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Last Updated: Mar 27, 2024
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Initial Value Theorem of Laplace Transform

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Prerita Agarwal
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Introduction

Laplace transform is very useful in the various fields of science and technology as Laplace transform replaces operations of calculus by operation of algebra. Initial and Final value theorems are basic properties of Laplace transform. These theorems were given by French mathematician and physicist Pierre Simon Marquis De Laplace. Initial and Final value theorem are collectively called Limiting theorems. In this article, we will be focusing on these two theorems.

Initial Value Theorem of Laplace Transform

Laplace Transform

The Laplace transform is a powerful mathematical tool used in engineering, physics, and applied mathematics to analyze linear time-invariant systems, particularly in the context of differential equations. It transforms a function of time, often a function describing a dynamic system, into a complex function of a complex variable. 

Let f(t) be a given function defined for all t ≥ 0  If we multiply f(t) by e-st, where s is a real or complex parameter, independent of t and integrate with respect to t from 0 to ∞, and if the resulting integral exists, the result will be a function of s, denoted by F(s).

F(s)= L{f(t)} = 0e−st f(t) dt

The function F(s) is called Laplace transform of the function f(s).

Key points about Laplace transform:

  • Linearity: L{af(t)+bg(t)}=aL{f(t)}+bL{g(t)}
     
  • Derivative property: L{f′(t)}=sF(s)−f(0+)
     
  • Integration property: L{∫0tf(τ)}=1/sF(s)
     
  • Time-shift property: L{f(ta)u(ta)}=easL{f(t)}
     
  • Convolution property: L{fg}=F(s)G(s)

Laplace transforms are particularly useful for solving linear ordinary differential equations (ODEs) and linear partial differential equations (PDEs) with constant coefficients. The transformed equations often simplify into algebraic equations, making them easier to solve. After finding the solution in the Laplace domain, an inverse Laplace transform is applied to obtain the solution in the time domain.

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Initial value theorem

Conditions for initial value theorem are 

  1. If t approaches to 0+, function f(t) should exist

 

i.e., lim t→0+ f(t) should exist

Function f(t) and its derivative f’(t) should be laplace transformable.
 

Statement

Initial value theorem is given by 

lim t→0+ f(t) = lim s→ sF(s) = f(0+)

Where F(s) is laplace transform of f(t).

Proof

We know that,

 𝐿[𝑓 ′ (𝑡)] = 𝑠 𝐿[𝑓(𝑡)] − 𝑓(0) = 𝑠𝐹(𝑠) − 𝑓(0) 

∴ 𝑠𝐹(𝑠) 

= 𝐿[𝑓 ′ (𝑡)] + 𝑓(0) 

= ∫0   e −𝑠𝑡𝑓 ′ (𝑡)𝑑𝑡 + 𝑓(0) 

Taking limit as 𝑠 → ∞ on both sides, 

we have 

lim𝑠→∞ 𝑠𝐹(𝑠) = lim 𝑠→∞ [∫0 ∞ e −𝑠𝑡𝑓 ′ (𝑡)𝑑𝑡 + 𝑓(0) ]

= lim𝑠→∞ [∫0 ∞  e −𝑠𝑡𝑓 ′ (𝑡)𝑑𝑡 ] + 𝑓(0) 

= ∫0  lim [ e −𝑠𝑡𝑓 ′ (𝑡)]𝑑𝑡 + 𝑓(0) 

= 0 + 𝑓(0)           ∵ e−∞ = 0 

= 𝑓(0) 

= lim    𝑓(𝑡) 

   𝑡→0

∴  lim𝑠→∞ 𝑠𝐹(𝑠) = lim 𝑡→0 𝑓(𝑡)           ( proved)

Final value theorem

Conditions for final value theorem are 

  1. Function f(t) and its derivative f’(t) should be laplace transformable.
     
  2. 𝑠𝐹(𝑠) has no pole on j-w axis and right half plane where 𝐹(𝑠) is the laplace transform of 𝑓(𝑡).

