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Young Won Lim2/2/15
Definitions of the Laplace Transform (1A)
Young Won Lim2/2/15
Copyright (c) 2014 Young W. Lim.
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Definitions (1A) 3 Young Won Lim2/2/15
Improper Integral
limb→+∞
∫a
b
f (x ) dx
lima→−∞
∫a
b
f (x) dx
I = ∫a
b
f ( x) dx limb→+∞
I
I = ∫a
b
f ( x) dx lima→−∞
I
∫a
+∞
f (x ) dx
∫−∞
b
f (x) dx
limc→b−
∫a
c
f (x) dx
limc→a+
∫c
b
f (x) dx
I = ∫a
c
f (x) dx limc→b−
I
I = ∫c
b
f (x) dx limc→a+
I
∫a
b−
f (x) dx
∫a+
b
f (x) dx
Hiding the limiting process
L
∞
converge
diverge
L
∞
converge
diverge
L
∞
converge
diverge
L
∞
converge
diverge
Definitions (1A) 4 Young Won Lim2/2/15
Improper Integral Examples
limb→+∞
∫a
b
f (x ) dx ∫a
+∞
f (x) dx ∫a
c−
f (x) dxlimb→c−
∫a
b
f (x) dx
http://en.wikipedia.org/wiki/http://en.wikipedia.org/wiki/
∫1
+∞ 1
x2 dx = limb→∞
∫1
b1
x2 dx = limb→∞ [−1
x ]1
b
= limb→∞
(−1b+
11 ) = 1 converge
∫0
11
√xdx = lim
a→0+∫a
11
√xdx = lim
a→0+
[2√x ]a1
= lima→0+
(2−2√b ) = 2 converge
limx→ 0
f (x ) = limx →0
1
√x= ∞ f (0)
Definitions (1A) 5 Young Won Lim2/2/15
An Improper Integration
F (s) = ∫0
∞
f (t)e−s t dt
Complex Number Real Number
ℜ{s} ℑ{s}
s = σ + iω t Integration Variable
Real Number Real Number
real part
imag part
The improper integral converges if the limit defining it exists.
Definitions (1A) 6 Young Won Lim2/2/15
F(s) : a function of s
F (s) = ∫0
∞
f (t)e−s t dt
g(s , t) = f (t )e−st
∂∂ t
G (s , t ) = g(s , t )
∫0
∞
g(s , t)dt = limb→∞
[G (s , t)]0b
During integration, complex variable s is treated as a constant
∫0
∞
g(s , t)dt = F (s)
In the result, the literal t vanishes
a function of s
G: an antiderivative of g with respect to t
For a given function f(t)
= limb→∞
[G (s , b) − G (s , 0)]
Definitions (1A) 7 Young Won Lim2/2/15
An Integration Function
F (s) = ∫0
∞
f (t)e−s t dt
F (s1) = ∫0
∞
f (t )e−s1t dt
F (s2) = ∫0
∞
f (t )e−s2 t dt
F (s3) = ∫0
∞
f ( t)e−s3 t dt
F (s3)
F (s2)
F (s1)
s3
s2
s1
Complex Numbers Complex Numbers
Complex Number Real Number
s = σ + iωt Real Number
Integration Variable
For a given function f(t)
Definitions (1A) 8 Young Won Lim2/2/15
F(s) : a Complex Function
F (s1) = ∫0
∞
f (t )e−s1t dtF (s1)s1
Complex Numbers
Complex Numbers
s1 = σ 1 + iω1
F (s1) = ℜ{F (s1)} + iℑ{F (s1)}
ℜ
ℑ
ℜ
ℑ
Complex Number Real Number
s = σ + iω
For a given function f(t)
σ1
ω1
ℑ{F (s1)}
ℜ{F (s1)}
Complex Function
Definitions (1A) 9 Young Won Lim2/2/15
f(t) : a real-valued or complex-valued function
t1
f (t 1) = ℜ{f (t 1)} + i ℑ{f (t 1)}
t ℜ{s}
ℑ{s}
t1ℑ{f (t1)}
ℜ{f (t 1)}
f
t1
Real Numbers
