Appendix 5
Tables of Laplace Transforms
Last update: February 2, 2008
Many Laplace transforms and their inverses have been derived and compiled. Extensive
compilations of Laplace transforms are available from many sources, including entire
books (see for example, Oberhettinger and Badii, 1973). In this appendix we reproduce
the Laplace transforms compiled in Churchill’s excellent textbook Operational
Mathematics (3rd edition, 1972). These transforms are supplemented with transforms
compiled by Carslaw and Jaeger (1959) and Hantush (1964). The transforms compiled in
Carslaw and Jaeger (1959) [Conduction of Heat in Solids, 2nd edition, 1959] arise
frequently in solute transport applications. M.S. Hantush included a short table of
Laplace transforms in his landmark publication Hydraulics of Wells (1964). These
transforms are also reproduced here because the transforms are of immediate application
to solutions for aquifer tests.
Laplace transforms.doc
Page 1 of 15
1. Laplace Transform Operations (adapted from Churchill, 1972)
f ( p ) ( Re p > α )
∞
1
∫e
− pt
F ( t ) dt = L { F }
F(t) ( t > 0 )
F (t )
0
γ +iβ
2
f ( p)
1
lim ∫ etz f ( z ) dz
2π i β →∞ γ −iβ
3
pf ( p ) − F ( 0 )
4
p n f ( p ) − p n −1 F ( 0 ) − p n − 2 F ' ( 0 ) − ... − F ( n −1) ( 0 )
∂F
∂t
∂n F
F ( n) ( t ) ≡ n
∂t
5
1
f ( p)
p
t
∫ F (τ ) dτ
0
6
e − ap f ( p )
7
f ( p − a)
8
F '(t ) ≡
( a > 0)
f '( p)
10
f ( n) ( p )
e F ( t ) a = constant
( −1)
n
t n F (t )
1
F (t )
t
f ( cp )
1 ⎛t⎞
F⎜ ⎟
c ⎝c⎠
∫e
− pt
( c > 0)
F ( t ) dt
0
a
15
0
∫e
− pt
SHIFT THEOREM
−tF ( t )
1 − e − ap
14
0
∫ f ( x ) dx
a
13
t
CONVOLUTION
p
12
t
∫ F (τ ) G ( t − τ ) dτ = ∫ F ( t − τ ) G (τ ) dτ
∞
11
if t < a
0
at
f ( p) g ( p)
9
F ( t − a ) if t > a
F ( t ) dt
0
( a > 0)
( a > 0)
1 + e − ap
∞
an
( k > 0)
∑
n+k
n =0 p
Laplace transforms.doc
F ( t ) if F ( t + a ) = F ( t )
F ( t ) if F ( t + a ) = − F ( t )
∞
an
∑ Γ (n + k ) t
n + k −1
n =0
Page 2 of 15
2. Laplace Transforms (adapted from Churchill, 1972)
f ( p)
1
2
1
p
1
p2
F(t) ( t > 0 )
1
t
3
1
pn
( n = 1, 2,...)
t n −1
( n − 1)!
3a
1
p v +1
( v > −1)
tv
Γ ( v + 1)
4
5
6
7
8
9
1
p
1
p p
p
11
12
Carslaw and Jaeger (1959; #2)
1
πt
2
⎛ 1⎞
−⎜ n + ⎟
⎝ 2⎠
( n = 1, 2,...)
Γ(k )
pk
1
p−a
1
( k > 0)
t
π
n
n−
1
2
2 t
1⋅ 3 ⋅ 5... ⋅ ( 2n − 1) π
t k −1
e at
( p − a)
te at
2
1
10
Hantush (1964; #2)
( n = 1, 2,...)