 

Statement

If  Laplace transforms of 𝑓(𝑡) and 𝑓 ′(𝑡) exist and 𝐹(𝑠) is laplace transform of 𝑓(𝑡) i.e. 𝐿[𝑓(𝑡)] = 𝐹(𝑠),then

lim t→ f(t) = lims→ 0 sF(s)

Proof

We know that 

𝐿{𝑓 ′ (𝑡)} 

= 𝑠 𝐿[𝑓(𝑡)] − 𝑓(0) 

= 𝑠𝐹(𝑠) − 𝑓(0) 

 

So, 𝑠𝐹(𝑠) = 𝐿[𝑓 ′ (𝑡)] + 𝑓(0) 

= ∫0  e- st f' (t) dt + f(0)

Taking limit s →0 on both sides of the above relation,

lims→0 sF(s) = lim s→0 [∫0 ∞  e-st f'(t)dt + f(0)]

                    = lims→0 [∫0 e-st f'(t) dt] + f(0)

                    =  ∫0 ∞  lim s→0 [e-st f'(t)] dt + f(0)

                    = ∫0   f'(t)dt + f(0)

                    = [f(t)]0 + f(0)

                    = f(∞) - f(0) + f(0)

                    = f(∞)

                    = lim t→∞ f(t)

Therefore, limt→∞ f(t) = lims→ 0 sF(s)

Application of Initial value theorem

These are some instances of initial value theory applications:

  • The initial value theorem allows us to derive a signal's initial value, or x(0), directly from its Z-transform, or X(z), without first having to determine X's inverse Z-transform (z).
     
  • The initial value theorem is a theorem used in mathematical analysis to connect frequency domain statements to time domain behaviour as time approaches zero.
     
  • The Initial Value Theorem can be used to calculate transient state gain.

Numerical Example

Let's consider a simple numerical example to illustrate the Laplace transform. Suppose we have a function f(t) = e-2t, and we want to find its Laplace transform.
The Laplace transform of f(t) is given by:

L{e−2t}=∫0​e−ste−2tdt

Simplifying this integral:

L{e−2t}=∫0​e−(s+2)tdt

Now, we can solve the integral:

L{e−2t}=-1/s+2e−(s+2)t|0∞​

L{e−2t}=limt→∞​(-1/s+2e−(s+2)t)−(-1/s+2)

Since limt→∞​e−(s+2)t=0(because s is assumed to have a positive real part), the result is:

L{e−2t}=1/s+2

So, the Laplace transform of f(t)=e−2t  is 1/s+2.

Frequently Asked Questions

What is initial and Final Value Theorem?

The Initial Value Theorem (IVT) and Final Value Theorem (FVT) are Laplace transform theorems used to determine a function's behavior as time approaches zero or infinity.

What is initial value theorem expression?

The Initial Value Theorem expression is given by lim s→∞ sF(s), where F(s) is the Laplace transform.

What are the rules for initial value theorem?

The rules for intial value theroem are that function must have a finite number of poles and the limit of sF(s) as s approaches each pole must exist.

What is the initial value theorem in PDC?

In Pole-Zero Analysis for Phase-locked Loops (PDC), the Initial Value Theorem evaluates the transient response, providing insights into the system's stability and initial conditions.

Conclusion

This article covered Initial and final value theorem of Laplace transform. We have also discussed the applications of intial value theorem. We also explained a numerical example based on the Laplace transform. 

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Topics covered
1.
Introduction
2.
Laplace Transform
3.
Initial value theorem
3.1.
Statement
3.2.
Proof
4.
Final value theorem
4.1.
Statement
4.2.
Proof
5.
Application of Initial value theorem
6.
Numerical Example
7.
Frequently Asked Questions
7.1.
What is initial and Final Value Theorem?
7.2.
What is initial value theorem expression?
7.3.
What are the rules for initial value theorem?
7.4.
What is the initial value theorem in PDC?
8.
Conclusion