tm ma hlektrologÐac pinakas perieqomenwn...tm ma hlektrologÐac shmei¸seic sta efarmosmèna...
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Tm ma HlektrologÐac
Shmei¸seic sta Efarmosmèna Majhmatik�
ParasÐdhc I.N.
PINAKAS PERIEQOMENWN
1. Metasqhmatismìc Laplace
1. Je¸rhma grammikìthtac
2. Je¸rhma metatìpishc wc proc s
3. Je¸rhma allag c klhmakac
4. Metasqhmatismìc Laplace twn parag¸gwn
5. Metasqhmatismìc Laplace twn parast�sewn tnf(t)
6. Metasqhmatismìc Laplace twn oloklhrom�twn
7. Metasqhmatismìc Laplace twn parast�sewn f(t)t
8. EpÐlush oloklhrwm�twn tÔpou∫∞0
f(t)t dt
9. Je¸rhma metatìpishc wc proc t Je¸rhma kajustèrhshc
10. Je¸rhma sunèlhxhc
11. Metasqhmatismìc Laplace twn periodik¸n sunart sewn
12. Sun�rthsh apokop c monadiaÐa sun�rthsh b matoc tou Heaviside. Metasqhmatismìc Laplace thcsun�rthshc apokop c
13. Metasqhmatismìc Laplace thc sun�rthshc Dèlta (Dirac) thc monadiaÐac kroÔsewc
2. Eidikèc sunart seic kai metasqhmatismìc Laplace
1. Gamma sun�rthsh olokl rwma tou Euler b-eÐdouc
2. Bhta sun�rthsh olokl rwma tou Euler pr¸tou eÐdouc
3. AntÐstrofoc metasqhmatismìc Laplace
1. Je¸rhma grammikìthtac
2. Je¸rhma allag c klhmakac
1
3. AntÐstrofoc metasqhmatismìc Laplace twn parag¸gwn
4. Je¸rhma pollaplasiasmoÔ epÐ s
5. Je¸rhma diaÐreshc dia s
6. AntÐstrofoc metasqhmatismìc Laplace twn oloklhrwm�twn
7. Je¸rhma sunèlhxhc gia ton antÐstrofo metasqhmatismì Laplace
8. Mèjodoi upologismìu tou antÐstrofou metasqhmatismou Laplace me analush se apl� kl�smata:
a. Mejodoc teqnasmatwn
b. Mèjodoc twn aprosdiìristwn suntelest¸n
g. Mèjodoc Heaviside
9. Grammik� kukl¸mata
4. EpÐlush grammik¸n diaforik¸n exis¸sewn (g.d.e.) kai susthm�twn g.d.e. me
stajeroÔc suntelestec me to metasqimatismì Laplace5. Seirèc Fourier
1. SÔgklish thc seir�c Fourier
2. Seirèc Fourier sto di�sthma [−π, π]
3. Seirèc Fourier sto aujaÐreto di�sthma [−l, l]
4. Je¸rhma Dirichlet
5. Seirèc Fourier �rtiwn kai peritt¸n sunart sewn.
6. Epèktash periodik -�rtia (sunhmitonik )
7. Epèktash periodik -peritt (hmitonik )
8. Ekjetik (migadik ) morf thc seir�c Fourier
9. Olokl rwma kai metasqhmatismìc Fourier. AntÐstrofoc metasqhmatismìc Fourier
10. Je¸rhma sunèlhxhc gia to metasqhmatismì Fourier
11. Efarmogèc tou metasqhmatismoÔ Fourier
2
1 METASQHMATISMOS LAPLACE
Orismìc 1: ean f(t) eÐnai mia sun�rthsh, orismènh kai oloklhr¸simh se k�je di�sthma [0, t] ìpou t > 0,
tìte o metasqhmatismìc Laplace sumbolÐzetai me L{f(t)} F (s) kai orÐzetai wc:
L{f(t)} = F (s) =∫ ∞
0e−stf(t)dt, s ∈ R (1.1)
Orismìc 2: mia sun�rthsh f(t) eÐnai ekjetik c t�xewc α, ean up�rqoÔn treic stajerèc α, M kai t0
tètoiec ¸ste e−αt|f(t)| ≤ M gia k�je t ≥ t0 an
limt→∞
f(t)eαt
= 0.
Orismìc 3: mia sun�rthsh f(t) lègetai tmhmatik� suneq c sto anoiktì di�sthma (a, b), ean h f(t) eÐnaisuneq c sto (a, b) ektìc Ðswc apì èna peperasmèno pl joc shmeÐwn.
Orismìc 4: mia sun�rthsh f(t) lègetai tmhmatik� suneq c sto kleistì di�sthma [a, b], ean eÐnai
tmhmatik� suneq c sto anoiktì di�sthma (a, b) kai jp�rqoun ta ìria apì arister� kai dexi�:
limt→a+0
f(t) kai limt→b−0
f(t).
Je¸rhma
E�n h f(t) eÐnai tmhmatik� suneq c se k�je kleistì di�sthma [0, b], b > 0 kai e�n h f(t) eÐnai ekjetik ct�xewc α, tìte to olokl rwma tou dexioÔ mèlouc thc (1.1) sungklÐnei, dhlad up�rqei o metasqhmatismìc
Laplace thc f(t) gia s > α.
PINAKAS 1
f(t) = L−1{F (s)} L{f(t)} = F (s)
1.π 1 1s , s > 0
2.π tn, n ∈ N n!sn+1 , s > 0
3.π eat 1s−a , s > a
4.π tneat n ∈ N n!(s−a)n+1 , s > a
5.π sin(kt) ks2+k2
6.π cos(kt) ss2+k2
7.π eat sin(kt) k(s−a)2+k2
8.π eat cos(kt) s−a(s−a)2+k2
9.π sinh(kt) ks2−k2 , s > |k|
10.π cosh(kt) ss2−k2 , s > |k|
11.π u(t− a) e−as
s
12.π tx, x ∈ R Γ(x+1)sx+1 , x + 1 > 0
3
Idiìthtec tou metasqhmatismoÔ Laplace
Je¸rhma 1 (grammikìthtac)
E�n L{f1(t)} = F1(s) kai L{f2(t)} = F2(s), tìte gia opoiesd pote stajerèc c1 kai c2 isqÔei
L{c1f1(t) + c2f2(t)} = c1F1(s) + c2F2(s) (1.2)
Je¸rhma 2 (metatìpishc wc proc s)E�n L{f(t)} = F (s), tìte gia k�je stajerh a eÐnai
L{eatf(t)} = F (s− a), s > a (1.3)
Askhsh 1. Na brejeÐ
L{
e3t cos 4t} =
LÔsh: 'Eqoume a = 3 kai f(t) = cos 4t Tìte
L{cos 4t} (6.π)=
s
s2 + 16= F (s) (1.4)
kai
F (s− a) = F (s− 3)(1.4)=
s− 3(s− 3)2 + 16
=s− 3
s2 − 6s + 25
Epomènwc
L{
e3t cos 4t} =s− 3
s2 − 6s + 25Je¸rhma 3 (allag c klhmakac )
L{f(at)} =1aF
(s
a
), L{f(t)} = F (s), a eÐnai stajerh 6= 0 (1.5)
Askhsh 1. An L{f(t)} = s2−s+1(2s+1)2(s−1)
na brejeÐ
L{f(2t)} =
LÔsh: 'Eqoume a = 2 kai
F (s) =s2 − s + 1
(2s + 1)2(s− 1)(1.6)
Tìte
L{f(2t)} (1.5)=
12F
{s
2
}(1.6)=
12
(s2
)2− s
2 + 1
(2 · s2 + 1)2( s
2 − 1)=
=12
s2
4 −s2 + 1
(s + 1)2( s2 − 1)
=14
