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0,
>
s
−st (t) dt
−st (t)|dt,
A
f
f
∫
τ
|e
∫
lim
θ---real,lim
,L,
0
A→∞
0
t
0
τ→0
=
≥
βL[g(t)].
−st (t) dt
αt t
,
0),
f
Me
∞
+
∫
≥
−as (s)(a
e
αL[f(t)]
≤
isin θ,
F
0,|f(t)|
=
sL(f(t))
=
F(s)
+
e
0
−
=
βg(t)]
f(0)
cos θ
→
(t)f(t−a))
=
−st (t)|dt
=
−
+
(t))
+ −
=
)
iθ
L[f(t)]
′
f
n−1 (0
−1 (s),e
f
a
′
a),L(u
τ
|e
∫
=
F
s
>
(t))
=

tau
>
−
f)(t)]
→
′
1, 2, 3, . . . (s
L
L(f(t))
(areal, Re(s)
+ L(f
=
0asRe(s)
≤

f(t)
).
n
−st (t)dt
∗

L[(1
f(0
s
f
=
]
=
′
=
τ
→

(t))
sL(f(t))
e
∫
n
(n)
α).
L(f(t))
n (t)),
t
at (t)),

α).L(f
)
>
[∫
τ
f
f
n
=

t
(s
=
L(e
(t))
f(t)
L((−1)
(
F(s)
̸=
>
=
′
L
α),L(f
=
(s)]
F(s−a)
− (Re(s)
t
0
+ L(g(t))
=
2
cg,
L
d F(s)
>
))
=
n
[F
)(Re(s)
).
−1
1
f
f(t
F(x)dx
1
(0
n
(n−1)
g
ds
+
+ −
dx.
0,L
2
∞
)
−x
∫
cf
f(0
−t (f(t
=
≥
f
1
s
e
t
g)
−
1
α).
s
f(t),
· · ·
∗
2
τ)dτ,c(f
e
0
>
=
−
=
π
n−2 (0+)
=
)dt
L(f(t))(Re
h),erf(t)
′
− 0
f
−
−1
f(τ)g(t
(t)δ(t
1
>
b
s
t
∫
0,L
∗
t
(f
−εs
=
s
iyt (x, y)dy,
>
+
.
a
0
=
L[1]L[f(t)]
g)
(t)]
s
,
εs
∗
(f
L[f
T
−sT
=
=
∗
xt
x+iy ts (s)ds.f(t)
(s)].(f
e
h)
1−e
dt
−st
∞
e
+
>
=
∫
ε
(g
∫
[F
L[f(t)]
−1
,
2
e
F
∗
1
h,f
1
t
e
0
(s)]L
c
2π
k
= ε
∗
=
g)
∫
x−iy
[F
−1
1
dt
−st
f(a),f(t)
∗
lim
∞
(f
∑
=
L
(t)e
h)
0,
y→∞
∞
=
ε
f
∗
>
∫
(g
=
=
).f(t)
s
∗
a)dt
f
ts (s)ds
0
−st (t) dt
=
−st (t)|dt,
Res(z
k
(t)]
F
−
A
f
f(t)δ(t
x+i∞
f
e
n
ε
L[f
∑
∫
b
∫
τ
k=1
∫
|e
∫
lim
x−i∞
θ---real,lim
).
a
=
1
,L,
0
f(t
0
x+iy ts (s)ds
2πi
A→∞
t
0
=
τ→0
=
F
βL[g(t)].
≥
−st (t) dt
0f(t)
αt t
,
∫
0),
f
∞
Me
x−iy
∫
+
1
e
αL[f(t)]
lim ∞,L[αf(t) e − g)(t) e (F(s)) α).L(f f(τ)dτ sL(f(t)) 1−e −∞ − ∫ (s)F √ ∗ F ∫ 0 e = t k=1 t ∗ (k−1)! = k−1 t −as (s)(a ≥ −
≤
isin θ,
2πi
F
=
0,|f(t)|
y→∞ = F(s) 0 cos θ + → βg(t)] = = e = sL(f(t)) f(0) −
(t)f(t−a))
−
+
0.L[f(t)] −1 (s),e iθ = ∫ τ ′ |e −st (t)|dt . . .Andriy Bandura, n−1 (0 + −
(t))
sL(f(t))
)
∞,L[αf(t)
′
f
α).L(f
f
a
a),L(u
=
F
>Ovchar,
L(f(t))
(areal, Re(s)
+ L(f
= L ≤ tau → > Ihor ′ (t)) − s f)(t)] =
f(t) −st (t)dt 0asRe(s) 1, 2, 3, . . . (s ). s n L[(1 ∗
Oksana
f(0 Vytvytska
∫ τ ′ e f → n = − (t)) = ] =
L(f(t)) at (t)), n (t)), sL(f(t)) (n) t f(τ)dτ ) α).
τ = f n t f = α).L(f [∫ (s >
F(s) = L(e L((−1) ′ (t)) > = L ( f(t) ̸=
.
F(s−a) = > α),L(f )) + L(g(t)) 0 = L t (s)F 2 (s)] ∗ cg,
