n =0
( = X ( z)) ;
xn
zn
ozn
( x n ) = ( x 0 , x1 ,...)
sin(α ± β ) = sin α ⋅ cos β ± cos α ⋅ sin β
cos(α ± β ) = cos α ⋅ cos β m sin α ⋅ sin β
xn
X (z )
xn
X (z )
1
z
z −1
e αn , α ∈ C
n
z
( z − 1) 2
sin ωn, ω ∈ C
z
z − eα
z sin ω
2
z − 2 z cos ω + 1
z ( z + 1)
( z − 1) 3
cos ωn, ω ∈ C
z ( z − cos ω )
z − 2 z cos ω + 1
z
z−a
1
n!
ez
n
2
a n , a ∈ C − {0}
Z [( x n −k )] = z − k ⋅ X ( z ) ,
2
1
k ∈N
k −1
x
Z [( x n + k )] = z k ⋅ X ( z ) − ∑ νν
ν =0 z
dX ( z )
Z [(nx n )] = − z
dz
Dla X ( z ) =
L( z )
(funkcja wymierna) : xn = ∑ resz i [ X ( z ) ⋅ z n −1 ]
M ( z)
i
SPLOT CIĄGÓW: (u n ) = ( x n ) ∗ ( y n )
n
u n = ∑ xν ⋅ y n −ν ,
ν =0
n = 0,1,2,... ;
Z [(u n )] = X ( z ) ⋅ Y ( z )
1
[cos(α − β ) − cos(α + β )]
2
1
cos α ⋅ cos β = [cos(α − β ) + cos(α + β )]
2
1
sin α ⋅ cos β = [sin(α − β ) + sin(α + β )]
2
sin α ⋅ sin β =
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