s2 − 2s + 4(s + 1)2(s− 2)
4
Je¸rhma 4 (metasqhmatismìc Laplace twn parag gwn)
E�n L{f(t)} = F (s), tìte
L{f ′(t)} = sF (s)− f(0), L{f ′′(t)} = s2F (s)− sf(0)− f ′(0),
L{f ′′′(t)} = s3F (s)− s2f(0)− sf ′(0)− f ′′(0),
L{f (n)(t)} = snF (s)− sn−1f(0)− sn−2f ′(0)− ...− sf (n−2)(0)− f (n−1)(0)
Je¸rhma 5 (metasqhmatismìc Laplace twn parast�sewn tnf(t))
L{tnf(t)} = (−1)nF (n)(s), ìpou L{f(t)} = F (s)
Askhsh 1 UpologÐste to
L{
t sin2 t}
=
LÔsh: 'Eqoume
f(t) = sin2 t ⇒ L{f(t)} = L{1− cos 2t
2
}=
12L{1} − 1
2L{cos 2t} (6.π)
=
=12
1s− 1
2s
s2 + 4
Je¸rhma 6 (metasqhmatismìc Laplace twn oloklhrwm�twn)
L{∫ t
0f(x)dx
}=
F (s)s
, ìpou L{f(t)} = F (s) (1.7)
Askhsh 1 UpologÐste to
L{∫ t
0exdx
}=
LÔsh: 'Eqoume f(x) = ex, tìte F (s) = L{ex} (3.π)= 1
s−1 , s > 1. 'Ara lìgw thc sqèshc (1.7)
lamb�noume
L{∫ t
0exdx
}=
1s(s− 1)
Je¸rhma 7 (metasqhmatismìc Laplace twn parast�sewn f(t)t )
L{f(t)
t
}=
∫ ∞
sF (s)ds, ìpou L{f(t)} = F (s) (1.8)
Askhsh 1. Na brejeÐ
L{et − 1
t
}=
LÔsh: 'Eqoume f(t) = et − 1, tìte F (s) = L{et − 1} = L{et} − L{1} (3.π)= 1
s−1 −1s kai lìgw thc
sqèshc (1.8) lamb�noume
L{et − 1
t
}=
∫ ∞
s
( 1s− 1
− 1s
)ds =
[ln(s− 1)− ln s
]∞s
=[ln
s− 1s
]∞s
=
=[ln
(1− 1
s
)]∞s
= ln 1− ln(1− 1
s
)= − ln
(1− 1
s
)5
Je¸rhma 8 (EpÐlush oloklhrwm�twn tÔpou∫∞0
f(t)t dt)
An L{f(t)} = F (s) tìte ∫ ∞
0
f(t)t
dt =∫ ∞
0F (s)ds (1.9)
Askhsh 1. DeÐxte ìti ∫ ∞
0
sin t
tdt =
π
2
LÔsh: 'Eqoume f(t) = sin t, tìte F (s) = L{sin t} (5.π)= 1
s2+1kai lìgw thc sqèshc (1.9) lamb�noume∫ ∞
0
sin t
tdt =
∫ ∞
0
1s2 + 1
ds = [arctan s]∞0 = arctan∞− arctan 0 =π
2− 0 =
π
2
Je¸rhma 1 (metasqhmatismìc Laplace twn periodik¸n sunart sewn)
'Estw ìti h f(t) eÐnai periodik sun�rthsh periìdou T , dhlad f(t + T ) = f(t). Tìte
L{f(t)} =
∫ T0 f(t)dt
1− e−sT
Sun�rthsh apokop c monadiaÐa sun�rthsh b matoc tou HeavisideorÐzetai wc ex c
u(t)η= η(t)
oρ.=
0 gia t < 0
1 gia t ≥ 0
genikìtera :
u(t− a)η= ua(t)
oρ.=
0 gia t < a
1 gia t ≥ a
H sun�rthsh tou Heaviside qrhsimopoieÐtai gia na parast soume me èna kai mìno analutikì tÔpo mia
sun�rthsh h opoÐa ekfr�zetai me perissìterouc analutikoÔc tÔpouc. P.q. h sun�rthsh
g(t) =
g1(t), 0 < t < a
g2(t), t ≥ a(1.10)
gr�fetai sth morf
g(t) = g1(t) + [g2(t)− g1(t)]u(t− a), (1.11)
kai h sun�rthsh
g(t) =
g1(t), 0 < t < a1
g2(t), a1 < t < a2
g3(t), t > a2
(1.12)
gr�fetai sth morf
g(t) = g1(t) + [g2(t)− g1(t)]u(t− a1) + [g3(t)− g2(t)]u(t− a2). (1.13)
6
O metasqhmatismìc Laplace thc sun�rthshc apokop c eÐnai
L{ua(t)} = L{u(t− a)} =e−as
s
Je¸rhma 2 (metatìpishc wc proc t kajustèrhshc)
L{f(t− a)u(t− a)} = e−asF (s), ìpou L{f(t)} = F (s). (1.14)
Askhsh 1. BreÐte ton metasqhmatismì Laplace thc sun�rthshc
g(t) =
0, t < 1
t2 − t, 1 ≤ t < 2
0, t ≥ 2
LÔsh: Qrhsimopoi¸ntac touc tÔpouc (1.12) kai (1.13) lamb�noume
g(t) = (t2 − t)u(t− 1) + [0− (t2 − t)]u(t− 2) ⇒ (1.15)
g(t) = (t2 − t)u(t− 1)− (t2 − t)u(t− 2) (1.16)
Tìte
L{g(t)} = L{(t2 − t)u(t− 1)} − L{(t2 − t)u(t− 2)} = (1.17)
UpologÐzoume to pr¸to mèroc L{(t2 − t)u(t − 1)} me ton tÔpo (1.14), ìpou a = 1 kai f(t − a) =f(t− 1) = t2 − t. Jètontac t− 1 = p èqoume t = p + 1 kai
f(p) = (p + 1)2 − (p + 1) = p2 + p ⇒ f(p) = p2 + p ⇒ f(t) = t2 + t ⇒
L{f(t)} = L{t2 + t} =2s3
+1s2
= F (s)
Epomènwc
L{(t2 − t)u(t− 1)} (1.14)= e−s
( 2s3
+1s2
). (1.18)
UpologÐzoume to deÔtero mèroc thc (1.17) ìpou a = 2 kai f(t−a) = f(t−2) = t2− t. Jètontac t−2 = p
èqoume t = p + 2 kai
f(p) = (p + 2)2 − (p + 2) = p2 + 3p + 2 ⇒ f(t) = t2 + 3t + 2 ⇒
L{f(t)} = L{t2 + 3t + 2} =2s3
+ 31s2
+2s
= F (s)
Epomènwc
L{(t2 − t)u(t− 2)} (1.14)= e−2s
( 2s3
+3s2
+2s
). (1.19)
Ap ton tÔpo (1.17) lìgw twn (1.18), (1.19) lamb�noume
L{g(t)} = e−s( 2
s3+
1s2
)− e−2s
( 2s3
+3s2
+2s
)7
je¸rhma 3 ( sunèlhxhc)
An oi sunart seic f(t), g(t) eÐnai suneqeÐc tmhmatik� suneqeÐc gia t ≥ 0 kai ekjetik c t�xhc, ìtan t →∞kai an L{f(t)} = F (s), L{g(t)} = G(s), tìte
L{f(t) ∗ g(t)} = L{f(t)}L{g(t)} = F (s)G(s), ìpou (1.20)
f(t) ∗ g(t) =∫ t
0f(x)g(t− x)dx
η=
∫ t
0g(x)f(t− x)dx (1.21)
lègetai sunèlhxh.
Askhsh 1. BreÐte th sunèlixh f(t) ∗ g(t) an f(t) = 3t, g(t) = t2
LÔsh. 'Eqoume f(x) = 3x, g(x− t) = (x− t)2. 'Ara
f(t) ∗ g(t)(1.21)=
∫ t
03x(x− t)2dx = 3
∫ t
0x(x2 − 2xt + t2)dx =
=3∫ t
0x3dx− 6t
∫ t
0x2dx + 3t2
∫ t
0xdx = 3
[x4
4
]t
0− 6t
[x3
3
]t
0+
+3t2[x2
2
]t
0=
34t4 − 2t4 + 3t2
t2
2=
t4
4
Askhsh 2. BreÐte ton metasqhmatismì Laplace,
L{∫ t
0sin 3(t− x)e5xdx
}=
qrhsimopoi¸ntac to je¸rhma sunèlhxhc.
LÔsh: PaÐrnontac upìyh mac ton tÔpo (1.21) blèpoume ìti∫ t
0sin 3(t− x)e5xdx = sin 3t ∗ e5t.
Kai epeid
L{sin kt} =k
s2 + k2, L{ekt} =
1s− k
,
efarmìzontac ton tÔpo (1.20) lamb�noume
L{∫ t
0sin 3(t− x)e5xdx
}=
3s2 + 9
· 1s− 5
.
