Help me to survive from a big dilemma(Unit step function)

This integral is divergent, and the step function does not affect this result. In summary, the value of the integral is \infty, and the step function does not need to be included in the answer.
  • #1
sinaphysics
8
0
Consider:

[tex] u(t)=\begin{cases} 1\quad \quad \quad \quad t>0 \\ 0\quad \quad \quad \quad t<0 \end{cases} [/tex]
Now I want to calculate this:

[tex] \int _{ 0 }^{ a }{ \frac { u(t)-u(t-a) }{ { t }^{ 2 } } } dt [/tex]
whereas: a>0
What is confusing me is this point that should our answer for the integral include the step function again?
 
Mathematics news on Phys.org
  • #2
sinaphysics said:
Consider:

[tex] u(t)=\begin{cases} 1\quad \quad \quad \quad t>0 \\ 0\quad \quad \quad \quad t<0 \end{cases} [/tex]
Now I want to calculate this:

[tex] \int _{ 0 }^{ a }{ \frac { u(t)-u(t-a) }{ { t }^{ 2 } } } dt [/tex]
whereas: a>0
What is confusing me is this point that should our answer for the integral include the step function again?

It's not clear what you mean when you say, "should our answer for the integral include the step function again?"

What I see is a step function which is active when t > 0 but which is turned off for t > a, so that there is a non-zero integrand in the interval 0 < t < a.
 
  • Like
Likes sinaphysics
  • #3
In this integral t is always between 0 and a so t is positive while t- a is negative. u(t)= 1 and u(t- a)= 0.

Your integral is just [itex]\int_0^a \frac{1}{t^2} dt[/itex]
 
  • Like
Likes sinaphysics

What is a unit step function?

A unit step function, also known as a Heaviside step function, is a mathematical function that is defined as 0 for negative input values and 1 for positive input values. It is represented by the symbol u(t) or θ(t), where t is the input variable.

How can a unit step function be used to help me survive from a big dilemma?

A unit step function can be used to model scenarios in which a decision needs to be made at a specific point in time. It can help you determine the optimal time to take action in order to survive from a big dilemma. For example, if you are facing a financial crisis, a unit step function can help you decide when to make a big purchase or investment.

Can a unit step function be applied to real-life situations?

Yes, unit step functions have many real-life applications in fields such as physics, engineering, economics, and biology. They can be used to model various scenarios, such as the spread of a disease, the growth of a population, or the charging of a capacitor in an electrical circuit.

What are some properties of a unit step function?

Some properties of a unit step function include:

  • It is discontinuous at t = 0.
  • It is a non-differentiable function.
  • The integral of a unit step function is a ramp function.
  • It is an even function, meaning that u(-t) = u(t).

Can a unit step function be generalized to multiple dimensions?

Yes, a unit step function can be extended to multiple dimensions, such as two or three dimensions. In these cases, it is known as a unit step surface or a unit step volume. It can be used to represent situations in which a decision needs to be made in multiple variables simultaneously, such as in optimization problems or control systems.

Similar threads

Replies
3
Views
997
  • Differential Equations
Replies
1
Views
676
  • Differential Equations
Replies
7
Views
308
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
Replies
3
Views
300
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
646
  • General Math
Replies
1
Views
7K
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
429
Back
Top