− (Re(s)
d F(s)
n
).
F(x)dx
(0
ds n + )(Re(s) ) f(t 1 (n−1) OPERATIONAL −1 [F 1 ∗ g = f 2 dx.
+ −
0,L
∞
∫
f(0 −t (f(t 1 − f t ≥ g) = cf ∫ t e −x
s
α).
1
s
f(t),
2
τ)dτ,c(f
e − · · · CALCULUS ∗ √ π 0 =
>
=
n−2 (0+) L(f(t))(Re (F(s)) − h),erf(t) = − 0 )dt
′
t
f
s 1 −1 ∫ t f(τ)g(t ∫ b (t)δ(t >
= s > 0,L = 0 + (f ∗ −εs . a t
L[1]L[f(t)] L[f T (t)] , s (s)].(f ∗ g)(t) h) = (f ∗ g) dt = 1−e εs xt e iyt (x, y)dy, =
F
−sT
1−e
−st
+
e
ε
(g
∫
L[f(t)] = (s)]L LECTURES ∫ ∞ x+iy ts (s)ds.f(t) k t k−1 t >
[F
,
−1
2
e
F
−∞
∗
1
h,f
1
e
0
c
2π
−1 [F 1 (f ∗ g) ∗ −st dt = ε = ∫ x−iy ∑ ∞ (k−1)!
L h) = (t)e f(a),f(t) lim k=1 0,
(g ∗ ∫ ∞ f ε = = y→∞ = >
f ∗ = 0 a)dt ).f(t) s
(t)] − ts (s)ds Res(z k A −st (t) dt −st (t)|dt,
F
f
L[f ε ∫ b f(t)δ(t ∫ x+i∞ e ∑ n k=1 ∫ e ∫ τ |e f
). a 1 x−i∞ = lim ,
f(t 0 = 2πi x+iy ts (s)ds = A→∞ 0 τ→0 0 ≥ t 0
F
0f(t) ∫ x−iy e ∫ −st (t) dt θ---real,lim Me αt t αL[f(t)] +
,
f
∞
=
1
lim 2πi = e isin θ, 0,|f(t)| ≤ = − a)) α).
y→∞ = F(s) 0 cos θ + → + βg(t)] a (t)f(t > + ).
1, 2, 3, . . . (s
0.L[f(t)] −1 (s),e = ∫ τ ′ |e −st (t)|dt ∞,L[αf(t) > a),L(u sL(f(t)) − f(0 (t)) = =
iθ
f
(n)
=
α).L(f
F
→
n (t)),
=
(t))
f
f(t) = L −st (t)dt ≤ tau 0asRe(s) (areal, Re(s) n ′ − (Re(s) > + L(g(t)) =
n
t
).
f
f(t
∫ τ ′ e f → at (t)), L((−1) > α),L(f )) (n−1) (0 ∫ ∞ F(x)dx
L(f(t)) = L(e = + ) − 1 f α). s
n
−t (f(t
τ = a) d F(s) + )(Re(s) 1 · · · − > 0,
s
L,F(s) − ≥ sL(f(t)) − f(0) − e n−2 (0+) − 1 L(f(t))(Re f(t), t ≥
n
1
f(0
ds
0),
βL[g(t)].F(s
′
f
F
)
e −as (s)(a = sL(f(t)) − + − s L[1]L[f(t)] = s −1 (F(s)) ∫ = t f(τ)g(t − τ)dτ, =
(t))
h)
′
f
L(f ′ (t)) = s n−1 (0 = > 0,L 0 (g + dt =
L(f − ∗ f)(t)] T (t)] , s g)(t) = h,f ∗ (t)e −st
n L(f(t)) ] = L[(1 L[f −sT ∗ g) ∗ ∫ ∞ f ε f(a),
s t = 1−e (s)].(f (f ∗ =
[∫ f(τ)dτ α).L[f(t)] −1 [F 2 h) = (t)] = 0 − a)dt =
L ) (s)]L (g ∗ 2 dx.L[f ε ∫ b f(t)δ(t ts (s)ds ).
F
0 (s > −1 [F 1 ∗ −x x+i∞ e k
( f(t) L cg,f ∫ t e 0 ). a ∫ n Res(z
t ̸= f ∗ 2 0 = f(t 1 x−i∞ ∑ k=1
L (s)F 2 (s)] ∗ g = = √ π t )dt = 2πi =
−1 [F 1 = cf h),erf(t) ∫ b (t)δ(t − 0 0f(t) x+iy ts (s)ds
F
L g) ∗ −εs t > ∫ e
a
c(f ∗ g) + (f 1−e εs . iyt (x, y)dy, 1 x−iy
(f ∗ −st dt = xt e F lim 2πi
∫ ε e ∫ ∞ e = y→∞
x+iy ts (s)ds.f(t)
1 1 −∞
ε 0 = 2π F > 0.
,
f(t) ∫ e t k−1 t
c
k
lim x−iy ∞ (k−1)!
y→∞ ∑ k=1
f(t) =