Alutec ask seic
BreÐte ton metasqhmatismì Laplace, qrhsimopoi¸ntac to je¸rhma sunèlhxhc:
1. L{∫ t0 sin 3(t− x)e5xdx} =
2. L{∫ t0 x5 cos 2(t− x)dx} =
3. L{∫ t0 x3 sin(t− x)dx} =
8
2 Eidikèc sunart seic kai metasqhmatismìc Laplace
Gamma sun�rthsh olokl rwma tou Euler b-eÐdouc
orÐzetai apì th sqèsh
Γ(x) =∫ ∞
0e−ttx−1dt, x > 0.
Idiìthtec thc Gamma sun�rthshc:
Γ(x + 1) =∫ ∞
0e−ttxdt (2.1)
Γ(x + 1) = xΓ(x) (2.2)
Γ(n + 1) = n! (2.3)
Γ(1) = 1, Γ(1
2
)=√
π (2.4)
Γ(x)Γ(1− x) =π
sinπx, 0 < x < 1 (2.5)
Γ(n +
12
)=
(2n− 1)!!√
π
2n(2.6)
ìpou
n! = 1 · 2 · 3 · 4 · ... · (n− 1) · n
(2n− 1)!! = 1 · 3 · 5 · 7 · 9 · ... · (2n− 3) · (2n− 1)
p.q.
3! = 1 · 2 · 3 = 6, 5! = 1 · 2 · 3 · 4 · 5 = 3! · 4 · 5 = 120,
7!! = 1 · 3 · 5 · 7 = 105, 9!! = 1 · 3 · 5 · 7 · 9 = 945
Askhsh 1. UpologÐste me th bo jeia thc G�mma sun�rthshc∫ ∞
0e−tt3dt =
LÔsh: Qrhsimopoi¸ntac touc tÔpouc (2.1) kai (2.3) ja èqoume∫ ∞
0e−tt3dt = Γ(3 + 1) = 3! = 6
Askhsh 2. UpologÐste me th bo jeia thc G�mma sun�rthshc∫ ∞
0e−t
√t7dt =
LÔsh: Qrhsimopoi¸ntac touc tÔpouc (2.1) kai (2.3) lamb�noume∫ ∞
0e−t
√t7dt =
∫ ∞
0e−tt7/2dt = Γ
(72
+ 1)
= Γ(4 +
12
)(2.6)=
=7!!√
π
24=
1 · 3 · 5 · 7√
π
16=
10516√
π
9
'Alutec Ask seic
UpologÐste me th bo jeia thc G�mma sun�rthshc ta parak�tw oloklhr¸mata
'Askhsh 1.
∫ ∞
0e−t
√t5dt =
'Askhsh 2.
∫ ∞
0e−t
√t9dt =
'Askhsh 3.
∫ ∞
0e−2t
√t3dt =
B ta sun�rthsh olokl rwma tou Euler pr¸tou eÐdouc
sumbolÐzetai me B(x, y) kai orÐzetai apì th sqèsh
B(x, y) =∫ 1
0tx−1(1− t)y−1dt, x > 0, y > 0. (2.7)
MporeÐ na upologizetai me th bo jeia thc Gamma sun�rthshc
B(x, y) =Γ(x)Γ(y)Γ(x + y)
. (2.8)
H Bhta sun�rthsh qrhsimopoieÐtai gia ton upologismì twn oloklhrwm�twn∫ π/2
0sinm x cosn xdx =
12B
(m + 12
,n + 1
2
), m, n > 0. (2.9)
Askhsh 1. UpologÐste me th bo jeia thc B ta sun�rthshc∫ 1
0t5(1− t)7dt =
LÔsh: SugkrÐnontac to olokl rwm� mac me ton tÔpo (2.7), brÐskoume ìti x − 1 = 5, y − 1 = 7. Tìte
x = 6, y = 8. Qrhsimopoi¸ntac ton orismì (2.7) kai ton tÔpo (2.8) lamb�noume∫ 1
0t5(1− t)7dt =B(6, 8) =
Γ(6)Γ(8)Γ(14)
=Γ(5 + 1)Γ(7 + 1)
Γ(13 + 1)=
5!7!13!
=
=1 · 2 · 3 · 4 · 5 · 7!
7!8 · 9 · 10 · 11 · 12 · 13=
18 · 3 · 11 · 4 · 13
=1
13728Askhsh 2. UpologÐste me th bo jeia thc B ta sun�rthshc∫ π/2
0sin5 x cos3 xdx =
LÔsh: SugkrÐnontac to olokl rwm� mac me ton tÔpo (2.9), brÐskoume ìti m = 5, n = 3 kai∫ π/2
0sin5 x cos3 xdx =
12B
(5 + 12
,3 + 1
2
)=
12
B(3, 2)(2.8)=
=12
Γ(3)Γ(2)Γ(3 + 2)
=12
Γ(2 + 1)Γ(1 + 1)Γ(4 + 1)
(2.3)=
12
2!1!4!
=124
10
'Alutec Ask seic
UpologÐste me th bo jeia thc B ta sun�rthshc ta parak�tw oloklhr¸mata
'Askhsh 1.
∫ π/2
0sin2 x cos3 xdx =
'Askhsh 2.
∫ 1
0
√t (1− t)3dt =
'Askhsh 3.
∫ π/2
0sin3 x cos4 xdx =
3 ANTISTROFOS METASQHMATISMOS LAPLACE
Mèsw tou metasqhmatismoÔ Laplace h sun�rthsh f(t) metasqhmatÐsthke sth sun�rthsh
L{f(t)} =∫ ∞
0e−stf(t)dt ≡ F (s), s ∈ R
Up�rqei monadiaÐoc metasqhmatismìc pou odigeÐ apì thn F (s) sthn f(t). O metasqhmatismìc autìc lègetai
antÐstrofoc metasqhmatismìc Laplace kai sumbolÐzetai me L−1{F (s)} = f(t).
3.1.Idiìthtec tou antÐstrofou metasqhmatismoÔ LAPLACE
Je¸rhma 1 (grammikìthtac)
E�n L−1{F1(s)} = f1(t) kai L−1{F2(s)} = f2(t), tìte gia opoiesd pote stajerèc c1 kai c2 isqÔei
L−1{c1F1(s) + c2F2(s)} = c1f1(t) + c2f2(t) (3.1)
Je¸rhma 2 (metatìpishc wc proc s)E�n L−1{F (s)} = f(t), tìte gia k�je stajerh a eÐnai
L−1{F (s− a)} = eatf(t), s > a (3.2)
Askhsh 1. Na brejeÐ
L−1{ s− 4
s2 − 6s + 10} =
LÔsh:
L−1{ s− 4
s2 − 6s + 10
}= L−1
{ s− 4s2 − 6s + 9 + 1
}= L−1
{ (s− 3)− 1(s− 3)2 + 1
}=
= L−1{[ s− 3
(s− 3)2 + 1− 1
(s− 3)2 + 1
]}(3.1)= L−1
{ s− 3(s− 3)2 + 1
}−
− L−1{ 1
(s− 3)2 + 1
}(7.π),(8.π)
= e3t cos t− e3t sin t
11
Je¸rhma 3 (allag c klhmakac )
L−1{
F (as)}
=1af( t
a
), ìpou L−1{F (s)} = f(t) (3.3)
Askhsh 1. Na brejeÐ
L−1{ 5
16s2 + 25
}=
LÔsh:
L−1{ 5
16s2 + 25
}= L−1
{ 5(4s)2 + 52
}(3.3)=
14
sin5t
4
epeid
L−1{ 5
s2 + 52
}(5.π)= sin 5t kai a = 4
Askhsh 2. Na brejeÐ
L−1{ 3s + 1
9s2 + 6s + 5
}=
LÔsh:
L−1{ 3s + 1
9s2 + 6s + 5
}=L−1
{ 3s + 1(9s2 + 6s + 1) + 4
}= L−1
{ 3s + 1(3s + 1)2 + 22
}=
(3.3)=
13e−
t3 cos
2t
3
epeid
L−1{ s + 1
(s + 1)2 + 22
}= e−t cos 2t kai a = 3
Je¸rhma 4 (AntÐstrofoc metasqhm. Laplace twn parag gwn)
L−1{Fn(s)} = (−1)ntnf(t) ìpou L−1{F (s)} = f(t) (3.4)
Askhsh 1. UpologÐste to
L−1{( 2
s2 − 4
)′}=
LÔsh: 'Eqoume
n = 1, F (s) =2
s2 − 4⇒ L−1
{ 2s2 − 4
}(9.π)= sinh 2t.
