Expressions with Trigonometric Functions

Expressions with Trigonometric Functions

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Auxiliary Sections > Integral Transforms > Tables of Inverse Laplace Transforms > Inverse Laplace
Transforms: Expressions with Trigonometric Functions
Inverse Laplace Transforms: Expressions with Trigonometric
Functions
No
Laplace transform, fe(p)
Inverse transform, f (x) =
1
2πi
1
sin(a/p)
√
p
¡√
¢ ¡√
¢
1
√
sinh 2ax sin 2ax
πx
2
sin(a/p)
√
p p
¡√
¢ ¡√
¢
1
√ cosh 2ax sin 2ax
πa
3
cos(a/p)
√
p
¡√
¢ ¡√
¢
1
√
cosh 2ax cos 2ax
πx
4
cos(a/p)
√
p p
5
¡ √ ¢ ¡√ ¢
1
√ exp − ap sin ap
p
6
¡ √ ¢ ¡√ ¢
1
√ exp − ap cos ap
p
¡√
¢ ¡√
¢
1
√ sinh 2ax cos 2ax
πa
³ a ´
1
√
sin
2x
πx
³
1
a ´
√
cos
2x
πx
Z
c+i∞
epx fe(p) dp
c−i∞
Inverse Laplace Transforms: Expressions with Inverse
Trigonometric Functions
No
1
Laplace transform, fe(p)
arctan
a
p
3
1
a
arctan
p
p
a
p arctan − a
p
4
arctan
2
2ap
+ b2
p2
1
Inverse transform, f (x) =
2πi
Z
c+i∞
epx fe(p) dp
c−i∞
1
sin(ax)
x
Si(ax),
where Si(x) is the integral sine
¤
1 £
ax cos(ax) − sin(ax)
2
x
¡ √
¢
2
sin(ax) cos x a2 + b2
x
References
Bateman, H. and Erdélyi, A., Tables of Integral Transforms. Vols. 1 and 2, McGraw-Hill Book Co., New York, 1954.
Doetsch, G., Einführung in Theorie und Anwendung der Laplace-Transformation, Birkhäuser 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.
Inverse Laplace Transforms: Expressions with Trigonometric Functions
c 2005 Andrei D. Polyanin
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