Unit Step Function or Series?

In summary, the conversation discusses finding a function that encompasses a given equation with different values at different intervals. The attempt at a solution involves defining a series and considering the use of discontinuity functions, specifically mentioning the floor function.
  • #1

Homework Statement


I have an equation that has the following values at different intervals:

It is:

r when 0<x<2Pi
r - (1)d when 2Pi<x<4Pi
r - (2)d when 4Pi<x<6Pi

And so on. I want to find a function that encompasses this whole function. Unit functions / discontinuity functions are fine; as long as I can take derivatives in the future.

2. The attempt at a solution

The furthest I could get is to define a series as follows:

r - n*d when 2nPi < x < 2(n+1)*Pi

At this point, my mind thinks discontinuity functions, but those would only work if 'n' was always a constant value, and didn't increase by 1 each iteration. Thank you!
 
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  • #2
cough, cough...the floor function.
 

1. What is a unit step function or series?

A unit step function or series is a mathematical function that is defined as 0 for all negative inputs and 1 for all positive inputs. It is represented by the symbol u(t) and is often used to model sudden changes or transitions in a system.

2. What is the difference between a unit step function and a unit step series?

The main difference between a unit step function and a unit step series is that a function is continuous, while a series is discrete. This means that a function can take on any value within a given range, while a series can only take on specific values.

3. How is a unit step function or series used in real-world applications?

Unit step functions and series have a wide range of applications in fields such as engineering, physics, and economics. They are often used to model physical systems, such as electrical circuits, and to analyze the behavior of systems over time.

4. Can a unit step function or series have multiple steps?

Yes, a unit step function or series can have multiple steps. This occurs when there are multiple sudden changes or transitions in a system, and each step represents a different value for the function.

5. What is the Laplace transform of a unit step function or series?

The Laplace transform of a unit step function or series is 1/s, where s is the complex variable in the Laplace domain. This transform is often used to simplify the analysis of systems that involve unit step functions or series.

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