Tìte
L−1{( 2
s2 − 4
)′} (3.4)= −t sinh 2t.
Je¸rhma 5 (pollaplasiasmoÔ epÐ s)
L−1{sF (s)} = f ′(t), ìpou L−1{F (s)} = f(t), f(0) = 0 (3.5)
Askhsh 1 UpologÐste
L−1{ 2s
s2 + 4
}=
12
LÔsh:
L−1{ 2s
s2 + 4
}= L−1
{s · 2
s2 + 22
}'Ara
F (s) =2
s2 + 22⇒ L−1{F (s)} = L−1
{ 2s2 + 22
}= sin 2t = f(t)
kai epeid sin 0 = 0, èqoume
L−1{ 2s
s2 + 4
}(3.5)= (sin 2t)′ = 2 cos 2t
Je¸rhma 6 (diaÐreshc dia s)
L−1{F (s)
s
}=
∫ t
0f(x)dx, ìpou L−1{F (s)} = f(x) (3.6)
Askhsh 1. UpologÐste
L−1{ 1
s(s2 + 9
}=
LÔsh:
L−1{ 1
s(s2 + 9)
}=
13L−1
{ 3(s2+9)
s
}= ...
SugkrÐnontac me ton tÔpo (3.6) blèpoume ìti
F (s) =3
s2 + 9⇒ L−1{F (s)} = L−1
{ 3s2 + 9
}= sin 3x = f(x)
'Ara
L−1{ 1
s(s2 + 9)
}=
13L−1
{ 3(s2+9)
s
}(3.6)=
13
∫ t
0sin 3xdx =
−19[cos 3x
]t
0=
19(1− cos 3t)
Je¸rhma 7(AntÐstrofoc metasqhm. Laplace twn oloklhrwm�twn)
L−1{∫ ∞
sF (u)du
}=
f(t)t
, ìpou L−1{F (u)} = f(t) (3.7)
Je¸rhma 8 (sunèlhxhc gia ton antÐstrofo metasq. Laplace)An L−1{F (s)} = f(t) kai L−1{G(s)} = g(t) tìte
L−1{F (s)G(s)} = L−1{F (s)} ∗ L−1{G(s)} = f(t) ∗ g(t) ìpou (3.8)
f(t) ∗ g(t) =∫ t
0f(x)g(t− x)dx
η=
∫ t
0g(x)f(t− x)dx (3.9)
3.2. Mèjodoi upologismìu tou antÐstrofou metasq. Laplace me analush se apl�
kl�smata
13
3.2.1. Mejodoc teqnasmatwn
Askhsh 1. UpologÐste
L−1{ 2
s3 + s
}=
LÔsh:
L−1{ 2
s3 + s
}=2L−1
{ 1s(s2 + 1)
}= 2L−1
{(1 + s2)− s2
s(s2 + 1)
}=
=2L−1{[ 1 + s2
s(s2 + 1)− s2
s(s2 + 1)
]}= 2L−1
{[1s− s
s2 + 1
]}=
=2L−1{1
s
}− 2L−1
{ s
s2 + 1
}= 2− 2 cos t
Askhsh 2. UpologÐste
L−1{ 1
(s + 2)(s + 3)
}=
LÔsh: Jètw u = s + 2. Tìte s = u− 2, s + 3 = u + 1 kai
L−1{ 1
(s + 2)(s + 3)
}= L−1
{ 1u(u + 1)
}= L−1
{(1 + u)− u
u(u + 1)
}=
=L−1{[ 1 + u
u(u + 1)− u
u(u + 1)
]}= L−1
{1u− 1
u + 1
}= L−1
{ 1s + 2
− 1s + 3
}=
=L−1{ 1
s + 2
}− L−1
{ 1s + 3
]}= L−1
{ 1s− (−2)
}− L−1
{ 1s− (−3)
]}=
(3.π)= e−2t − e−3t
3.2.2. Mèjodoc twn aprosdiìristwn suntelest¸n
Askhsh 1. AnaptÔxte se aplì kl�sma to
s
(s + 1)(s + 2)
LÔsh:s
(s + 1)(s + 2)=
a
s + 1+
b
s + 2=
a(s + 2) + b(s + 1)(s + 1)(s + 2)
(3.10)
Tìte
a(s + 2) + b(s + 1) = s (3.11)
Jètontac sthn (3.11) thn tim s = −1 lamb�noume a = −1Jètontac sthn (3.11) thn tim s = −2 lamb�noume −b = −2 ⇒ b = 2. AntikajistoÔme tic timec autècsthn (3.10) kai ètsi paÐrnoume
s
(s + 1)(s + 2)=
−1s + 1
+2
s + 2
14
3.2.3. Mèjodoc Heaviside
H mèjodoc aut efarmìzetai gia rhtèc sunart seic tÔpou F (s) = P (s)Q(s) ìpou P (s), Q(s) eÐnai
polu¸numa.
(i) 'Estw ìti s1, s2, s3 eÐnai oi aplèc ( di�forec metaxÔ touc ) pragmatikèc migadikèc
rÐzec thc Q(s), tìteP (s)Q(s)
=A1
s− s1+
A2
s− s2+
A3
s− s3(3.12)
ìpou oi suntelestèc Ai upologÐzontai me dÔo trìpouc
Ai =P (si)Q′(si)
i = 1, 2, 3 (3.13)
Ai = lims→si
P (s)Q(s)
(s− si), i = 1, 2, 3 (3.14)
'Etsi lamb�noume
L−1{F (s)} = L−1{P (s)
Q(s)
}= A1e
s1t + A2es2t + A3e
s3t (3.15)
(ii) 'Estw ìti Q(s) èqei pollaplèc rÐzec: p.q.
Q(s) = (s− s1)(s− s2)2 (3.16)
dhlad èqoume thn perÐptwsh thc dipl c rÐzac s2. Tìte ja èqoume thn an�lush:
P (s)Q(s)
=A
s− s1+
B1
(s− s2)2+
B2
s− s2(3.17)
ìpou
A =P (s1)Q′(s1)
A = lims→s1
P (s)Q(s)
(s− s1) (3.18)
B1 = lims→s2
(s− s2)2P (s)Q(s)
, B2 = lims→s2
((s− s2)2
P (s)Q(s)
)′(3.19)
'Etsi lamb�noume
L−1{F (s)} = L−1{P (s)
Q(s)
}= Aes1t + B1te
s2t + B2es2t (3.20)
Sthn perÐptwsh thc tripl c rÐzac s2 tou Q(s) p.q.
Q(s) = (s− s1)(s− s2)3 (3.21)
ja èqoume thn an�lush:
P (s)Q(s)
=A
s− s1+
B1
(s− s2)3+
B2
(s− s2)2+
B3
s− s3(3.22)
15
ìpou o suntelest c A upologÐzetai me èna ap touc tÔpouc (3.18) kai oi suntelestèc B1, B2, B3 upologÐzontai
wc ex c
B1 = lims→s2
(s− s2)3P (s)Q(s)
(3.23)
B2 = lims→s2
((s− s2)3
P (s)Q(s)
)′(3.24)
B3 = lims→s2
((s− s2)3
P (s)Q(s)
)′′(3.25)
'Etsi lamb�noume
L−1{F (s)} = L−1{P (s)
Q(s)
}= Aes1t +
12B1t
2es2t + B2tes2t + B3e
s2t (3.26)
Askhsh 1. Na upologisteÐ o antÐstrofoc metasqhmatismìc Laplace
L−1{ 1
3s2 − 2s− 1
}=
LÔsh: Ed¸ èqoume P (s) = 1, Q(s) = 3s2 − 2s − 1. 'Estw ìti Q(s) = 3s2 − 2s − 1 = 0 ⇒ ∆ =(−2)2 − 4 · 3 · (−1) = 16 ⇒
s1,2 =−(−2)±
√16
2 · 3=
2± 46
⇒ s1 = 1, s2 = −13.