( p − a)
Γ(k )
( k > 0)
k
( p − a)
n
1
( p − a )( p − b )
p
1
t n −1e at
( n − 1)!
t k −1e at
1
( eat − ebt )
a −b
13
( p − a )( p − b )
1
ae at − bebt )
(
a −b
14
1
( p − a )( p − b )( p − c )
b − c ) e at + ( c − a ) ebt + ( a − b ) ect
(
−
( a − b )( b − c )( c − a )
Laplace transforms.doc
Page 3 of 15
f ( p)
15
16
17
18
19
20
F(t) ( t > 0 )
a
p + a2
p
2
p + a2
1
2
p − a2
p
2
p − a2
1
2
p ( p + a2 )
sin at
2
cos at
1
sinh at
a
cosh at
1
(1 − cos at )
a2
1
p2 ( p2 + a2 )
1
( at − sin at )
a3
1
21
(p
2
+a
)
1
( sin at − at cos at )
2a 3
)
t
sin at
2a
)
1
( sin at + at cos at )
2a
2 2
p
22
(p
2
+a
2 2
p2
23
(p
2
+a
2 2
p2 − a2
24
(p
2
+ a2 )
25
(p
2
+a
t cos at
2
p
2
)( p
2
1
26
27
28
29
30
( p − a)
2
+b
2
p−a
( p − a)
2
+ b2
3a 2
p3 + a3
4a 3
p 4 + 4a 4
p
4
p + 4a 4
Laplace transforms.doc
+b
2
)
(a
2
≠ b2 )
cos at − cos bt
b2 − a 2
1 at
e sin bt
b
e at cos bt
αt
⎛
at 3
at 3 ⎞
− 3 sin
e − at − e 2 ⎜⎜ cos
⎟
2
2 ⎟⎠
⎝
sin at cosh at − cos at sinh at
1
sin at sinh at
2a 2
Page 4 of 15
f ( p)
31
32
F(t) ( t > 0 )
1
p − a4
p
4
p − a4
1
( sinh at − sin at )
2a 3
1
( cosh at − cos at )
2a 2
4
8a 3 p 2
33
(p
2
+a
(1 + a t ) sin at − at cos at
2 2
)
2 3
n
34
35
1 ⎛ p −1 ⎞
⎜
⎟
p⎝ p ⎠
p
( p − a) p − a
Ln ( t ) =
Ln(t) is the Laguerre polynomial of degree n
1
πt
36
p −a − p −b
37
1
p +a
39
p
p + a2
1
πt
πt
40
1
p ( p − a2 )
41
1
p ( p + a2 )
b −a
2
44
45
2
a π
( p − a ) (b +
p +a
p
Laplace transforms.doc
2
∫ exp ( λ ) d λ
2
0
2
)
)(
a t
e
− a 2t
∫
eλ d λ
2
0
( )
( )
e a t erfc a t
1
e − at erf
b−a
b2 − a 2
p(p−a
π
a t
e− a t
( )
)
p+b
2
2a
2
2
e a t ⎡b − a erf a t ⎤ − b eb t erfc b t
⎣
⎦
1
( p + a)
−
( )
2
1
( )
2
+ ae a t erf a t
1 a 2t
e erf a t
a
2
(
− e at )
bt
( )
1
p
(e
2 πt
2
1
− ae a t erfc a t
πt
3
p
p − a2
43
e at (1 + 2at )
1
38
42
et d n n − t
(t e )
n ! dt n
p +b
)
(
b−a t
)
2 ⎡b
2
⎤
e a t ⎢ erf a t − 1⎥ + eb t erfc b t
⎣a
⎦
( )
( )
Page 5 of 15
f ( p)
46
(1 − p )
p
47
n+
F(t) ( t > 0 )
n!
H 2n
( 2n ) ! π t
n
Hn(x) is the Hermite polynomial
1
2
Hn ( x) = e
(1 − p )
p + 2a
p
50
53
54
e
Γ (k )
( p + a) ( p + b)
4
( k > 0)
4
1
p + a p + b ( p + b)
(
(
4
p+a + p+b
p+a + p
)
)
2k
( k > 0)
−2v
( v > −1)
p p+a
1
55
56
p2 + a2 − p
)
( t)
1
+ a2 )
Jn
1
( a + b )t
2
is Bessel’s function of the first kind
⎛ a −b ⎞
I0 ⎜
t⎟
⎝ 2 ⎠
⎛ t ⎞
π⎜
⎟
⎝ a −b⎠
−
1
( a + b )t
2
k−
1
2
e
−
1
( a + b )t
2
I
1
k−
2
⎛ a −b ⎞
t⎟
⎜
⎝ 2 ⎠
⎡ ⎛ a −b ⎞
⎛ a − b ⎞⎤
⎢ I 0 ⎜ 2 t ⎟ + I1 ⎜ 2 t ⎟ ⎥
⎠
⎝
⎠⎦
⎣ ⎝
k − 12 ( a +b ) t ⎛ a − b ⎞
e
Ik ⎜
t⎟
t
⎝ 2 ⎠
1 − 12 at ⎛ 1 ⎞
e I v ⎜ at ⎟
av
⎝2 ⎠
v
p2 + a2
(p
2
58
(
p2 + a2 − p
k
( v > −1)
( k > 0)
57
Laplace transforms.doc
2
J 0 ( at )
p2 + a2
(
( )
1 − at
e I1 ( at )
t
p + 2a + p
( a − b)
−
te
p + 2a − p
52
⎛ d n ⎞ −x
⎜ dx n ⎟ e
⎝
⎠
I n ( x ) = I − n J n ( Ix )
−1
1
p+a p+b
51
2
ae − at ⎡⎣ I1 ( at ) + I 0 ( at ) ⎤⎦
where
49
x
n!
