Second shifting theorem of laplace transform

In summary, the shifted unit step function is used in defining the second shifting theorem in order to ensure that the Laplace transform is only calculated for functions defined on [0, \infty). This is necessary because the Laplace transform requires an integral from 0 to \infty. Multiplying by u(t-a) allows us to redefine the value of f(t-a) to be 0 for 0\le x< a, so that it is still defined on [0, \infty). This allows for the use of negative, positive, or zero values for "a", which can shift the function in the time domain. The Laplace transform is not defined for any other values outside of [0, \infty
  • #1
asitiaf
21
0
1. why do we need to use shifted unit step function in defining second shifting theorem?
2. why don't we instead calculate laplace transform of a time shifted function just by replacing t by t-a?
3. everywhere in the books as well as internet i see second shifting theorem defined for f(t-a).u(t-a),
why not just f(t-a)?
4. what is the value of laplace transforms for negative limits?
 
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  • #2
The Laplace transform is only defined for functions defined on [itex][0, \infty)[/itex] (since it requires an integral from 0 to [itex]\infty[/itex]). If f is defined on [itex][0, \infty)[/itex], f(t- a) is defined on [itex][a, \infty)[/itex]. Multiplying by u(t- a) just redefines the value to be 0 for [itex]0\le x< a[/itex] so that it is still defined on [itex][0, \infty)[/itex].

I don't know what you mean by "negative limits". If you are referring to the limits of integration or the values on which the function is defined, as I said above, they must be 0 and [itex]\infty[/itex]. The "value of the Laplace transform" is simply not defined for any other values.
 
  • #3
My question is why do we use unit time shifted step function?
I never said f is defined between zero and infinity.
f can also have t- values.
Then f(t-a) does not mean f is defined between a and infinity.
f(t-a) just means, f(t) shifted by "a" time, where "a" can be negative, positive or zero.
In case "a" is negative, its the advance time and the function gets shifted towards left (for t-).
 

Related to Second shifting theorem of laplace transform

1. What is the Second Shifting Theorem of Laplace Transform?

The Second Shifting Theorem of Laplace Transform is a mathematical principle that allows for the transformation of a function in the time domain to the frequency domain. It states that if a function f(t) has a Laplace transform F(s), then the Laplace transform of the function f(t-a) is e^(-as)F(s). In simpler terms, it means that shifting a function in the time domain by a constant amount results in a multiplication of the Laplace transform by an exponential factor in the frequency domain.

2. How is the Second Shifting Theorem used in practice?

The Second Shifting Theorem is commonly used in engineering and scientific fields to analyze the behavior of systems over time. It is especially useful in solving differential equations and in signal processing applications. By transforming a function from the time domain to the frequency domain, it becomes easier to analyze and manipulate the function using mathematical tools such as integration, differentiation, and convolution.

3. What is the difference between the First and Second Shifting Theorems?

The First and Second Shifting Theorems are both used for transforming functions from the time domain to the frequency domain. The main difference between them is that the First Shifting Theorem applies to functions that are shifted to the right, while the Second Shifting Theorem applies to functions that are shifted to the left. In other words, the First Shifting Theorem is used when a function is delayed, while the Second Shifting Theorem is used when a function is advanced.

4. Are there any limitations to the Second Shifting Theorem?

Like most mathematical theorems, the Second Shifting Theorem has some limitations. It can only be applied to functions that have a Laplace transform, and the function must be piecewise continuous and of exponential order. Additionally, the shifting amount (a) must be a positive real number.

5. Can the Second Shifting Theorem be extended to multiple shifts?

Yes, the Second Shifting Theorem can be extended to multiple shifts. If a function f(t) has a Laplace transform F(s), then the Laplace transform of the function f(t-a-b) is e^(-as)e^(-bs)F(s). This can be generalized for any number of shifts, with each shift resulting in an additional exponential factor in the frequency domain. However, all the limitations of the Second Shifting Theorem still apply to multiple shifts.

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