Epeid s1 6= s2, èqoume thn perÐptwsh (i) me dÔo aplèc rÐzec, ìpou, lìgw tou tÔpou (3.12), A3 = 0 kai
13s2 − 2s− 1
=A1
s− 1+
A2
s + 13
Gia na qrhsimopoi soume ton tÔpo (3.13), pr¸ta brÐskoume P (s1) = P (s2) = 1 epeid P (s) = 1,
Q′(s) = (3s2−2s−1)′ = 6s−2, Q′(s1) = Q′(1) = 6·1−2 = 4, Q′(s2) = Q′(−1
3
)= 6·
(−1
3
)−2 = −4.
Tìte
A1 =1
Q′(s1)=
14, A2 =
1Q′(s2)
=1−4
= −14
Kai epitèlouc efarmìzontac ton tÔpo (3.15) lamb�noume
L−1{ 1
3s2 − 2s− 1
}=
14et − 1
4e−t/3
Askhsh 2. Na upologisteÐ o antÐstrofoc metasqhmatismìc Laplace
L−1{ 1
(s + 2)(s− 2)
}=
LÔsh: Ed¸ èqoume P (s) = 1, Q(s) = (s+2)(s− 2). 'Estw ìti Q(s) = (s+2)(s− 2) = 0 ⇒ s+2 =0, s− 2 = 0 ⇒
s1 = −2, s2 = 2.
16
Epeid s1 6= s2, èqoume thn perÐptwsh (i) me dÔo aplèc rÐzec, ìpou, lìgw tou tÔpou (3.12), A3 = 0 kai
1(s + 2)(s− 2)
=A1
s + 2+
A2
s− 2
T¸ra qrhsimopoioÔme ton tÔpo (3.14)
A1 = lims→s1
P (s)Q(s)
(s− s1) = lims→−2
1(s + 2)(s− 2)
(s + 2) = lims→−2
1s− 2
=1
−2− 2= −1
4
A2 = lims→s2
P (s)Q(s)
(s− s2) = lims→2
1(s + 2)(s− 2)
(s− 2) = lims→2
1s + 2
=1
2 + 2=
14
Kai epitèlouc efarmìzontac ton tÔpo (3.15) lamb�noume
L−1{ 1
(s + 2)(s− 2)
}= −1
4e−2t +
14e2t
Askhsh 3. Na upologisteÐ o antÐstrofoc metasqhmatismìc Laplace
L−1{ 1
(s− 1)(s− 2)2}
=
LÔsh: Ed¸ èqoume P (s) = 1, Q(s) = (s − 1)(s − 2)2. SugkrÐnontac me ton tÔpo (3.16), brÐskoume
s1 = 1, s2 = 2, h teleutaÐa eÐnai dipl rÐza. Tìte efarmìzontac ton tÔpo (3.17) èqoume thn ex c an�lush:
1(s− 1)(s− 2)2
=A
s− 1+
B1
(s− 2)2+
B2
s− 2
ìpou lìgw tou deÔterou mèrouc tou tÔpou (3.18) kai (3.19) lamb�noume
A = lims→s1
P (s)Q(s)
(s− s1) = lims→1
1(s− 1)(s− 2)2
(s− 1) = lims→1
1(s− 2)2
=1
(1− 2)2= 1
B1 = lims→s2
(s− s2)2P (s)Q(s)
= lims→2
(s− 2)21
(s− 1)(s− 2)2= lim
s→2
1s− 1
= 1
B2 = lims→s2
((s− s2)2
P (s)Q(s)
)′= lim
s→2
((s− 2)2
1(s− 1)(s− 2)2
)′= lim
s→2
( 1s− 1
)′= lim
s→2
((s− 1)−1
)′=
=− lims→2
(s− 1)−2 = −1
Kai epitèlouc efarmìzontac ton tÔpo (3.20 brÐskoume
L−1{F (s)} = L−1{P (s)
Q(s)
}= Aes1t + B1te
s2t + B2es2t = et + te2t − e2t
'Alutec Ask seic
Na upologisteÐ o antÐstrofoc metasqhmatismìc Laplace
Askhsh 1. y(t) = L−1{
1(s−3)(s−4)
}= LÔsh y(t) = e4t − e3t
Askhsh 2. y(t) = L−1{
1(s−1)(s+1)
}= LÔsh y(t) = 1
2(et − e−t)
Askhsh 3. y(t) = L−1{
s(s−1)(s+1)
}= LÔsh y(t) = 1
2(et + e−t)
Askhsh 4. y(t) = L−1{
s(s−1)(s+1)2
}= LÔsh y(t) = 1
4(2te−t + et − e−t)
Askhsh 5. y(t) = L−1{
s+1(s+4)(s−3)2
}= LÔsh y(t) = 1
4(2te3t + e3t + e−4t)
17
4 LÔsh grammik¸n diaforik¸n exis¸sewn (g.d.e.) kai susthm�twn
g.d.e. me stajeroÔc suntelestec me to metasqimatismì Laplace
Askhsh 1. Na lujeÐ to prìblhma arqik¸n tim¸n
y′ + 2y = e3t, y(0) = 0 (4.1)
me th bo jeia tou metasqhmatismoÔ Laplace.LÔsh: Jètw L{y(t)} = Y (s). Tìte L{y′(t)} = sY (s)− y(0) = sY (s) kai apì thn (4.1) brÐskoume
L{y′ + 2y} = L{e3t} (3.π)⇒ L{y′}+ 2L{y} =1
s− 3⇒
sY (s) + 2Y (s) =1
s− 3⇒ (s + 2)Y (s) =
1s− 3
⇒
Y (s) =1
(s + 2)(s− 3)=
A
s + 2+
B
s− 3=
A(s− 3) + B(s + 2)(s + 2)(s− 3)
(4.2)
opìte epeid to pr¸to kl�sma eÐnai Ðson me to teleutaÐo
A(s− 3) + B(s + 2) = 1 (4.3)
Ap ton tÔpo (4.3) gia s = 3 ⇒ 5B = 1 ⇒ B = 1/5.
Ap ton tÔpo (4.3) gia s = −2 ⇒ −5A = 1 ⇒ A = −1/5.
AntikajustoÔme tic timèc A = −1/5, B = 1/5 sthn (4.6) kai eqoume
Y (s) = −15
1s + 2
+15
1s− 3
(4.4)
opìte, paÐrnontac ton antÐstrofo metasqhmatismì tou Laplace, brÐskoume th lÔsh tou probl matoc (4.1)
y(t) = −15e−2t +
15e3t.
Askhsh 2. Na lujeÐ to prìblhma arqik¸n tim¸n
y′′ + 2y′ + y = 3e−t, y(0) = y′(0) = 1 (4.5)
me th bo jeia tou metasqhmatismoÔ Laplace.LÔsh: Jètw L{y(t)} = Y (s). Tìte L{y′(t)} = sY (s) − y(0) = sY (s) − 1, L{y′′(t)} = s2Y (s) −sy(0)− y′(0) = s2Y (s)− s− 1 kai apì thn (4.5) brÐskoume
L{y′′ + 2y′ + y} = L{3e−t} (3.π)⇒ L{y′′}+ 2L{y′}+ L{y} = 31
s + 1⇒
s2Y (s)− s− 1 + 2[sY (s)− 1] + Y (s) = 31
s + 1⇒
(s2 + 2s + 1)Y (s) = 31
s + 1+ s + 1 + 2 ⇒
18
Y (s) =3
(s + 1)3+
1s + 1
+2
(s + 1)2(4.6)
opìte, paÐrnontac ton antÐstrofo metasqhmatismì tou Laplace, brÐskoume th lÔsh tou probl matoc (4.5)
y(t) = L−1{3
22
(s + 1)2+1+
1s + 1
+ 21
(s + 1)1+1
}=
32t2e−t + 2te−t ⇒
y(t) = e−t(3
2t2 + 2t + 1
)Askhsh 3. Na lujeÐ to sÔsthma arqik¸n tim¸nx′ + y = 0
y′ + x = 0 , x(0) = 1 y(0) = −1(4.7)
me th bo jeia tou metasqhmatismoÔ Laplace.LÔsh: Jètw L{x(t)} = X(s), L{y(t)} = Y (s). Tìte L{x′(t)} = sX(s)−x(0) = sX(s)−1, L{y′(t)} =sY (s)− y(0) = sY (s) + 1. Ap to sÔsthma (4.7) brÐskoumeL{x′ + y} = L{0}
L{y′ + x} = L{0}⇒
L{x′}+ L{y} = 0
L{y′}+ L{x} = 0⇒
sX(s)− 1 + Y (s) = 0
sY (s) + 1 + X(s) = 0⇒
sX(s) + Y (s) = 1
X(s) + sY (s) = −1⇒
Y (s) = 1− sX(s)
X(s) + sY (s) = −1⇒
Y (s) = 1− sX(s)
X(s) + s[1− sX(s)] = −1⇒
Y (s) = 1− sX(s)
(1− s2)X(s) = −1− s⇒
Y (s) = 1− sX(s)
X(s) = − 1+s1−s2
⇒
Y (s) = 1− sX(s)
X(s) = 1s−1
⇒
Y (s) = 1− ss−1
X(s) = 1s−1
⇒
Y (s) = s−1−ss−1
X(s) = 1s−1
⇒
Y (s) = − 1s−1
X(s) = 1s−1
opìte, paÐrnontac ton antÐstrofo metasqhmatismì tou Laplace, brÐskoume th lÔsh tou sust matoc (4.7)
x(t) = et, y(t) = −et
'Alutec Ask seic
Na lujeÐ to prìblhma arqik¸n tim¸n
Askhsh 1. y′ + 2y = sin t, y(0) = 0 LÔsh y(t) =Askhsh 2. y′ − y = e−3t, y(0) = 0 LÔsh y(t) = 1
4(et − e−3t)Askhsh 3. y′ − 2y = e4t, y(0) = 0 LÔsh y(t) = 1
2(e4t − e2t)Askhsh 4. y′ − y = e−2t, y(0) = 1 LÔsh y(t) = 1
3(4et − e−2t)Askhsh 5. y′′ + 4y = et, y(0) = 0, y′(0) = −1 LÔsh y(t) = 1
5(et − cos 2t− 3 sin 2t)Askhsh 6. y′′′ + y′ = 1, y(0) = y′(0) = y′′(0) = 0 LÔsh y(t) = t− sin t)Askhsh 7. y′′ + 2y′ + y = sin t, y(0) = 0, y′(0) = −1 LÔsh y(t) = 1
2(e−t + te−t − cos t)
19
Na lujeÐ to sÔsthma arqik¸n tim¸n
Askhsh 8.