H 2 n +1
π ( 2n + 1) !
−
p n +1 p
48
( t)
)
k
( k > 0)
a v J v ( at )
π ⎛ t ⎞
⎜ ⎟
Γ ( k ) ⎝ 2a ⎠
k−
1
2
J
k−
1
2
( at )
ka k
J k ( at )
t
Page 6 of 15
59
f ( p)
(p−
p2 − a2
)
v
p2 − a2
1
60
(s
61
e − kp
p
62
e − kp
p2
63
e − kp
pµ
64
1 − e − kp
p
65
2
− a2 )
k
67
68
69
70
71
72
73
( v > −1)
( k > 0)
a v I v ( at )
π ⎛ t ⎞
⎜ ⎟
Γ ( k ) ⎝ 2a ⎠
k−
1
2
I
k−
1
2
( at )
⎧0 when 0 < t < k ⎫
H (t − k ) = ⎨
⎬
⎭
⎩1 when t > k
when 0 < t < k
=0
Heaviside step function
= t − k when t > k
=0
when 0 < t < k
( µ > 0)
1
=
p (1 − e − kp )
µ −1
(t − k )
=
Γ(µ )
when t > k
= 1 when 0 < t < k
=0 when t > k
1
1 + coth kp
2
2p
1
66
F(t) ( t > 0 )
p ( e kp − a )
⎡t ⎤
1 + ⎢ ⎥ = n when ( n − 1) k < t < nk
⎣k ⎦
=0
M ( 2k , t ) = ( -1)
1
tanh kp
p2
1
p sinh kp
H ( 2k , t )
1
coth kp
p
k
πp
coth
2
2
2k
p +k
Laplace transforms.doc
when 0 < t < k
= 1 + a + a 2 + ... + a n −1 when nk < t < ( n + 1) k ( n = 1, 2,...)
1
tanh kp
p
1
p (1 + e − kp )
1
p cosh kp
( n = 1, 2,...)
n-1
when 2k ( n − 1) < t < 2kn ( n = 1, 2,...)
1
1 1 − ( -1)
when ( n − 1) k < t < nk
M (k, t ) + =
2
2
2
n
F ( t ) =2 ( n − 1) when ( 2n − 3) k < t < ( 2n − 1) k ( t > 0 )
M ( 2k , t + 3k ) +1=1 + ( -1)
n
when ( 2n − 3) k < t < ( 2n − 1) k ( t > 0 )
F ( t ) =2n − 1 when 2k ( n − 1) < t < 2kn
sin kt
Page 7 of 15
f ( p)
74
1
( p + 1)(1 − e−π p )
75
1 −p
e
p
2
F(t) ( t > 0 )
1
( sin t + sin t )
2
(
k
J 0 2 kt
k
76
1 −p
e
p
77
1 p
e
p
1
78
79
1
p p
1
p p
1
e
k
p
−
k
p
e
80
81
1 p
e
pµ
82
e− k
83
1 −k
e
p
1
πk
( µ > 0)
k
( µ > 0)
( k ≥ 0)
p
83A
1 −k
e
p−a
84
1 −k
e
p
ae − k
86
(
e− k
87
p
p
s a+ p
)
( k ≥ 0)
( k ≥ 0)
p
(
p a+ p
Laplace transforms.doc
⎛t⎞
⎜ ⎟
⎝k⎠
)
sinh 2 kt
