Page 741 - Handbook Of Integral Equations

1 c+i∞ px ˜
˜
No Laplace transform, f(p) Inverse transform, f(x)= e f(p) dp
2πi
c–i∞
n
e –ax 2
– exp(a k x)
(2n +1)a 2n (2n +1)a 2n+1
1 k=1
72 2n+1 2n+1 , n =0, 1, ...
p + a × a k cos(b k x) – b k sin(b k x) ,
π(2k – 1)
a k = a cos ϕ k , b k = a sin ϕ k , ϕ k =
2n +1
n
e ax 2
+ exp(a k x)
(2n +1)a 2n (2n +1)a 2n+1
1 k=1
73 2n+1 2n+1 , n =0, 1, ...
p – a × a k cos(b k x) – b k sin(b k x) ,
2πk
a k = a cos ϕ k , b k = a sin ϕ k , ϕ k =
2n +1
Q(p)
, n
P(p) Q(a k ) exp a k x ,

74 P(p)=(p – a 1 ) ... (p – a n ); P (a k )

Q(p) is a polynomial of degree k=1
(the prime stand for the differentiation)
≤ n – 1; a i ≠ a j if i ≠ j
Q(p)
, n m k
P(p) Φ kl (a k ) x m k –l exp a k x ,

P(p)=(p – a 1 ) m 1 ... (p – a n ) m n ; (m k – l)! (l – 1)!
75 k=1 l=1
Q(p) is a polynomial of degree d l–1 Q(p) P(p)
< m 1 + m 2 + ··· + m n – 1; Φ kl (p)= dp l–1 , P k (p)= m k
a i ≠ a j if i ≠ j P k (p) (p – a k )
Q(p)+ pR(p) n
, Q(ia k ) sin(a k x)+ a k R(ia k ) cos(a k x)
P(p) a k P k (ia k ) ,
2
2
2
2
76 P(p)=(p + a ) ... (p + a ); k=1
n
1
Q(p) and R(p) are polynomials P m (p)= P(p) , i = –1
2
2
of degree ≤ 2n – 2; a l ≠ a j , l ≠ j p + a 2 m
5.3. Expressions With Square Roots
1 c+i∞ px ˜
˜
No Laplace transform, f(p) Inverse transform, f(x)= e f(p) dp
2πi
c–i∞
1 1
1 √ √
p πx
√ e bx – e ax
2 p – a – p – b √
2 πx 3
1 1 –ax
3 √ √ e
p + a πx

p + a –ax/2
4 – 1 1 2 ae I 1 1 2 ax + I 0 1 2 ax
p
√ –ax
p + a e 1/2 –bx 1/2 1/2
5 √ +(a – b) e erf (a – b) x
p + b πx
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 725