x′ = x− 2y
y′ = x− y , x(0) = 0, y(0) = 1
Askhsh 9.
x′ = x + 3y
y′ = x− y , x(0) = 0, y(0) = 1
Askhsh 10.
x′ = 2x− 3y
y′ = −2x + y , x(0) = 8 , y(0) = 3
5 Seirèc Fourier
5.1 Seir’es Fourier sto di�sthma [−π, π]Orismìc 5.1.1. Onom�zoume Trigwnometrik seir� mia seir� thc morf c
a0
2+
∞∑n=1
(an cos nx + bn sinnx) (5.1)
ìpou a0, an, bn (n ∈ N∗) eÐnai stajeroÐ arijmoÐ, en¸ x ∈ R.
Orismìc 5.1.2. Onom�zoume n Merikì 'Ajroisma thc seir�c (5.1) th 2π periodik sun�rthsh
Sn(x) =a0
2+
n∑k=1
(ak cos kx + bk sin kx) (5.2)
Je¸rhma 5.1.1. An mia sun�rthsh f(x) orismènh kai oloklhr¸simh sto [−π, π] anaptÔssetai seTrigwnometrik seir�, dhlad
f(x) =a0
2+
∞∑n=1
(an cos nx + bn sinnx) (5.3)
tìte aut h an�ptuxh eÐnai monadik .
Orismìc 5.1.3. Oi arijmoÐ a0, an, bn (n ∈ N∗) sthn (5.3) prosdiorÐzontai kat� monadikì trìpo apì
touc tÔpouc
a0 =1π
∫ π
−πf(x)dx (5.4)
an =1π
∫ π
−πf(x) cos nxdx (5.5)
bn =1π
∫ π
−πf(x) sinnxdx (5.6)
20
kai lègontai suntelestèc Fourier kai h Trigwnometrik seir� (5.3) me tètoiouc suntelestèc onom�zetai
seir� Fourie.Orismìc 5.1.4. Mia sun�rthsh F (x) periìdou T = 2π kai orismènh se ìlo ton R, lègetai periodik epèktash miac sun�rthshc f(x) orismènhc sto di�sthma [−π, π] e�n isqÔei
F (x) = f(x), ∀x ∈ [−π, π]
Orismìc 5.1.5. 'Estw to Merikì 'Ajroisma Sn(x) dÐnetai me ton tÔpo (5.2) ìpou a0, an, bn (n ∈ N∗)prosdiorÐzontai apì touc tÔpouc (5.4)-(5.6). An up�rqei to ìrio
limn→∞
Sn(x) = f(x), ∀x ∈ [−π, π] (5.7)
tìte lème ìti h seir� Fourie thc sun�rthshc f(x) sugklÐnei shmiak¸c proc thn f(x) sto [−π, π] kai hf(x) lègetai 'Ajroisma thc seir�c aut c.
Parat rhsh 5.1.1. An mia seir� Fourie sugklÐnei proc thn f(x) sto [−π, π], dhlad an isqÔei h (5.7)kai h F (x) eÐnai periodik epèktash thc sun�rthshc f(x) se ìlo to R, tìte h seir� Fourie sugklÐnei
proc thn F (x) se ìlo to R, dhlad up�rqei ìrio
limn→∞
Sn(x) = F (x), ∀x ∈ R
5.2 Seirèc Fourier sto aujaÐreto di�sthma [−l, l]Gia mia tuqoÔsa sun�rthsh f(x) orismènh se èna aujaÐreto di�sthma [−l, l] omoÐwc eis�getai h seir� kai
oi suntelestèc Fourie me touc tÔpouc
f(x) =a0
2+
∞∑n=1
(an cos
nπx
l+ bn sin
nπx
l
)(5.8)
a0 =1l
∫ l
−lf(x)dx (5.9)
an =1l
∫ l
−lf(x) cos
nπx
ldx n ∈ N∗ (5.10)
bn =1l
∫ l
−lf(x) sin
nπx
ldx n ∈ N∗ (5.11)
Orismìc 5.2.1. Mia sun�rthsh f(x) lègetai suneq c kat� tm mata sto di�sthma [a, b] an eÐnai suneqeÐc parousi�zei asunèqeia 1-ou eÐdouc se èna peperasmèno pl joc shmeÐwn sto di�sthma [a, b].Orismìc 5.2.2. Lème ìti mia sun�rthsh f(x), x ∈ [a, b] ikanopoieÐ tic sunj kec Dirichlet an èqei
peperasmèno pl joc megÐstwn kai elaqÐstwn sto di�sthma [a, b] kai eÐnai suneq c kat� tm mata sto
di�sthma autì.
To epìmeno je¸rhma mac dÐnei thn ap�nthsh pìte mia sun�rthsh f(x) orismènh se èna aujaÐreto di�sthma
[−l, l] eÐnai anaptÔximh kat� seir� Fourier, dhlad pìte isqÔoun oi tÔpoi h (5.8)-(5.11).
21
Je¸rhma DirichletAn mia sun�rthsh f(x) ikanopoieÐ tic sunj kec Dirichlet sto di�sthma [−l, l], tìte1. h antÐstoiqh Trigwnometrik seir� Fourier sugklÐnei sthn f(x) se k�je shmeÐo sunèqeiac x ∈ [−l, l]thc f(x), dhlad isqÔei
limn→∞
Sn(x) = f(x) (5.12)
2. h antÐstoiqh Trigwnometrik seir� Fourier se k�je shmeÐo asunèqeiac x0 ∈ [−l, l] thc f(x), sugklÐneisthn tim
12
[lim
x→x−0
f(x) + limx→x+
0
f(x)]
dhlad isqÔei
limn→∞
Sn(x0) =12
[lim
x→x−0
f(x) + limx→x+
0
f(x)]
(5.13)
Pìrisma
An h sun�rthsh f(x) plhreÐ tic sunj kec Dirichlet kai eÐnai periodik , tìte se ìlo to R1. h antÐstoiqh Trigwnometrik seir� Fourier sugklÐnei sthn f(x) se k�je shmeÐo sunèqeiac thc f(x),2. h antÐstoiqh Trigwnometrik seir� Fourier se k�je shmeÐo asunèqeiac x0 thc f(x), sugklÐnei stohmi�jroisma twn pleurik¸n orÐwn thc f(x) sto x0.