( µ −1)
2
( µ −1)
2
J µ −1 2 kt
(
)
(
)
I µ −1 2 kt
⎛ k2 ⎞
exp ⎜ − ⎟
2 π t3
⎝ 4t ⎠
⎛ k ⎞
erfc ⎜
⎟
⎝2 t ⎠
1 at ⎡ − k a
⎧ k
⎫
⎧ k
⎫⎤
e ⎢e
erfc ⎨
− at ⎬ + e k a erfc ⎨
+ at ⎬⎥
2 ⎣
⎩2 t
⎭
⎩2 t
⎭⎦
⎛ k2 ⎞
exp ⎜ − ⎟
πt
⎝ 4t ⎠
1
( k ≥ 0)
p
e− k
( a > 0)
p
⎛t⎞
⎜ ⎟
⎝k⎠
sin 2 kt
k
( k > 0)
p
p p
1
πk
k
1
cosh 2 kt
πt
1 −p
e
pµ
85
cos 2 kt
πt
k
)
( k ≥ 0)
2
⎛ k2 ⎞
⎛ k ⎞
exp ⎜ − ⎟ − k erfc ⎜
⎟
π
⎝2 t ⎠
⎝ 4t ⎠
t
2
k
⎛
−e ak e a t erfc ⎜ a t +
2 t
⎝
⎞
⎛ k ⎞
⎟ + erfc ⎜
⎟
⎠
⎝2 t ⎠
2
k ⎞
⎛
e ak e a t erfc ⎜ a t +
⎟
2 t⎠
⎝
Page 8 of 15
f ( p)
88
89
90
91
e
F(t) ( t > 0 )
=0
− k p( p + a )
p ( p + a)
e− k
=e
=0
p2 −a2
−k
(
(
)
)
+ 2kt )
= I0 a t 2 − k 2
p2 − a2
e
(
⎛1
⎞
I 0 ⎜ a t 2 − k 2 ⎟ when t > k
⎝2
⎠
when 0 < t < k
= J0 a t 2 − k 2
p2 + a2
e− k
1
− at
2
=0
p2 + a2
when 0 < t < k
p2 +a2 − p
)
p2 + a2
(
( k ≥ 0)
J0 a t 2
when t > k
when 0 < t < k
when t > k
=0
92
e − kp − e − k
p2 + a2
=
=0
93
e− k
p −a
2
2
− e − kp
av e− k
94
p +a
2
95
1
ln p
p
96
1
ln p
pk
97
98
99
100
101
2
(
=
p2 + a2
p +a + p
2
2
( v > −1)
ak
t −k
2
2
t −k
)
when t > k
when 0 < t < k
ak
2
(
J1 a t 2 − k 2
2
(
I1 a t 2 − k 2
)
when t > k
when 0 < t < k
1
v
2
(
⎛t−k ⎞
2
2
=⎜
⎟ Jv a t − k
t
+
k
⎝
⎠
)
when t > k
Γ ' (1) − ln t ⎡⎣ Γ ' (1) = −0.5772 ⎤⎦
( k > 0)
ln p
( a > 0)
p−a
ln p
p2 + 1
p ln p
p2 + 1
1
ln (1 + kp ) ( k > 0 )
p
p−a
ln
p −b
Laplace transforms.doc
)
v
=0
when 0 < t < k
t
k −1
⎧⎪ Γ ' ( k )
ln t ⎫⎪
−
⎨
⎬
2
⎪⎩ ⎣⎡ Γ ( k ) ⎦⎤ Γ ( k ) ⎭⎪
e at ⎡⎣ ln a + E1 ( at ) ⎤⎦
E1(t) is the exponential-integral function
cos t Si t – sin t Ci t
Si is the sine-integral function
Ci is the cosine-integral function
sin t Si t –cos t Ci t
⎛t⎞
E1 ⎜ ⎟
⎝k⎠
1 bt at
(e − e )
t
