Page 732 - Handbook Of Integral Equations

4.6. Expressions With Trigonometric Functions

∞

˜
No Original function, f(x) Laplace transform, f(p)= e –px f(x) dx
0
a
1 sin(ax) 2 2
p + a
a πp
2 |sin(ax)|, a >0 2 2 coth
p + a 2a
2n
a (2n)!
2n
3 sin (ax), n =1, 2, ... 2 2 2 2 2 2
p p +(2a) p +(4a) ... p +(2na)
a 2n+1 (2n + 1)!
4 sin 2n+1 (ax), n =1, 2, ...
2
2 2
p + a 2 p +3 a ... p +(2n +1) a
2
2 2
2
a
n! p n+1 k 2k+1 2k+1
n
5 x sin(ax), n =1, 2, ... n+1 (–1) C n+1
2
p + a 2 0≤2k≤n p
a
1
6 sin(ax) arctan
x p
1 2 1 2 –2
7 sin (ax) 4 ln 1+4a p
x
1 2 1 2 –2
8 sin (ax) a arctan(2a/p) – p ln 1+4a p
4
x 2
√
√ πa –a/p
9 sin 2 ax √ e
p p
1 √
10 sin 2 ax π erf a/p
x
p
11 cos(ax) 2 2
p + a
2
p +2a 2
2
12 cos (ax)
p p +4a 2
2
n! p n+1 k 2k 2k
a
n
13 x cos(ax), n =1, 2, ... n+1 (–1) C n+1
2
p + a 2 0≤2k≤n+1 p
1 1 2 –2
14 1 – cos(ax) 2 ln 1+ a p
x
2
1 p + b 2
1
15 cos(ax) – cos(bx) ln
x 2 p + a 2
2
√ √
16 x cos 2 ax 1 π 1/2 –5/2 (p – 2a)e –a/p
p
2
1 √
17 √ cos 2 ax π/p e –a/p
x
2abp
18 sin(ax) sin(bx) 2 2 2 2
p +(a + b) p +(a – b)
2 2 2
b p – a + b
19 cos(ax) sin(bx)
2
2
p +(a + b) 2 p +(a – b) 2
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
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