Laplace Transform
If f(t) is a function of t, defined for all t > 0, the Laplace transform of f(t), denote by the symbol
L{ f (t) } is defined by:
where s may be real or complex. In circuit applications we assume s = σ + jω.
The operation L{f(t)} transforms a function f(t) in the time domain into a
function F(s) in the complex frequency domain or simply the s domain.

Laplace tables
Solved examples:
Solved problems of Laplace transform
Example 1a: Obtain the Laplace transform of f(t) = At where A is a constant.


Example 2a: Obtain the Laplace transform of f(t) = e^{−at} where a is a constant.  
Example 3a: Obtain the Laplace transform of f(t) = te^{−at} where a is a constant.


Example 4a: Find the Laplace transform of f(t) = sin(ωt)
Apply integration by parts second time on the cos term on the right side of the integral:


Example 5a: Find the Laplace transform of f(t) = A cos ωt
Apply integration by parts second time on the cos term on the right side of the integral:


Example 6a: Find the Laplace transform of f(t) = e^{−at} cos ωt
The Laplace trensform is defined by the integral:
The integrals on both sides of the equation are the same and can be added.


Example 7a: Find the Laplace transform of f(t) = sin (ωt + θ)
Apply integration by parts second time on the integral at the right side to get:


Example 8a: Find the Laplace transform of f(t) = cosh ωt
The second term reach zero at infinity, the first term is zero only when s>ω


Example 9a: Find the Laplace transform of the derivative of f^{'}(t) = df(t) / dt.


Example 10a: Find the Laplace transform of: f(t) = e^{−t} (t^{2} + 12t + 10)