Page 9 of 15
f ( p)
102
103
104
1
ln (1 + k 2 p 2 )
p
1
ln ( p 2 + a 2 ) ( a > 0 )
p
1
ln ( p 2 + a 2 ) ( a > 0 )
2
p
105
ln
106
ln
107
108
p2 + a2
p2
p2 − a2
p2
k
arctan
p
1
k
arctan
p
p
2
p2
2 ln a − 2 Ci ( at )
2
⎡ at ln a + sin at − at Ci ( at ) ⎤⎦
a⎣
2
(1 − cos at )
t
Si ( kt )
erfc ( kp )
( k > 0)
ek
110
1 k 2 p2
e erfc ( kp )
p
111
e kp erfc kp
112
1
erfc
p
113
1 kp
e erfc
p
114
⎛t⎞
−2Ci ⎜ ⎟
⎝k⎠
2
(1 − cosh at )
t
1
sin kt
t
109
(
F(t) ( t > 0 )
( k > 0)
( k > 0)
kp
)
(
kp
⎛ t2 ⎞
exp ⎜ − 2 ⎟
k π
⎝ 4k ⎠
⎛ t ⎞
erf ⎜ ⎟
⎝ 2k ⎠
1
k
π t (t + k )
=0
when 0 < t < k
= (π t )
)
⎛ k ⎞
erf ⎜
⎜ p ⎟⎟
⎝
⎠
−1/ 2
when t > k
1
( k > 0)
π (t + k )
(
1
sin 2k t
πt
k
⎛ k ⎞
1 p
e erfc ⎜
⎜ p ⎟⎟
p
⎝
⎠
)
2
115
116
117
π e− kp I 0 ( kp )
e − kp I1 ( kp )
1
πt
t
= ⎡⎣t ( 2k − t ) ⎤⎦
−1/ 2
=0
=
when 0 < t < 2k
when t > 2k
k −t
π k t ( 2k − t )
=0
Laplace transforms.doc
e −2 k
when 0 < t < 2k
when t > 2k
Page 10 of 15
f ( p)
118
e ap E1 ( ap )
119
1
− pe ap E1 ( ap )
a
120
121
⎛π
⎞
⎜ − Si p ⎟ cos p + Ci p sin p
⎝2
⎠
⎛π
⎞
⎜ − Si p ⎟ sin p − Ci p cos p
⎝2
⎠
Laplace transforms.doc
F(t) ( t > 0 )
1
t+a
1
( a > 0)
(t + a )
2
( a > 0)
1
t +1
t
2
t +1
2
Page 11 of 15
3. Laplace Transforms from Carslaw and Jaeger (1959)
Carslaw and Jaeger (1959) write q =
p
κ
.
κ and x are always real and positive. α and h are unrestricted.
F(t)
f(p)
x
− qx
e
−
x
4κ t
6
e
7
e − qx
q
⎛κ ⎞
⎜ ⎟
⎝ πt ⎠
8
e − qx
p
erfc
9
e − qx
pq
x
x
⎛ κt ⎞2 −
2 ⎜ ⎟ e 4κ t − x erfc
2 κt
⎝π ⎠
10
e − qx
p2
⎛
x2
t
+
⎜
⎝ 2κ
2 πκ t 3
e − qx
11
1
1+ n
2
1/ 2
e
−
(t > 0)
2
x2
4κ t
x
2 κt
1
2
1
x
⎞
x
⎛ t ⎞ 2 − 4κ t
− x⎜
⎟ erfc
⎟ e
2 κt
⎝ πκ ⎠
⎠
1
x
n
( 4t ) 2 i n erfc
2 κt
n = 0,1, 2,...