Complex Numbers
t1
t
f (t)
t1
f (t1)
f
t1
Real Numbers
Real Numbers
f (t 1)
f (t 1)
Real-valued Function
Complex-valued Function
f (t 1)
Definitions (1A) 10 Young Won Lim2/2/15
Complex Function Plot
ℜ
ℑ
ℜ{F (s1)}
ℜ
ℑ
ℑ{F (s1)}
∣F (s1)∣
arg {F (s1)}
s1
s1
s1 = σ 1 + iω1 F (s1) = ℜ{F (s1)} + iℑ{F (s1)}
ℜ
ℑ
ℜ
ℑF
s1 = σ 1 + iω1 F (s1) = ℜ{F (s1)} + iℑ{F (s1)}
ℜ
ℑ
ℜ
ℑF
Definitions (1A) 11 Young Won Lim2/2/15
Two Functions: f(t) & F(s)
t
f (t)
ℜ{s}
ℑ{s}
ℜ{F (s1)}
ℜ{s}
ℑ{s}
ℑ{F (s1)}
∣F (s1)∣
arg {F (s1)}
s1
s1
Real number domain function f(t) Complex number domain function F(s)
For a given function f(t)
there exits a unique F(s)
t-domain function f(t)
s-domain function F(s)
f (t ) F (s)
Definitions (1A) 12 Young Won Lim2/2/15
Laplace Transform
Lf a(x ) F A (s)
f b(t ) FB (s)
f c( t) FC (s)
∫0
∞
f 1(t)e−st dt = F1(s)
∫0
∞
f 2(t)e−st dt = F2(s)
∫0
∞
f 3(t)e−st dt = F3(s)
1
e−a t
1s
1s+a
s
s2+k2
L
L
s1 = σ 1 + iω1
F (s1) = ℜ{F (s1)} + iℑ{F (s1)}
ℜ
ℑ
ℜ
ℑ
σ1
ω1
ℑ{F (s1)}
ℜ{F (s1)}
t1
f (t 1)
t
f (t)
t1
f (t1)
f a(x ) F A(s)Real-valued Function Complex Function
cos(k t)
Definitions (1A) 13 Young Won Lim2/2/15
Laplace transforms of 1 and exp(–at)
L1
e−a t
1s
1s+a
L
F(s) = ∫0
∞
1⋅e−s t dt = limb→∞ [−1
se−st ]
0
b
= limb→∞ [−1
se−sb
+1se−s 0]
limb→∞
e−s b = 0−s < 0 s > 0 F(s) =1s
F(s) = ∫0
∞
e−a t⋅e−s t dt = limb→∞ [− 1
(s+a)e−(s+a) t ]
0
b
= limb→∞ [− 1
(s+a)e−(s+a)b +
1(s+a)
e−(s+a)0]limb→∞
e−(s+a )b= 0−(s+a) < 0 s >−a F(s) =
1(s+a)
Definitions (1A) 14 Young Won Lim2/2/15
Laplace transforms of exp(+at) and exp(–at)
e+a t 1s−a
L F (s) = ∫0
∞
e+a t⋅e−s t dt = limb→∞ [− 1
(s−a)e−(s−a )t ]
0
b
= limb→∞ [− 1
(s−a)e−(s−a )b +
1(s−a)
e−(s−a )0]limb→∞
e−(s−a)b= 0−(s−a) < 0 s >+a F (s) =
1(s−a)
e−a t 1s+a
L F(s) = ∫0
∞
e−a t⋅e−s t dt = limb→∞ [− 1
(s+a)e−( s+a) t ]
0
b
= limb→∞ [− 1
(s+a)e−( s+a)b +
1(s+a)
e−(s+a)0]limb→∞
e−(s+a )b= 0−(s+a) < 0 s >−a F(s) =
1(s+a)
Definitions (1A) 15 Young Won Lim2/2/15
Laplace transforms of cosh(kt) and sinh(kt)
sinh (k t) k
s2−k2L
cosh(k t ) s
s2−k2L F (s) = ∫
0
∞ (e+k t+e−k t )
2⋅e−s t dt
s >−k1
(s+k)cosh(k t ) =
(e+k t + e−k t )
2
sinh (k t ) =(e+k t − e−k t)
2
=12∫0
∞
e+k t⋅e−st dt +
12∫0
∞
e−k t⋅e−s t dt
s >+k1
(s−k)
F (s) =12 ( 1
(s−k)+
1(s+k )) =
s
(s2−k2
)
F (s) = ∫0
∞ (e+k t−e−k t )
2⋅e−s t dt
s >−k1
(s+k)
=12∫0
∞
e+k t⋅e−st dt −
12∫0
∞
e−k t⋅e−s t dt
s >+k1
(s−k)
F (s) =12 ( 1
(s−k)−
1(s+k )) =
k
(s2−k 2
)
Definitions (1A) 16 Young Won Lim2/2/15
Laplace transforms of cos(kt) and sin(kt)
sin(k t ) k
s2+k2L
cos(k t ) s
s2+k2L F (s) = ∫
0
∞ (e+ j k t+e− j k t)
2⋅e−s t dt
s > 0
cos(k t ) =(e+ jk t + e− j k t)
2
