Page 754 - Handbook Of Integral Equations

∞
ˇ
No Original function, f(x) Cosine transform, f c (u)= f(x) cos(ux) dx
0
1

4 J 0 (ax), a >0 √ a – u 2 if 0 < u < a,
2
0 if a < u

 cos ν arcsin(u/a)

√ if 0 < u < a,

 2 2
a – u
ν
5 J ν (ax), a >0, ν > –1  a sin(πν/2)
 – if a < u,

ξ(u + ξ) ν
√
2
where ξ = u – a 2
 ν cos ν arcsin(u/a) if 0 < u < a,
 –1
1 ν
6 J ν (ax), a >0, ν >0 a cos(πν/2)
x  √ ν if a < u
2
ν u + u – a 2

√ 2 2 ν–1/2
π a – u

–ν 1 if 0 < u < a,
7 x J ν (ax), a >0, ν > – 1
ν
2 (2a) Γ ν +
2

0 if a < u
0 √ if 0 < u < a,

x ν+1 J ν (ax),  2 ν+1 πa u
ν
8 1 if a < u
a >0, –1< ν < – ν+3/2
2  1 (u – a 2
Γ –ν –
2
2
√ 1 a 2
9 J 0 a x , a >0 sin
u 4u
1 √ 4 a 2
10 √ J 1 a x , a >0 sin 2
x a 8u
a
√
11 x ν/2 J ν a x , a >0, –1< ν < 1 ν u –ν–1 sin a 2 – πν
2
2 4u 2
√

2
 cos b a – u 2
√
2
12 J 0 a x + b 2 √ 2 2 if 0 < u < a,
a – u

0 if a < u
0 if 0 < u < a,

13 Y 0 (ax), a >0 1
– √ if a < u
2
u – a 2
0 if 0 < u < a,

 ν √
ν 1 (2a) π
14 x Y ν (ax), a >0, |ν| <
2 – if a < u
ν+1/2
Γ – ν (u – a
 1 2 2
2
√ π √ 2 2
2
15 K 0 a x + b 2 , a, b >0 √ exp –b u + a
2 u + a 2
2
•
References for Supplement 6: 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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