Have you ever heard about laplace transform?

If not yet, you are surely living in some other world! 

This concept is quite widely used in the field of technology or science to replace multiple algebra operations. 

When we are talking about the Laplace transform, we should mention its two vital concepts that are the initial value theorem and the final value theorem! 

Both known by the name of limiting theorems, the theorems hold high importance for the aspirants. 

If you too want to equip your knowledge bank with the apt information of both the theorems, you have particularly landed on the right blog post. 

Here, we are going to cover every minute detail regarding these theorems!

So let’s get started! 

Laplace transform 

This is an essential mathematical tool or concept which is helpful for converting the different or varied equations in case of time domain into the given algebraic equations with the frequency of the s domain. 

x(t) would be denoted as the mathematical domain function. 

Hence, in this case the given laplace transform would be:

L [x (t)] = X [s] = ഽ ∞    x[t] e -st dt . . 

                              - ∞

                               .[1]

Here you are given with a function which is defined to get real parameters which are independent of that function in respect to the  ∞. Thus, to give the integral value of the resultant function. 

Initial value theorem 

Let’s talk about the key concept of laplace transform i.e the initial value theorem. The theorem is also referred to as denoted by the IVT. It enables a user to get the initial value at the respective time t = [ 0+] for any given laplace transform function. 

Though, it will also save your time as you don’t need to work hard in order to find f[t] which can be a tedious process. 

Conditions to note down for this theorem are:

• The f[t] function and its respective f[t] derivative should be in the form of laplace transform
• If a given time t approaches towards the [ 0+] , the function f[t] needs to exist

In this case the respective function f [t] would be 0 for the given t>0. Though, it will not contain any impulse or any singularities in the high order at the point of its origin. 

Statement of this initial value theorem would be as follows:

F [t] ⇐ [ L] ⇒F [ s]

Here, in this case the f[t] or F[t] would be your laplace pair transform. 

In this case the theorem would be:

lim f [t] = lim s F [s] = f [ 0 +] 

t →0+            s  →∞

Proof of this theorem will be:

Laplace transform in is the function of f[t]

Lf[t] = ഽ∞   e -st f [t] dt = F[ s] 

            0

Of its derivative f ‘ [t] is 

Lf’ [t] = ഽ∞ e -st  f’[ t]dt = sF[s] - f [ 0 -] —-------------- [ 1] 

              0- 

This will be the type of differential theorem. 

Consider its integral part with e -st as the indeterminate with is splitted with two integral part] 

The final theorem you will get is;

ഽ∞    e -st f’ [ t]dt + f [0+] = sF[ s] —-----------

   0 +

We can easily write the above mentioned condition by taking the intention of integration limits [ 0- to ∞]. However, we can consider, negative value of its limits which will result in the positive value. 

Note: It is good to note that the laplace transform can only be applicable in the case of casual functions. 

On considering the infinity on both the sides we will say that;

f [0+] = lim sF[s] 

            s →∞

lim f [t] = lim sF[ s] = f[0+]

T →0+   s→∞

Hence, we have proved the initial value theorem. 

Final value theorem 

When we are talking about the final value theorem, we do know that this theorem too holds a great importance in the concept of laplace transform. In the case of this theorem, we find the final value. 

The theorem is best to use for the determination of the final value in the respective time domain while applying its given zero frequency with the components of the respective domain system in its place so you can ideally find its given value. 

Though, in some cases, this final theorem will be needed to get the final value which appears as a fine value to use in various conditions. Though, it may not be the final value in 

the case of your respective time domain. 

This is also an important theorem, which takes into account the following conditions:

• The given function f[t] will be used with the derivative f’[t] with the laplace transformable
• s[F]s will have no pole on its j-w axis in its right half where the F[s] would be the given laplace transform of the f[t] 

If the laplace transform in the case of f[t] or f’[t] do exists, F[s] can get the laplace transform as L [f(t)] = F[s].

Then we will get;

lim f [t]  = lim sF[s]

t →∞        s  → 0 

Let’s have a proof of it:

We know in the case that the:

L {f’ [ t]} 

= s L[f(t)] - f [0] 

= sF[s] - f[0] 

So, in this case:

sF[s] = L [f’(t)] + f[0] 

ഽ ∞  e - stf’ [t]dt + f[0] 

    0 

Here, we are taking the limit of s - 0 on both sides of the above-mentioned relation. 

Additional learning - device management in operating system

Another vital computer concept that should be of key importance for every computer aspirant is  device management in the operating system. It is a vital computer component which allows the user to view or control the computer hardware while managing all the respective operations. 

Wrapping up 

Initial value theorem and the final value theorem are the key concepts of mathematics that should be known by every aspirant. 

With this blog we tried to keep these concepts clear for you. Apply them ideally and enhance your learning.