Page 730 - Handbook Of Integral Equations

∞
˜
No Original function, f(x) Laplace transform, f(p)= e –px f(x) dx
0

1 –ax 2
5 1 – e (p +2a) ln(p +2a)+ p ln p – 2(p + a) ln(p + a)
x 2
√
2 1/2 2 1
6 exp –ax , a >0 (πb) exp bp erfc(p b), a =
4b
√
2 1/2 3/2 1
7 x exp –ax 2b – 2π b p erfc(p b), a =
4b
√
8 exp(–a/x), a ≥ 0 2 a/pK 1 2 ap
√ 1 √ √
3
9 x exp(–a/x), a ≥ 0 π/p 1+2 ap exp –2 ap
2
1 √
10 √ exp(–a/x), a ≥ 0 π/p exp –2 ap
x
1 √
11 √ exp(–a/x), a >0 π/a exp –2 ap
x x
√
ν/2
ν–1
12 x exp(–a/x), a >0 2(a/p) K ν 2 ap
√
–1 1/2 –3/2 a/p
13 exp –2 ax p – (πa) p e erfc a/p
1 √ 1/2 a/p
14 √ exp –2 ax (π/p) e erfc a/p
x
4.4. Expressions With Hyperbolic Functions
∞

˜
No Original function, f(x) Laplace transform, f(p)= e –px f(x) dx
0
a
1 sinh(ax) 2 2
p – a
2a 2
2
2 sinh (ax)
2
3
p – 4a p
1 1 p + a
3 sinh(ax) ln
x 2 p – a

4 x ν–1 sinh(ax), ν > –1 1 Γ(ν) (p – a) –ν – (p + a) –ν
2
√
√ πa a/p
5 sinh 2 ax √ e
p p
√ √ a/p 1/2 –2
p
6 x sinh 2 ax π 1/2 –5/2 1 p + a e erf a/p – a p
2
1 √
p
7 √ sinh 2 ax π 1/2 –1/2 a/p erf a/p
e
x
1 2 √
p
8 √ sinh ax 1 2 π 1/2 –1/2 e a/p – 1
x
p
9 cosh(ax) 2 2
p – a
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 714