Page 752 - Handbook Of Integral Equations

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

ln x – π ln(4u)+ C + π ,
2 √ 2u 2
x
C = 0.5772 ... is the Euler constant
π πν

–ν
πν
3 x ν–1 ln x, 0 < ν <1 Γ(ν) cos u ψ(ν) – tan – ln u
2 2 2

4 ln a + x , a >0 2 cos(au) Si(au) – sin(au) Ci(au)
a – x u
2 2 π –au
5 ln 1+ a /x , a >0 1 – e
u
2
a + x 2 π –bu –au
6 ln , a, b >0 e – e
2
b + x 2 u
1
2
2
aC + a ln(u + a )+ u arctan(u/a)
–ax
7 e ln x, a >0 – 2
2
u + a 2
a π

8 ln 1+ e –ax , a >0 2 – –1
2u 2u sinh πa u
a π
–ax –1
9 ln 1 – e , a >0 2 – coth πa u
2u 2u
6.6. Expressions With Trigonometric Functions
∞
ˇ
No Original function, f(x) Cosine transform, f c (u)= f(x) cos(ux) dx
0
 1
 π if u < a,
sin(ax) 2
1 , a >0 1 π if u = a,
x  4
0 if u > a
(u + a) –ν – |u + a| –ν sign(u – a)
2 x ν–1 sin(ax), a >0, |ν| <1 π 1
4Γ(1 – ν) cos πν
2
1 –ab
x sin(ax) πe cosh(bu) if u < a,
3 , a, b >0 2 1 –bu
2
x + b 2 – πe sinh(ab)if u > a
2
1 –2 –ab
sin(ax) 2 πb 1 – e cosh(bu) if u < a,
4 2 2 , a, b >0 1 –2 –bu
x(x + b ) πb e sinh(ab) if u > a
2
1 a + u a – u
5 e –bx sin(ax), a, b >0 +
2
2
2 (a + u) + b 2 (a – u) + b 2
1 2 1 2
6 sin (ax), a >0 ln 1 – 4 a

x 4 u 2

1 2 1 π(2a – u)if u <2a,
7 sin (ax), a >0 4
x 2 0 if u >2a
a
1 π √
8 sin , a >0 J 0 2 au
x x 2
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 736