Parat rhsh 5.2.1. Opoiad pote sun�rthsh f(x), x ∈ [a, b] periodik mh mporeÐ na analujeÐ
kat� seir� Fourier sto [a, b] efìson plhreÐ tic sunj kec Dirichlet h periodik thc epèktash F (x) se ìloto R.
5.3 Seirèc Fourier �rtiwn kai peritt¸n sunart sewn.
1. An h sun�rthsh f(x) eÐnai �rtia orismènh sto di�sthma [−π, π], dhlad an f(−x) = f(x) gia k�jex ∈ [−π, π], tìte h seir� Fourier thc f(x) eÐnai
f(x) =a0
2+
∞∑n=1
an cos nx (5.14)
ìpou gia k�je n ∈ N∗ èqoume
a0 =2π
∫ π
0f(x)dx, an =
2π
∫ π
0f(x) cos nxdx, bn = 0 (5.15)
2. An h sun�rthsh f(x) eÐnai peritt sto di�sthma [−π, π], dhlad an f(−x) = −f(x) gia k�je
x ∈ [−π, π], tìte h seir� Fourier thc f(x) eÐnai
f(x) =∞∑
n=1
bn sinnx (5.16)
22
ìpou gia k�je n ∈ N∗ èqoume
a0 = 0, an = 0, bn =2π
∫ π
0f(x) sinnxdx. (5.17)
3. An h sun�rthsh f(x) eÐnai �rtia orismènh sto di�sthma [−l, l] tìte h seir� Fourier thc f(x) eÐnai
f(x) =a0
2+
∞∑n=1
an cosnπx
l(5.18)
ìpou gia k�je n ∈ N∗ èqoume
a0 =2l
∫ l
0f(x)dx, an =
2l
∫ l
0f(x) cos
nπx
ldx, bn = 0 (5.19)
4. An h sun�rthsh f(x) eÐnai peritt sto di�sthma [−l, l] tìte h seir� Fourier thc f(x) eÐnai
f(x) =∞∑
n=1
bn sinnπx
l(5.20)
ìpou gia k�je n ∈ N∗ èqoume
a0 = 0, an = 0, bn =2l
∫ l
0f(x) sin
nπx
ldx. (5.21)
5.4 Periodikèc epekt�seic -�rtia (sunhmitonik ) kai peritt (hmitonik )
An th f(x), pou plhreÐ tic sunj kec Dirichlet kai eÐnai orismènh sto hmidi�sthma [0, l], epekteÐnoume stodi�sthma [−l, l] ètsi ¸ste h epèktas thc F (x) sto [−l, l] na eÐnai �rtia, tìte gia thn f(x) isqÔoun oi
tÔpoi (5.18) kai (5.19) ìpou to an�ptugma (5.18) onom�zetai sunhmitonik seir� Fourier thc f(x). H
periodik �rtia epèktas F (x) me perÐodo T = 2l thc f(x) se ìlo to R orÐzetai wc ex c
F (x) =
f(x) gia x ∈ [0, l]
f(−x) gia x ∈ [−l, 0]
F (x + 2l) = F (x) gia k�je x ∈ R
'Estw mia sun�rthsh f(x) plhreÐ tic sunj kec Dirichlet kai eÐnai orismènh sto hmidi�sthma [0, l] . An
epekteÐnoume th f(x) sto di�sthma [−l, l] ètsi ¸ste h epèktas thc F (x) sto [−l, l] na eÐnai peritt , tìte
gia thn f(x) isqÔoun oi tÔpoi (5.20) kai (5.21) ìpou to an�ptugma (5.20) onom�zetai hmitonik seir�
Fourier thc f(x). H periodik peritt epèktas F (x) me perÐodo T = 2l thc f(x) se ìlo to R orÐzetai
wc ex c
F (x) =
f(x) gia x ∈ [0, l]
−f(−x) gia x ∈ [−l, 0]
F (x + 2l) = F (x) gia k�je x ∈ R
23
Parat rhsh 5.4.1. Profan¸c mia sun�rthsh f(x), ìpou x ∈ [0, l] kai l jewroÔme hmiperÐodo,
mporei na epektajeÐ periodik� me �peirouc trìpouc.
5.5 Ekjetik (migadik ) morf thc seir�c FourierMe antikatast�seic tou Euler
sinx =12i
(eix − e−ix
), cos x =
12(eix − e−ix
)(5.22)
apì ton tÔpo (5.8) gia f(x), x ∈ [−l, l] prokÔptei h ekjetik (migadik ) morf thc seir�c Fourier
f(x) =+∞∑
n=−∞cne
inπxl , x ∈ [−l, l] (5.23)
ìpou oi suntelestèc
cn =12l
∫ l
−lf(x)e−
inπxl , gia n = 0,±1,±2,±3, ... (5.24)
ja eÐnai migadikoÐ kai isqÔoun oi sqèseic
c−n = cn, c0 =12a0, cn =
12(an − ibn), c−n =
12(an + ibn)
5.6 Olokl rwma kai metasqhmatismìc Fourier
Sthn Parat rhsh 5.2.1. m�jame ìti opoiad pote sun�rthsh f(t), t ∈ [a, b] periodik mh mporeÐ
na anaptuqjeÐ se seir� Fourier sto [a, b] efìson plhreÐ tic sunj kec Dirichlet h periodik thc epèktash
F (t) se ìlo to R.
T¸ra prokÔptei to er¸thma, an h f(t) den eÐnai periodik se ìlo to R, up�rqei an�ptugma gia thn f(t)se ìlo to R den up�rqei ; H ap�nthsh eÐnai jetik . Sthn perÐptwsh aut h seir� Fourier metatrèpetaise olokl rwma Fourier (5.25) pou sunep�getai ap th seir� Fourier thc f(t) sto di�sthma [−l, l] kaj¸cl → +∞ kai isqÔei to ex c je¸rhma
Je¸rhma 5.6.1.
An h sun�rthsh f(t) plhreÐ tic sunj kec Dirichlet se k�je peperasmèno di�sthma tou �xwna [0,+∞) kaito olokl rwma
∫ +∞−∞ |f(t)|dt up�rqei, tìte
1. se k�je shmeÐo sunèqeiac thc f(t) to olokl rwma Fourier thc f(t) sugklÐnei sthn f(t), dhlad isqÔei o tÔpoc
f(t) =1√π
∫ +∞
0
[A(ω) cos ωt + B(ω) sinωt
]dω (5.25)
ìpou
A(ω) =1√π
∫ +∞
−∞f(t) cos ωtdt (5.26)
B(ω) =1√π
∫ +∞
−∞f(t) sinωtdt (5.27)
24
2. se k�je shmeÐo asunèqeiac thc f(t) to olokl rwma Fourier thc f(t) sugklÐnei sto hmi�jroisma twn
orÐwn apì ta dexi� kai arister� thc f(t).
Me antikatast�seic tou Euler (5.22) ap ton tÔpo (5.25) èpetai h migadik morf tou oloklhr¸matoc
Fourier
f(t) =12π
∫ +∞
−∞
∫ +∞
−∞f(t)eiω(t−x)dxdω (5.28)
H sun�rthsh
F{f(t)} =1√2π
∫ +∞
−∞e−iωtf(t)dt ≡ F (ω) (5.29)
lègetai metasqhmatismìc Fourier thc f(t) kai sumbolÐzetai me F{f(t)} F (ω), en¸ h sun�rthsh
f(t) =1√2π
∫ +∞
−∞eiωtF (ω)dω ≡ F−1{F (ω)} (5.30)
kaleÐtai antÐstrofoc metasqhmatismìc Fourier thc f(t) kai sumbolÐzetai me f(t) F−1{F (ω)}.Me ton metasqhmatismì Fourier metabaÐnoume apì thn perioq tou qrìnou t sthn perioq thc suqnìthtac
ω kai me ton antÐstrofo metasqhmatismì èqoume thn antÐstrofh poreÐa. O metasqhmatismìc Fourierefarmìzetai gia thn epÐlush twn oloklhrwtik¸n kai diaforik¸n exis¸sewn sta opoÐa an�gwntai ta
hlektrik� kukl¸mata.
je¸rhma ( sunèlhxhc)
An oi pragmatikèc sunart seic f1(t), f2(t) eÐnai suneqeÐc tmhmatik� suneqeÐc sto R tìte up�rqoun oi
metasqhmatismoÐ touc kat� Fourier
F{f1(t)} = F1(s), F{f2(t)} = F2(s)
kai isqÔei
F{f1(t) ∗ f2(t)} = F{f1(t)}F{f2(t)} = F1(s)F2(s), ìpou (5.31)
f1(t) ∗ f2(t) =∫ t
0f1(x)f2(t− x)dx
η=
∫ t
0f2(x)f1(t− x)dx (5.32)
lègetai sunèlhxh.
