Laplace Transform Table
2
EqWorld http://eqworld.ipmnet.ru Auxiliary Sections > Integral Transforms > Tables of Laplace Transforms > Laplace Transforms: Expressions with Trigonometric Functions Laplace Transforms: Expressions with Trigonometric Functions No Original function, f (x) Laplace transform, e f (p) = Z ∞ 0 e -px f (x) dx 1 sin(ax) a p 2 + a 2 2 |sin(ax)|, a > 0 a p 2 + a 2 coth ‡ πp 2a · 3 sin 2n (ax), n =1, 2, ... a 2n (2n)! p £ p 2 + (2a) 2 ∕£ p 2 + (4a) 2 ∕ ... £ p 2 + (2na) 2 ∕ 4 sin 2n+1 (ax), n =1, 2, ... a 2n+1 (2n + 1 )! £ p 2 + a 2 ∕£ p 2 + 3 2 a 2 ∕ ... £ p 2 + (2n + 1 ) 2 a 2 ∕ 5 x n sin(ax), n =1, 2, ... n! p n+1 ( p 2 + a 2 ) n+1 X 0≤2k≤n (-1) k C 2k+1 n+1 ‡ a p · 2k+1 6 1 x sin(ax) arctan ‡ a p · 7 1 x sin 2 (ax) 1 4 ln ( 1+4 a 2 p -2 ) 8 1 x 2 sin 2 (ax) a arctan(2a/p) - 1 4 p ln ( 1+4 a 2 p -2 ) 9 sin ( 2 √ ax ) √ πa p √ p e -a/p 10 1 x sin ( 2 √ ax ) π erf (p a/p ) 11 cos(ax) p p 2 + a 2 12 cos 2 (ax) p 2 + 2 a 2 p ( p 2 + 4 a 2 ) 13 x n cos(ax), n =1, 2, ... n! p n+1 ( p 2 + a 2 ) n+1 X 0≤2k≤n+1 (-1) k C 2k n+1 ‡ a p · 2k 14 1 x £ 1- cos(ax) ∕ 1 2 ln ( 1+ a 2 p -2 ) 15 1 x £ cos(ax) - cos(bx) ∕ 1 2 ln p 2 + b 2 p 2 + a 2
description
EqWorldhttp://eqworld.ipmnet.ruAuxiliary Sections > Integral Transforms > Tables of Laplace Transforms > Laplace Transforms: Expressions with Trigonometric FunctionsLaplace Transforms: Expressions with Trigonometric FunctionsNo 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Original function, f (x) sin(ax) |sin(ax)|, sin2n (ax), sin2n+1 (ax), xn sin(ax), 1 sin(ax) x 1 sin2 (ax) x 1 sin2 (ax) x2 √ sin 2 ax √ 1 sin 2 ax x cos(ax) cos2 (ax) xn cos(ax), 1 1 − cos(ax) x 1 cos(ax) − cos(bx) x n = 1, 2, .
Transcript of Laplace Transform Table
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Auxiliary Sections > Integral Transforms > Tables of Laplace Transforms > Laplace Transforms: Expressionswith Trigonometric Functions
Laplace Transforms: Expressions with Trigonometric Functions
No Original function, f (x) Laplace transform, f (p) =∫ ∞
0e−pxf (x) dx
1 sin(ax)a
p2 + a2
2 |sin(ax)|, a > 0a
p2 + a2 coth( πp
2a
)
3 sin2n(ax), n = 1, 2, . . .a2n(2n)!
p[p2 + (2a)2
][p2 + (4a)2
]. . .
[p2 + (2na)2
]
4 sin2n+1(ax), n = 1, 2, . . .a2n+1(2n + 1)![
p2 + a2][
p2 + 32a2]. . .
[p2 + (2n + 1)2a2
]
5 xn sin(ax), n = 1, 2, . . .n! pn+1
(p2 + a2
)n+1
∑
0≤2k≤n
(−1)kC2k+1n+1
( a
p
)2k+1
61x
sin(ax) arctan( a
p
)
71x
sin2(ax) 14 ln
(1 + 4a2p−2)
81x2 sin2(ax) a arctan(2a/p) − 1
4 p ln(1 + 4a2p−2)
9 sin(2√
ax) √
πa
p√
pe−a/p
101x
sin(2√
ax)
π erf(√
a/p)
11 cos(ax)p
p2 + a2
12 cos2(ax)p2 + 2a2
p(p2 + 4a2
)
13 xn cos(ax), n = 1, 2, . . .n! pn+1
(p2 + a2
)n+1
∑
0≤2k≤n+1
(−1)kC2kn+1
( a
p
)2k
141x
[1 − cos(ax)
] 12 ln
(1 + a2p−2)
151x
[cos(ax) − cos(bx)
] 12
lnp2 + b2
p2 + a2
No Original function, f (x) Laplace transform, f (p) =∫ ∞
0e−pxf (x) dx
16√
x cos(2√
ax)
12 π1/2p−5/2(p − 2a)e−a/p
171√x
cos(2√
ax) √
π/p e−a/p
18 sin(ax) sin(bx)2abp[
p2 + (a + b)2][
p2 + (a − b)2]
19 cos(ax) sin(bx)b(p2 − a2 + b2)
[p2 + (a + b)2
][p2 + (a − b)2
]
20 cos(ax) cos(bx)p(p2 + a2 + b2)
[p2 + (a + b)2
][p2 + (a − b)2
]
21 ebx sin(ax)a
(p − b)2 + a2
22 ebx cos(ax)p − b
(p − b)2 + a2
Notation: erfcz is the complementary error function.
ReferencesBateman, H. and Erdelyi, A., Tables of Integral Transforms. Vols. 1 and 2, McGraw-Hill Book Co., New York, 1954.Doetsch, G.,Einfuhrung in Theorie und Anwendung der Laplace-Transformation, Birkhauser Verlag, Basel–Stuttgart, 1958.Ditkin, V. A. and Prudnikov, A. P., Integral Transforms and Operational Calculus, Pergamon Press, New York, 1965.Polyanin, A. D. and Manzhirov, A. V., Handbook of Integral Equations, CRC Press, Boca Raton, 1998.
Laplace Transforms: Expressions with Trigonometric Functions
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