Page 729 - Handbook Of Integral Equations

4.2. Expressions With Power-Law Functions

∞
˜
No Original function, f(x) Laplace transform, f(p)= e –px f(x) dx
0
1
1 1
p
0if 0 < x < a,

1 –ap –bp
2 1if a < x < b, e – e
p
0if b < x.
1
3 x 2
p
1 ap
4 –e Ei(–ap)
x + a
n!
n
5 x , n =1, 2, ...
p n+1
√
1 ⋅ 3 ... (2n – 1) π
n–1/2
6 x , n =1, 2, ...
2 p
n n+1/2
1 π ap √
7 √ e erfc ap
x + a p
√
x π √ ap √
8 – π ae erfc ap
x + a p
√
1/2 ap
–1/2
9 (x + a) –3/2 2a – 2(πp) e erfc ap
√
1/2
1/2 ap
10 x 1/2 (x + a) –1 (π/p) – πa e erfc ap
√
–1/2 ap
11 x –1/2 (x + a) –1 πa e erfc ap
ν
12 x , ν > –1 Γ(ν +1)p –ν–1
ν
13 (x + a) , ν > –1 p –ν–1 –ap Γ(ν +1, ap)
e
ap
ν
–1
ν
14 x (x + a) , ν > –1 ke Γ(–ν, ap), k = a Γ(ν +1)
ap
2
15 (x +2ax) –1/2 (x + a) ae K 1 (ap)
4.3. Expressions With Exponential Functions
∞

˜
No Original function, f(x) Laplace transform, f(p)= e –px f(x) dx
0
1 e –ax (p + a) –1
2 xe –ax (p + a) –2
e
3 x ν–1 –ax , ν >0 Γ(ν)(p + a) –ν
1 –ax –bx
4 e – e ln(p + b) – ln(p + a)
x

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 713