How to Calculate the Laplace Transform of a Function

The Laplace transform is an integral transform which allows a differential equation to be converted into a (hopefully) simpler algebraic equation, making it easier to solve.

While you can use tables of Laplace Transforms, it is never a bad idea to know how to do the transform yourself.

1. 1
   Consider "Laplace Transforms" -- in part it is a system to convert time dependent domain relationships to a set of equations expressed in terms of the Laplace operator 's'. Then, the solution of the original problem is effected by "complex-algebra manipulations" in the 's' or Laplace domain rather than the time domain:^[1]
   • "Applying Laplace Transforms is analogous to using logarithms to simplify certain types of mathematical operations. By taking logarithms, numbers are transformed into powers of 10 or e (natural logarithms). As a result of the transformations, mathematical multiplications and divisions are replaced by additions and subtractions respectively."^[1]
2. 2
   "Similarly, apply Laplace Transforms to the analysis of systems which can be described by linear, ordinary time differential equations overcomes some of the complexities encountered in the time-domain solution of such equations.", and, also:^[1]
   • The Laplace Transform involves integrating from 0 to infinity of a time variable f(t) arrived at by multiplying f(t) by e ^-st.
   • f(t) is your applied function which must be defined for all positive values of t.
   • s is a complex algebra variable defined by: s = a +jω where j = sqrt(-1), so you will be partly using imaginary numbers.

1. 1
   Carry out the integration using integration by parts. Depending on your function, f(t) you may need to carry out integration by parts multiple times in order to fully integrate the integral.
   

   If you are calculating the bilateral Laplace transform replace the 0 limit with -∞
2. 2
   Put the limits into your result. Write out the equation replacing t with infinity then write out the negative of the same equation, this time replacing t with 0. Simplify this down as much as you can, remembering the following values:
   
3. 3
   Check your answer using a table of Laplace transforms.

Discontinuous Functions

A discontinuous function can be written as:

where c is a constant and a and b can be either constants or functions of t. While this example has only two parts, there can be any finite number.

1. 1
   Write out the sum of the Laplace transforms of each part of the discontinuous function, using the limits specified, rather than the usual 0 to ∞.
   
2. 2
   Calculate the Laplace transforms as shown above. Remember to substitute in the correct limits, rather than 0 and ∞.
   

   This example assumes a and b to be constants, the result will be more complicated if they are functions of t
3. 3
   Simplify the result as much as possible.
   

   This example assumes a and b to be constants, the result will be more complicated if they are functions of t

Using Properties of Laplace Transforms

1. 1
   Attempt to derive a Laplace transform of a function, if it closely resembles one or more other functions, which you do know the transform of. For example:
   • The Laplace transform of a linear combination of functions is the same linear combination of the Laplace transforms.
   • The Laplace transform of tf(t) is equal to -F'(s), where F(s) is the Laplace transform of f(t) and F'(s) is its derivative.
   • The Laplace transform of f'(t) is equal to sF(s)-f(0).
   • The Laplace transform of e^(at)f(t) is equal to F(s-a).
   • The Laplace transform of a convolution of two functions f and g is equal to the product of their Laplace transforms.
2. 2
   Use the various known properties of Laplace transforms to be able to derive them using the steps above. It is also useful to know the meaning behind each property.
3. 3
   Examine this simplified general statement, "The Laplace Transform of f(t) equals function F of s" and write:^[2] laplace{f(t)} = F(s)
   • Similarly, the Laplace transform of a function g(t) would be written: laplace{g(t)} = G(s)

• Laplace Transforms have many applications in mathematics, physics, optics, electrical engineering, control engineering, signal processing, and probability theory. It's invention in about 1782 was in work on probability. In physics, it is used for analysis of linear systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems.^[3]

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• How to Find the Range of a Function in Math