p
2
∞
i n erfc {u} = ∫ i n −1erfc {β } d β ,
u
the nth repeated integral of the complementary error function with
i erfc {u} = erfc {u}
0
1
12
e − qx
q+h
13
e − qx
q (q + h)
x
2
⎧ x
⎫
⎛ κ ⎞ 2 − 4κ t
− hκ e hx +κ th erfc ⎨
+ h κt ⎬
e
⎜ ⎟
⎝πt ⎠
⎩2 κt
⎭
2
⎧ x
⎫
κ ehx +κ th erfc ⎨
+ h κt ⎬
⎩2 κt
⎭
14
e − qx
p ( q + h)
2
1
x
1
⎧ x
⎫
− e hx +κ th erfc ⎨
+ h κt ⎬
erfc
h
2 κt h
⎩2 κt
⎭
2
1
15
e − qx
pq ( q + h )
Laplace transforms.doc
2
2 ⎛ κ t ⎞ 2 − 4xκ t (1 + hx )
x
1
−
+ 2 e hx +κ th
erfc
⎜ ⎟ e
2
h⎝ π ⎠
h
2 κt h
2
⎧ x
⎫
+ h κt ⎬
* erfc ⎨
⎩ 2 κt
⎭
Page 12 of 15
F(t)
f(p)
κ
16
( −h )
e − qx
q n +1 ( q + h )
n
2
κ
⎧ x
⎫
+ h κt ⎬ −
e hx +κ th erfc ⎨
n
⎩ 2 κt
⎭ ( −h )
*i r erfc
19
2
2
p ( q + h)
2
a
a
1 α t ⎧⎪ − x κ
⎡ x
⎤ x κ
⎡ x
⎤ ⎫⎪
− α t ⎥ + e erfc ⎢
+ αt ⎥⎬
e ⎨e
erfc ⎢
2 ⎪⎩
⎣2 κt
⎦
⎣2 κt
⎦ ⎭⎪
e − qx
p −α
1 − qx
e
p 3/ 4
21
1
K 2 v ( qx )
p1/ 2
1
1 ⎛⎜ x
1
π⎜
⎝ 2tκ 2
1
2 πt
I v ( qx ') K v ( qx ) ( x > x ')
I v ( qx ) K v ( qx ') ( x < x ')
e
−
⎞ 2 − x2
2
⎟ e 8κ t K 1 ⎛⎜ x ⎞⎟
⎟
8κ t ⎠
4 ⎝
⎠
x2
8κ t
⎛ x2 + x' 2 ⎞
⎟
4κ t ⎟⎠
1 −⎜⎜⎝
e
2t
⎛ x2 ⎞
Kv ⎜
⎟
⎝ 8κ t ⎠
⎛ xx ' ⎞
Iv ⎜
⎟
⎝ 2κ t ⎠
( v ≥ 0)
2
23
K 0 ( qx )
1 − 4xκ t
e
2t
24
1 x/ p
e
p
I 0 ⎡⎣ 2 xt ⎤⎦
25
2
2
1
x
2 ⎛ κ t ⎞ 2 − 4xκ t 1
− ⎜ ⎟ e
− 2 (1 − hx − 2h 2κ t ) e hx +κ th
erfc
2
h
h
2 κt h ⎝ π ⎠
⎧ x
⎫
+ h κt ⎬
*erfc ⎨
⎩ 2 κt
⎭
e − qx
20
22
r
2 κt
1
18
⎡ −2h κ t ⎤
∑
⎣
⎦
r =0
2
⎛ κ 3t ⎞ 2 − 4xκ t
−2 h ⎜
+ κ (1 + hx + 2h 2κ t ) e hx +κ th
⎟ e
⎝ π ⎠
⎧ x
⎫
*erfc ⎨
+ h κt ⎬
⎩ 2 κt
⎭
e − qx
(q + h)
n −1
x
1
17
(t > 0)
1
⎧
⎫
exp ⎨ xp − x ⎡⎣( p + a )( p + b ) ⎤⎦ 2 ⎬
⎩
⎭
e
−
1
( a + b )( t + x )
2
1
⎡⎣( p + a )( p + b ) ⎤⎦ 2
1
⎧1
⎫
I 0 ⎨ ( a − b ) ⎡⎣t ( t + 2 x ) ⎤⎦ 2 ⎬
⎩2
⎭
∞
26
p
1
v −1
2
(
Kv x p
)
Laplace transforms.doc
x − v 2v −1 ∫ e − u u v −1du
x2
4t
Page 13 of 15
F(t)
f(p)
v
27
⎡ p − p2 − x2 ⎤
⎣
⎦
28
1
1 2⎫
⎧⎪ ⎡
⎤ ⎪
exp ⎨ x ⎢( p + a ) 2 − ( p + b ) 2 ⎥ ⎬
⎦ ⎪⎭
⎪⎩ ⎣
1
1
1
1 2v
⎡
2
2
2
( p + a ) ( p + b ) ⎢( p + a ) + ( p + b ) 2 ⎤⎥
⎣
⎦
( v > 0)
