Page 751 - Handbook Of Integral Equations

∞
ˇ
No Original function, f(x) Cosine transform, f c (u)= f(x) cos(ux) dx
0
x 1 π 2
8 ax 2 – 2
–1
2
e – 1 2u 2a sinh πa u
1 1 1 1 1 –2πu

9 – + – ln 1 – e
x
x 2 x e – 1 2

1 π u 2
2
10 exp –ax exp –
2 a 4a
1 a π – 2au √ √
√
11 √ exp – e cos 2au – sin 2au
x x 2u
1 a π √ √
12 √ exp – e – 2au cos 2au
x x x a
6.4. Expressions With Hyperbolic Functions
∞
ˇ
No Original function, f(x) Cosine transform, f c (u)= f(x) cos(ux) dx
0
1 π
1 , a >0 1 –1
cosh(ax) 2a cosh 2 πa u
1 πu
2 2 , a >0 2 1 –1
cosh (ax) 2a sinh 2 πa u
1 –1 1 –1
cosh(ax) π cos 2 πab cosh 2 πb u

3 , |a| < b –1 –1
cosh(bx) b cos πab + cosh πb u
–1
1 π sinh a bu
4
cosh(ax) + cos b a sin b sinh πa u
–1
2
1 π b – u 2 abu
2
5 exp –ax cosh(bx), a >0 exp cos
2 a 4a 2
x π 2
6
sinh(ax) 4a cosh 2 1 πa u
2
–1
2
–1
sinh(ax) π sin πab
7 , |a| < b
–1
sinh(bx) 2b cos πab –1 + cosh πb u
1 1 –1
8 tanh(ax), a >0 ln coth 4 πa u
x
6.5. Expressions With Logarithmic Functions
∞
ˇ
No Original function, f(x) Cosine transform, f c (u)= f(x) cos(ux) dx
0
ln x if 0 < x <1, 1

1 – Si(u)
0 if 1 < x u

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 735