Page 743 - Handbook Of Integral Equations

1 c+i∞ px ˜
˜
No Laplace transform, f(p) Inverse transform, f(x)= e f(p) dp
2πi
c–i∞
–n
–n –1
25 p + a + p na x J n (ax)
2
2
–n –n –1
2
2
26 p – a + p na x I n (ax)
n
(x/a) J n (ax)
2

27 p + a 2 –n–1/2
1 ⋅ 3 ⋅ 5 ... (2n – 1)
n

2 2 –n–1/2 (x/a) I n (ax)
28 p – a
1 ⋅ 3 ⋅ 5 ... (2n – 1)
5.4. Expressions With Arbitrary Powers
1 c+i∞ px ˜
˜
No Laplace transform, f(p) Inverse transform, f(x)= e f(p) dp
2πi
c–i∞
1 ν–1 –ax
–ν
1 (p + a) , ν >0 x e
Γ(ν)
ν

1/2 1/2 –2ν –1 1 1
2 (p + a) +(p + b) , ν >0 ν x exp – (a + b)x I ν 2 (a – b)x
2
(a – b)
√
–ν π x ν–1/2 a + b a – b

3 (p + a)(p + b) , ν >0 exp – x I ν–1/2 x
Γ(ν) a – b 2 2
√
π

2 2 –ν–1/2 1 ν
4 p + a , ν > – 1 x J ν (ax)
ν
2 (2a) Γ(ν + )
2
√
π ν
2

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

6 p p + a 2 –ν–1/2 , ν >0 1 x J ν–1 (ax)

ν
(2a) Γ ν +
2
√
a π

2 2 –ν–1/2 ν
7 p p – a , ν >0 ν 1 x I ν–1 (ax)
(2a) Γ ν +
2
2 2 1/2 –ν
(p + a ) + p =
–ν –1
8 –2ν 2 2 1/2 ν νa x J ν (ax)
a (p + a ) – p , ν >0
2 2 1/2 –ν
(p – a ) + p =
–ν –1
9 –2ν 2 2 1/2 ν νa x I ν (ax)

a p – (p – a ) , ν >0
2 2 1/2 –ν 1–ν –1 –ν –2
10 p (p + a ) + p , ν >1 νa x J ν–1 (ax) – ν(ν +1)a x J ν (ax)
2 2 1/2 –ν 1–ν –1 –ν –2
11 p (p – a ) + p , ν >1 νa x I ν–1 (ax) – ν(ν +1)a x I ν (ax)
2 2 –ν
p + a + p –ν
12 , ν > –1 a J ν (ax)
2
p + a 2
–ν
2
2
p – a + p
–ν
13 , ν > –1 a I ν (ax)
2
p – a 2
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
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