Why is the upper limit zero in this Laplace transformation problem?

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In summary, the conversation discusses a Laplace transformation problem with a lower limit constant and upper limit infinity. The variable is represented by t and the question is raised about why the upper limit is zero. The possibility of the limit being zero is explored and it is concluded that if s is greater than i, then the limit will be zero. The concept of an integral not existing for i<s is also discussed, with the understanding that s and i represent complex numbers. The conversation then delves into the details of the exponential component of the problem, with a focus on the imaginary unit and its impact on the limit at infinity. The conclusion is that the anti-derivative goes to 0 due to the negative exponential, despite the oscillating component
  • #1
neelakash
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I was performing a Laplace transformation problem,where I happened to face:

{lower limit constant and upper limit infinity} exp [(i-s)t] the variable being t.

I am not sure if the upper limit gives zero,but if I assume that the answer becomes correct.

Can anyone please tell me why the upper limit is zero here?
 
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  • #2
OK,I am writing it in LaTeX:

[tex]\int^\infty_{2\pi/3}[/tex] [tex]\ e^{(i-s)t} [/tex] [tex]\ dt [/tex]
 
  • #3
Friends,I do not know why the exponential dd not appear.Please assume this and let me know:
 
  • #4
Well, obviously, the anti-derivative is
[tex]\int e^{(i-s)t}dt= \frac{1}{i-s}e^{(i-s)t}[/tex]
IF s> i, then the limit as t goes to infinity will be 0. If [itex]s\le i[/itex] the integral does not exist.
 
  • #5
I see...Thank you.

Can you please tell me why the integral does not exist for i<s?
 
  • #6
Moreover, what does an expression like [itex]s \leq i[/itex] mean, considering that the complex numbers are not ordered?

If [itex]s = \sigma + i\omega[/itex], then

[tex]\begin{array}{rcl}\frac 1 {i - s} e^{(i -s)t} & = & \frac1 {i - \sigma - i\omega} e ^{(i- \sigma - i\omega)t}\\&&\\ &=& \frac 1 {-\sigma + i(1-\omega)} e^{i(1-\omega)t} e^{-\sigma t}\end{array}[/tex]

which has a finite limit at infinity only if [itex]\sigma > 0[/itex], and hence [itex]\Re(s) > 0[/itex].

Hmm...maybe that's what was meant originally, then.
 
  • #7
Yes,I afree.
If you take the modulus,the ghost of exp[it] runs away and the thing goes to zero as t tends to infinity
 
  • #8
My mistake. I had not realized that "i" was the imaginary unit! In that case, separate it into two parts. [itex]e^{(i-s)t}= e^{it}e^{-st}[/itex]. The [itex]e^{it}[/itex] part oscilates (it is a sin, cos combination) while e^{-st} will go to 0 as t goes to infinity because of the negative exponential. The entire product goes to 0 very quickly so the anti-derivative goes to 0.

(The "ghost" of eit- I like that.)
 
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What does "upper limit gives zero" mean?

"Upper limit gives zero" is a phrase commonly used in scientific research, particularly in the field of statistics. It refers to the idea that when the upper limit of a measurement or variable is reached, the value or result is considered to be zero or nonexistent.

Why is this concept important in scientific research?

This concept is important because it helps researchers to understand the limitations of their data or experiments. It also allows for more accurate analysis and interpretation of results, as well as highlighting potential errors or biases in the data.

How does "upper limit gives zero" relate to statistical significance?

In statistical analysis, the concept of "upper limit gives zero" is often used to determine if a result is statistically significant or not. If the upper limit of a measurement or variable is reached, it suggests that the difference or relationship being studied is not significant and may be due to chance.

Can the "upper limit gives zero" concept be applied to all scientific disciplines?

Yes, the concept of "upper limit gives zero" can be applied to all scientific disciplines, as it is a fundamental principle in the interpretation of data and results. It is particularly relevant in fields such as biology, chemistry, physics, and social sciences.

Are there any exceptions to the "upper limit gives zero" concept?

There may be some exceptions to this concept in certain scenarios, such as when dealing with extremely small or large values, or when using different statistical methods. However, the general principle of "upper limit gives zero" holds true in most cases and is an important consideration in scientific research.

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