Page 734 - Handbook Of Integral Equations

∞
˜
No Original function, f(x) Laplace transform, f(p)= e –px f(x) dx
0
1
9 Si(x) arccot p
p
1
10 Ei(–x) – ln(p +1)
p
1
11 J 0 (ax)
2
p + a 2
a ν
12 J ν (ax), ν > –1 ν
p + a 2 p + p + a 2
2
2
n
2
13 x J n (ax), n =1, 2, ... 1 ⋅ 3 ⋅ 5 ... (2n – 1)a n p + a 2 –n–1/2

ν 1 ν –1/2 1 ν 2 2 –ν–1/2

14 x J ν (ax), ν > – 2 π Γ ν + a p + a
2 2
2
ν
15 x ν+1 J ν (ax), ν > –1 2 ν+1 –1/2 Γ ν + 3 a p p + a 2 –ν–3/2

π
2
√ 1 –a/p
16 J 0 2 ax e
p
√
√ √ a
17 xJ 1 2 ax e –a/p
p 2
√
p
18 x ν/2 J ν 2 ax , ν > –1 a ν/2 –ν–1 –a/p
e
1
19 I 0 (ax)
2
p – a 2
a ν
20 I ν (ax), ν > –1 ν
2
2
p – a 2 p + p – a 2
ν
ν
ν
2

21 x I ν (ax), ν > – 1 2 π –1/2 Γ ν + 1 a p – a 2 –ν–1/2
2 2
ν
2

22 x ν+1 I ν (ax), ν > –1 2 ν+1 –1/2 Γ ν + 3 a p p – a 2 –ν–3/2
π
2
√ 1 a/p
23 I 0 2 ax e
p
1 √ 1 a/p
24 √ I 1 2 ax √ e – 1
x a
√
ν/2
p
25 x I ν 2 ax , ν > –1 a ν/2 –ν–1 a/p
e
2 Arsinh(p/a)
26 Y 0 (ax) –
π p + a 2
2

2 2
ln p + p – a – ln a
27 K 0 (ax)
2
p – a 2
•
References for Supplement 4: G. Doetsch (1950, 1956, 1958), H. Bateman and A. Erd´ elyi (1954), V. A. Ditkin and
A. P. Prudnikov (1965).
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
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