sin(k t ) =(e+ j k t − e− j k t)
2 j
=12∫0
∞
e+ j k t⋅e−s t dt +
12∫0
∞
e− jk t⋅e−s t dt
s > 01
(s− jω)
F (s) =12 ( 1
(s− jω)+
1(s+ jω)) =
s
(s2+k 2
)
F (s) = ∫0
∞ (e+ j k t−e− j k t)
2 j⋅e−s t dt
s > 01
(s+ jω)
=12 j
∫0
∞
e+ j k t⋅e−s t dt −
12 j
∫0
∞
e− j k t⋅e−s t dt
s > 01
(s− jω)
F (s) =12 ( 1
(s− jω)−
1(s+ jω)) =
k
(s2+k2
)
1(s+ jω)
Definitions (1A) 17 Young Won Lim2/2/15
Laplace transform of cos(kt)
s
s2+k2
cos(k t) L F(s) = ∫0
∞
cos(k t )⋅e−s t dt = limb→∞ {[ 1k sin (k t )e−s t ]
0
b
−∫0
∞
−sk
sin (k t)⋅e−s t dt }sk∫0
∞
sin (k t)⋅e−s t dt = limb→∞
sk {[−1
kcos (k t)e−s t ]
0
b
−∫0
∞ sk
cos (k t)⋅e−s t dt }limb→∞ {[1k sin (k t)e−s t ]
0
b
}
limb→∞ {[−1
kcos (k t )e−s t ]
0
b
} = limb→∞
{−1kcos (k b)e−s b
+1kcos (k 0)e−s0} =
1k
= limb→∞
{1k sin (k b)e−s b−
1k
sin (k 0)e−s0} = 0
F(s) = ∫0
∞
cos(k t )⋅e−s t dt = limb→∞
sk {1k −
sk∫0
∞
cos(k t)⋅e−st dt }
F(s) =s
k 2 −s2
k2 F(s) (1+s2
k 2 )F (s) =s
k 2 ( s2+k2
k 2 )F (s) =s
k 2
F(s) =s
(s2+k2
)
Definitions (1A) 18 Young Won Lim2/2/15
Laplace transform of sin(kt)
k
s2+k2
sin(k t ) L F (s) = ∫0
∞
sin (k t )⋅e−s t dt = limb→∞ {[−1
kcos (k t)e−s t ]
0
b
−∫0
∞
+skcos (k t )⋅e−st dt }
sk∫0
∞
cos(k t )⋅e−s t dt = limb→∞
sk {[+ 1
ksin (k t)e−s t ]
0
b
−∫0
∞ sk
sin (k t)⋅e−st dt }limb→∞ {[−1
kcos (k t )e−s t ]
0
b
}
limb→∞ {[+ 1
ksin (k t)e−st ]
0
b
} = limb→∞
{+ 1k
sin (k b)e−s b −1ksin (k 0)e−s0} = 0
= limb→∞
{−1k
cos(k b)e−s b +1kcos (k 0)e−s0} =
1k
F (s) = ∫0
∞
sin (k t )⋅e−s t dt = limb→∞ {1k −
s2
k 2∫0
∞
sin (k t )⋅e−s t dt }F (s) =
1k
−s2
k 2 F (s) (1+s2
k2 )F (s) =1k ( s
2+k 2
k 2 )F (s) =1k
F (s) =k
(s2+k 2)
Definitions (1A) 19 Young Won Lim2/2/15
Integration by parts
f (x)g(x ) f ' (x )g(x ) + f ( x)g ' (x )dd x
ddx
( f g) =d fdx
g + fd gdx
f (x )g(x ) f ' (x )g(x ) + f ( x)g ' (x )
f g = ∫ f ' g dx + ∫ f g ' dx
∫⋅d x
∫ f (x )g ' (x) dx f (x )g(x ) − ∫ f ' (x )g(x ) dx=
Definitions (1A) 20 Young Won Lim2/2/15
Region of Convergence
L1 1s ∫
0
∞
e−st dt = limb→∞ [−1
se−st ]
0
b
= limb→∞ [−1
se−sb
+1se−s 0]
−s < 0
limb→∞
e−(s+a )b= 0−(s+a) < 0 s >−a F(s) =
1(s+a)
= limb→∞ [−1
se−s b +
1s ]
−(σ+iω) < 0
limb→∞
e−s b= lim
b→∞
e−(σ+iω)b= lim
b→∞
e−bσ e+i bω= 0
−σ < 0
−ℜ{s} < 0
σ > 0t > 0
|e+i bω|= 1
Right-sided function Right-sided ROC
Definitions (1A) 21 Young Won Lim2/2/15
Laplace Transform
∫t1
t 2
e+k t dt = [ 1k ek t ]t 1
t 2
=1k⋅(ek t 2 − ek t1)
∫0
∞
e+k t dt = [ 1k ek t ]0
∞
=1k⋅(e k⋅∞ − ek⋅0)
ℜ{k } < 0 ek⋅∞→ 0
ℜ{k } = 0 ek⋅∞→ e jω+
1k⋅(e jω − 1)
−1k
k = σ + jω
Definitions (1A) 22 Young Won Lim2/2/15
Laplace Transform
Right-sided function
Left-sided function
exponential order
exponential order