29
( v > 0)
( p −α )
vx v I v ( xt )
t
1
v −
t2 e
1
( a + b )t
2
1
1⎫
⎧1
Iv ⎨ ( a − b ) t 2 (t + 4x ) 2 ⎬
⎩2
⎭
v
2
x
⎛
+⎜t +
⎝ 2 κα
31
32
33
(a ≠ κ h )
1
ln p
p
α
κ
x
1
2
κ2
+
1
1
)
Laplace transforms.doc
⋅e
x
α
κ
⎡ x
⎤⎪⎫
+ αt ⎥⎬
erfc ⎢
⎣ 2 κt
⎦ ⎪⎭
hκ 2 − α 2
2
hκ
⎡ x
⎤
− 2
+ h κt ⎥
e hx + h κ t erfc ⎢
h κ −α
⎣ 2 κt
⎦
− ln ( Ct )
(
α
⎡ x
⎤⎪⎫
+ αt ⎥⎬
erfc ⎢
⎣2 κt
⎦ ⎭⎪
1
⎧
α
−x
1 αt ⎪ κ 2
⎡ x
⎤
e ⎨ 1
e κ erfc ⎢
− αt ⎥
1
2 ⎪ 2
⎣ 2 κt
⎦
2
⎩ hκ + α
−e
e − qx
( p − α )( q + h )
p Kv x p
α
⎞ −x κ
⎡ x
⎤
erfc ⎢
− αt ⎥
⎟e
⎠
⎣2 κt
⎦
1 α t ⎛ κ ⎞ 2 ⎪⎧ − x κ
⎡ x
⎤
− αt ⎥
e ⎜ ⎟ ⎨e
erfc ⎢
2 ⎝ α ⎠ ⎪⎩
⎣2 κt
⎦
e − qx
q ( p −α )
1
v
2
v
α
⎞ x κ
⎡ x
⎤⎪⎫
e
erfc ⎢
+ αt ⎥⎬
⎟
⎠
⎣2 κt
⎦ ⎭⎪
1
30
1
( a − b) (t + 4x )2
1 α t ⎧⎪⎛
x
e ⎨⎜ t −
2 ⎪⎩⎝ 2 κα
e − qx
(t > 0)
xv
( 2t )
v +1
e
−
ln C = γ = 0.5772...
x2
4t
Page 14 of 15
4. Laplace Transforms for well hydraulics (Hantush, 1964)
f ( p)
3
6
7
1
p ( p + a)
1
⎡1 − exp ( −at ) ⎤⎦
a⎣
{
⎧ k2 ⎫
exp ⎨− ⎬
πt
⎩ 4t ⎭
⎛ k ⎞
erfc ⎜
⎟
⎝ 4t ⎠
1
exp − k p
p
{
1
exp − k p
p
1
8
F(t)
n
1+
2
}
{
p
1
}
exp −k p
(t > 0)
}
n
⎛ k ⎞
⎟
⎝ 4t ⎠
( 4t ) 2 i n erfc ⎜
inerfc is nth repeated integral of erfc
(
⎧ k2 ⎫
1
exp ⎨− ⎬
2t
⎩ 4t ⎭
)
9
K0 k p
10
1
K0 k p
p
(
1 ⎛ k2 ⎞
W⎜ ⎟
2 ⎝ 4t ⎠
)
W is the Theis well function
(
K0 k p + a
12
1
K0 k p + a
p
(
⎧
1
k2 ⎫
exp ⎨ −at − ⎬
2t
4t ⎭
⎩
)
11
⎞
1 ⎛ k2
W ⎜ ,k a ⎟
2 ⎝ 4t
⎠
)
W is the well function for leaky aquifers (Hantush, 1964; p. 321)
13
(
1
K0 k p + a p
p
)
1 ⎛ k 2 ka ⎞
H⎜ , ⎟
2 ⎝ 4t 4 ⎠
H is a well function defined by Hantush (164; p. 312)
14
( )
pK ( k p )
K (k p )
p (k p ) K (k p )
K (k p + a )
pK ( k p + a )
K0 k p
0
15
1
0
1
16
1
0
0
1
Laplace transforms.doc
1
⎛ t k⎞
A⎜ 2 , ⎟
⎝ k1 k1 ⎠
A is the flowing well function for leaky aquifers
(Hantush, 1964; p. 309)
1 ⎛ t k⎞
S⎜ , ⎟
2 ⎝ k12 k1 ⎠
S is the function defined by Hantush (1964; p. 318)
⎛ t k
⎞
Z ⎜ 2 , , k1 a ⎟
⎝ k1 k1
⎠
Z is the function defined by Hantush (1964; p. 325)
Page 15 of 15