The Laplace transform is a mathematical method that is used to convert timedomain functions into frequencydomain functions by simplifying the differential equations. Here we will cover the details parameters and the process of using this transform. So let’s get started
What is Laplace Transform?
Laplace transformation techniques are used for solving differential equations. In this process differential equations that are in time, domains are transformed into algebraic equations of the frequency domain.
Then solving algebraic equations in the frequency domain the result is converted into the time domain for getting the required solution of differential equations.
In simple words, Laplace transformation is a shortcut technique for finding the solution of differential equations.
History of Laplace Transforms
The Laplac transform is called due to French mathematician and astronomer Pierre Simon Laplace, transform functions in different mathematical domains for solving intractable problems.
He used this transform for additions to probability theory. That became common after WWII. This transform made by Oliver Heaviside, an English Electrical Engineer. Some other famous scientists like Niels Abel, Mathias Lerch, and Thomas Bromwich used it in the 19th century.
Properties of Laplace Transform
Linearity Property 
A f1(t) + B f2(t) ⟷ A F1(s) + B F2(s)

Frequency Shifting Property 
es0t f(t)) ⟷ F(s – s0)

Integration  t∫0 f(λ) dλ ⟷ 1⁄s F(s) 
Multiplication by Time 
T f(t) ⟷ (−d F(s)⁄ds)

Complex Shift Property 
f(t) e−at ⟷ F(s + a)

Time Reversal Property  f (t) ⟷ F(s) 
Time Scaling Property 
f (t⁄a) ⟷ a F(as)

Laplace Transform Table
f(t)  L(f(t)) = F(s) 
$1  1/s 
tn at t = 1,2,3,…  n!/s(n+1) 
√(t)  √π/2s(3/2) 
sin(at)  a/(s2+a2) 
t sin(at)  2as/(s2+a2)2 
sin(at+b)  (s sin(b)+ a cos(b)/(s2+a2) 
sinh(at)  a/(s2a2) 
e(at)sin(bt)  b/((sa)2+b2) 
e(ct)f(t)  F(sc) 
f'(t)  sF(s) – f(0) 
e(at)  1/(s − a) 
tp, at p>1  Γ(p+1)/s(p+1) 
t(n1/2) at n = 1,2,.. 
(1.3.5…(2n1)√π)/(2n s(n+1/2)

cos(at)  s/(s2+a2) 
t cos(at) 
(s2a2)/(s2+a2)2

cos(at+b) 
(s cos(b)a sin(b)/(s2+a2)

cosh(at)  s/(s2a2) 
e(at)cos(bt) 
(sa)/((sa)2+b2)

tnf(t) at n = 1,2,3..  (1)n Fn s 
f”(t) 
s2F(s) − sf(0) − f'(0)

Bilateral Laplace Transform
The Laplace transform is also known as the bilateral Laplace transform. It is also called 2 sided Laplace transform that can done through extending limts of integration to real axis.
So common unilateral Laplace transform is a special type of Bilateral Laplace transform that hs function transformed multiplied by the Heaviside step function.
The bilateral Laplace transform equation is
How to use a Laplace transform?
 Take Laplace transform of differential equation through the use of derivative property
 Put initial conditions in the resulting equation
 Solve output variables
Inverse Laplace Transform
In the inverse Laplace transform there is transform F(s) used and find what function initially exists. The inverse transform of function Fs is given as
f(t) = L^{1}{F(s)}
Such as two Laplace transform is as F(s) and G(s), the inverse Laplace transform will be as:
L^{1}{aF(s)+bG(s)}= a L^{1}{F(s)}+bL^{1} {G(s)}
 What is the Laplace transform simplified?
 We can consider that Laplace transform as a black box that gets functions and provides functions in new variables. Write the L{f(t)}=F(s) for the Laplace transform of f(t).
 That is commonly written as lowercase letters for functions in the time domain and upper case letters for functions for the frequency domain.
Which is easier Laplace or Fourier?
 Laplace transformers are used in place of Fourier transforms since their internal is simple. Fourier analysis is best for checking frequency components, spectrum, etc. Fourier transform is a frequency spectrum
What is the Laplace formula?
 Laplace equation is 2nd order partial derivate and used as a boundary condition for solving different difficult equations in Physics. The laplace equation is mathematically written equations for divergence gradient of scalar function is zero such as ▽^{2}f=0.
What is the Laplace of 1?
 Laplace transform of 1 was equal to 1/s.
Where is Laplace used?
 Laplace transform used in engineering and physics output of linear time invariant system can be find through convolving its unit inpulse with input signal.
What are the different types of Laplace transforms?
 onesided Laplace transformation and twosided Laplace transformation.
What are the applications of Laplace transforms?
 it used for conversion of complicated differential equations into simple types of polynomials.
 it used for covnerion derivates in different domain variables and then transform polynomials to differential equations through use of inverse Laplace transform.
 it also part of telecom sectors for sending signals to both side of medium. Such as when signals are transmit through phone first transformed in time varying wave and then superimposed to medium
What is the Laplace transform of sin t?
 As we know s Laplace transform of sin at = a/(s^2 + a^2).
What is the difference between ztransform and Laplace transform?
 The Z transform trnasfomrs equations ofdiscrete times systems to algebric equations that make simple discrete time system analsysi. lapclace transform and Z transform are common but different that Laplace work with continuous time signals and systems
Can you multiply Laplace transforms?
 The main drawback is that the Laplace transform of product of 2 functions is not multiple of Laplcae transforms. But Laplace transform of convolution of 2 functions is product of their Laplace transforms.
How to solve Laplace transform?
 First step for use of laplac transform is to sovler IVP take transform of each term in different equations. The use of proper formula from table of laplae shown above. collect all the terms that have a Y(s) Y ( s ) in them