e−a t u( t)
e+b t u(−t )e−a t u(−t )
e+b t u(t )
a > 0 b > 0
Laplace transform exists
Laplace transform exists
exponential order
exponential order
Laplace transform exists
Laplace transform exists
Definitions (1A) 23 Young Won Lim2/2/15
Exponential Order
F (s) = ∫0
∞
f (t)e−st d t
Laplace Transform
the growth rate of a function f(t)∣ f (t)∣ ≤ M eα t ,
s > 0 ℜ(s) > 0for the integral converges if f(t) does not grow too rapidly
a function has exponential order
Exponential Order α
there exist constants and such that for some
M > 0 αt > t 0
f α
t > t 0
∫0
∞
∣ f ( t)∣e−σ t d t < ∞ for some σ
∫0
∞
∣ f ( t)e−st ∣ d t = ∫0
∞
∣ f (t)e−x t e−i y t∣ d t = ∫0
∞
∣ f ( t)e−xt∣ d t < ∫0
∞
∣ f (t)∣e−σ t d t < ∞ for
s > σ
ℜ(s) > σ
exponential order σ absolutely converges for f (t ) F (s) = ∫0
∞
f ( t)e−s t d t
s > σ
Definitions (1A) 24 Young Won Lim2/2/15
Convergence of the Laplace Transform
f(t) continuous on [0, ∞)
f(t) = 0 for t < 0
f(t) has exponential order α
f'(t) piecewise continuous on [0, ∞)
F(s) converges absolutely
for Re(s) > α
= ∫0
∞
∣f ( t)∣ e−xt d t < ∞∫0
∞
∣f (t )e−st∣ d t
x > α
∫0
∞
∣f (t )e−s t∣ d t < ∞
( ∣e−st∣ =∣e−xt ∣∣e−iyt ∣= e−xt )
{f ( t) e−xt } = g( t)
absolutely integrable for
Use Fourier Inversion
F (s) = ∫0
∞
f (t)e−st d t
Laplace Transform
= ∫0
∞
{f (t )e−xt} e−iy t d t
F(s) converges absolutely
for Re(s) > α
∫0
∞
∣f (t )e−s t∣ d t < ∞
Definitions (1A) 25 Young Won Lim2/2/15
Fourier Transform
F (x , y) = ∫0
∞
{ f (t)e−xt} e−iyt d t
g (t ) = f (t )e−xtFourier Transform
F (x , y) = ∫0
∞
g (t) e−iyt d t
g (t ) =12π
∫−∞
+∞
F (x , y)eiyt d y
f (t) =12π
∫−∞
+ ∞
F ( x , y)e xt eiyt d y
Inverse Fourier Transform
x > αabsolutely integrable forg (t ) = f (t )e−xt
f (t ) =12π
∫−∞
+ ∞
F (x , y)e( x+ iy)t d y
s = x + i y
(x > α)
d s = i d yfixed =
12π i ∫
x−i∞
x+ i∞
F (s)es t d s
s = x + i∞y =+ ∞
s = x − i∞y =−∞
= limy→∞
12π i ∫
x−i y
x+ i y
F (s)es t d s
x − i y
xα
Definitions (1A) 26 Young Won Lim2/2/15
Forward and Inverse Laplace Transform
Forward Laplace Transform
f (t ) F (s)
f (t ) F (s)
F (s) = ∫0
∞
f (t)e−s t dt
f (t) =1
2π j ∫σ− j∞
σ+ j∞
F (s)e+s t ds
Inverse Laplace Transform
Young Won Lim2/2/15
References
[1] http://en.wikipedia.org/[2] http://planetmath.org/[3] M.L. Boas, “Mathematical Methods in the Physical Sciences”[4] E. Kreyszig, “Advanced Engineering Mathematics”[5] D. G. Zill, W. S. Wright, “Advanced Engineering Mathematics”[6] T. J. Cavicchi, “Digital Signal Processing”[7] F. Waleffe, Math 321 Notes, UW 2012/12/11[8] J. Nearing, University of Miami[9] http://scipp.ucsc.edu/~haber/ph116A/ComplexFunBranchTheory.pdf