Page 758 - Handbook Of Integral Equations

7.5. Expressions With Logarithmic Functions

∞
ˇ
No Original function, f(x) Sine transform, f s (u)= f(x) sin(ux) dx
0
ln x if 0 < x <1, Ci(u) – ln u – C ,
1
1 u
0 if 1 < x
C = 0.5772 ... is the Euler constant
ln x 1
2 – π(ln u + C)
2
x
ln x π π
3 √ – ln(4u)+ C –
x 2u 2
πu –ν ψ(ν)+ π 2 cot πν – ln u
2
4 x ν–1 ln x, |ν| <1 πν
2Γ(1 – ν) cos
2
π

5 ln a + x , a >0 sin(au)
a – x u
2
(x + b) + a 2 2π –au
6 ln , a, b >0 e sin(bu)
2
(x – b) + a 2 u
2
1
C
2
2
7 e –ax ln x, a >0 a arctan(u/a) – u ln(u + a ) – e u
2
u + a 2
1 2 2 u
8 ln 1+ a x , a >0 –π Ei –
x a
7.6. Expressions With Trigonometric Functions
∞
ˇ
No Original function, f(x) Sine transform, f s (u)= f(x) sin(ux) dx
0
sin(ax) 1
1 , a >0 u + a
x 2 ln u – a
1
sin(ax) πu if 0 < u < a,
2 , a >0 2 1
x 2 πa if u > a
2
|u – a| –ν – |u + a| –ν
3 x ν–1 sin(ax), a >0, –2< ν <1 π 1 , ν ≠ 0
4Γ(1 – ν) sin πν
2
1 –1 –ab
sin(ax) 2 πb e sinh(bu)if 0 < u < a,
4 , a, b >0 1 –1 –bu
2
x + b 2 2 πb e sinh(ab) if u > a
sin(πx) sin u if 0 < u < π,
5
1 – x 2 0 if u > π
a 1 1
–ax
6 e sin(bx), a >0 –
2 a +(b – u) 2 a +(b + u) 2
2
2
2
1 (u + b) + a 2
–1 –ax
7 x e sin(bx), a >0 ln
4 (u – b) + a 2
2
 1
 π if 0 < u <2a,
1 2 4
8 sin (ax), a >0 1 π if u =2a,
x  8
0 if u >2a
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 742