Lumènec Ask seic
'Askhsh 1. Na anaptuqjeÐ se seir� Fourier h sun�rthsh
f(x) =
1 giax ∈ [−π, 0]
3 giax ∈ [0, π]
25
LÔsh: H dojeÐsa sun�rthsh plhreÐ tic sunj kec Dirichlet. Qrhsimopoi¸ntac touc tÔpouc (5.4)-(5.6)
brÐskoume touc suntelestèc Fourier:
a0 =1π
∫ π
−πf(x)dx =
1π
∫ 0
−π1dx +
1π
∫ π
03dx =
1π
[x]0−π +1π
[3x]π0 =
=1π
(0 + π) +1π
(3π − 0) =1π· 4π = 4
an =1π
∫ π
−πf(x) cos nxdx =
1π
∫ 0
−π1 cos nxdx +
1π
∫ π
03 cos nxdx =
1π
[sinnx
n
]0
−π+
1π
[3 sinnx
n
]π
0= 0
bn =1π
∫ π
−πf(x) sinnxdx =
1π
∫ 0
−π1 sinnxdx +
1π
∫ π
03 sinnxdx =
1π
[− cos nx
n
]0
−π+
1π
[− 3 cos nx
n
]π
0=
=−1nπ
[1− cos n(−π)]− 3nπ
(cos nπ − 1) =−1nπ
[1− (−1)n + 3
((−1)n − 1
)]=−2nπ
[(−1)n − 1
]⇒
bn =
0 gia n = 2k, k ∈ N∗
4(2k−1)π gia n = 2k − 1, k ∈ N
Antikajust¸ntac tic timèc a0, an, bn ston tÔpo (5.8) brÐskoume thn ap�nthsh
f(x) = 2 +∞∑
k=1
4(2k − 1)π
sin(2k − 1)x = 2 +4π
∞∑k=1
sin(2k − 1)x2k − 1
'Askhsh 2. Na anaptuqjeÐ se seir� Fourier h sun�rthsh
f(x) = cos ax, gia x ∈ [−π, π], a /∈ Z
LÔsh: H dojeÐsa sun�rthsh eÐnai �rtia orismènh sto di�sthma [−π, π] kai plhreÐ tic sunj kec Dirichlet.Qrhsimopoi¸ntac touc tÔpouc (5.15) brÐskoume bn = 0 kai
a0 =2π
∫ π
0cos axdx =
2π
[sin ax
a
]π
0=
2 sin aπ
aπ
an =2π
∫ π
0cos ax cos nxdx =
2π
12
∫ π
0[cos(a + n)x + cos(a− n)x]dx =
1π
1a + n
[sin(a + n)x]π0+
+1π
1a− n
[sin(a− n)x]π0 =sin(a + n)ππ(a + n)
+sin(a− n)ππ(a− n)
=sin(aπ + nπ)
π(a + n)+
sin(aπ − nπ)π(a− n)
=
=2(−1)n+1 sin aπ
π(n2 − a2), n ∈ N∗
epeid sin(aπ − nπ) = sin(aπ + nπ) = (−1)n sin aπ gia k�je n ∈ N.Antikajust¸ntac tic timèc a0, an, bn ston tÔpo (5.14) brÐskoume thn ap�nthsh
cos ax =2 sin aπ
2aπ+
2a
π
∞∑k=1
(−1)n+1 sin aπ
π(n2 − a2)cos nx =
2 sin aπ
π
[ 12a
+ a∞∑
k=1
(−1)n+1 cos nx
n2 − a2
]gia k�je n ∈ N.
26
'Askhsh 3. Na brejeÐ h sunhmitonik seir� Fourier thc sun�rthshc
f(x) = x, gia x ∈ [0, π]
LÔsh: Efìson y�qnoume th sunhmitonik seir� Fourier thc sun�rthshc f(x) = x, tìte gia thn f(x)isqÔoun oi tÔpoi (5.18) kai (5.19) me l = π. Qrhsimopoi¸ntac touc tÔpouc (5.19) brÐskoume bn = 0 kai
a0 =2π
∫ π
0xdx =
2π
[x2
2
]π
0= π
an =2π
∫ π
0x cos nxdx =
2π
1n
∫ π
0x(sinnx)′dx =
2πn
[x sinnx]π0 −2
πn
∫ π
0sinnxdx =
=0− 2πn
[− 1
ncos nx
]π
0=
2πn2
[cos πn− cos 0] =2
πn2[(−1)n − 1] ⇒
an =
0 gia n = 2k, k ∈ N∗
−4(2k−1)2π
gia n = 2k − 1, k ∈ N
lìgw thc paragontik c olokl rwshc kai cos nπ = (−1)n. Antikajust¸ntac tic timèc a0, an, bn ston
tÔpo (5.18) brÐskoume
x =π
2−
∞∑k=1
−4(2k − 1)2π
cos(2k − 1)x =π
2− 4
π
∞∑k=1
cos(2k − 1)x(2k − 1)2
'Askhsh 4. Na brejeÐ h seir� Fourier thc sun�rthshc
f(x) = |x|, gia x ∈ [−1, 1]
LÔsh: Efìson y�qnoume th seir� Fourier thc �rtiac sun�rthshc f(x) = |x|, tìte gia thn f(x) isqÔounoi tÔpoi (5.18) kai (5.19) me l = 1. Qrhsimopoi¸ntac touc tÔpouc (5.19) brÐskoume bn = 0 kai
a0 =2∫ 1
0xdx = 2
[x2
2
]1
0= 1
an =2∫ 1
0x cos nπxdx =
2nπ
∫ 1
0x(sinnπx)′dx =
2nπ
[x sin nπx]10 −2
nπ
∫ 1
0sinnπxdx =
=0− 2nπ
[− 1
nπcos nπx
]1
0=
2(nπ)2
[cos nπ − cos 0] =2
(nπ)2[(−1)n − 1] ⇒
an =
0 gia n = 2k, k ∈ N∗
−4(2k−1)2π2 gia n = 2k − 1, k ∈ N
lìgw thc paragontik c olokl rwshc kai cos nπ = (−1)n. Antikajust¸ntac tic timèc a0, an, bn ston
tÔpo (5.18) brÐskoume
|x| = 12−
∞∑k=1
−4(2k − 1)2π2
cos(2k − 1)x =π
2− 4
π2
∞∑k=1
cos(2k − 1)x(2k − 1)2
27
'Askhsh 5. Na brejeÐ to olokl rwma Fourier thc sun�rthshc
f(t) =
1, gia − 1 ≤ t ≤ 1
0, gia |t| > 1
LÔsh: qrhsimopoi¸ntac touc tÔpouc (5.26, (5.27, brÐskoume
A(ω) =1√π
∫ +∞
−∞f(t) cos ωtdt =
1√π
∫ 1
−1cos ωtdt =
1√π
[sinωt
ω
]1
−1=
1√π
2 sinω
ω
B(ω) =1√π
∫ +∞
−∞f(t) sinωtdt =
1√π
∫ 1
−1sinωtdt =
1√π
[− cos ωt
ω
]1
−1= 0
Kai t¸ra sÔmfwna me ton tÔpo (5.25) lamb�noume
f(t) =1√π
∫ +∞
0
[A(ω) cos ωt + B(ω) sinωt
]dω =
2π
∫ +∞
0
sin ω cos ωt
ωdω
28
'Alutec Ask seic
Na brejoÔn oi seirèc Fourier twn parak�tw sunart sewn
1. f(x) = sin ax, x ∈ [−π, π]
LÔsh : f(x) =2 sinπa
π
∞∑n=1
(−1)n+1n sinnx
n2 − a2
2. f(x) =
−1 gia − π ≤ x < 0
1 gia 0 ≤ x < π
LÔsh : f(x) =2π
∞∑k=1
sin(2k − 1)x2k − 1
3. Na brejeÐ h hmitonik seir� Fourier thc sun�rthshc
f(x) = x, x ∈ [0, 1] LÔsh : f(x) =2π
∞∑n=1
(−1)n+1 sinnπx
n
29