Heavisidefüggvényre

The Heaviside step function, often denoted as H(t) or θ(t), is a discontinuous function that plays a significant role in various fields of engineering, physics, and mathematics. It is defined as having a value of zero for negative input values and a value of one for non-negative input values. Mathematically, it is expressed as:

H(t) = 0 if t < 0

H(t) = 1 if t >= 0

At t=0, the value of the function is often defined as 1/2, though this convention is not universal and the value at this single point can be arbitrary for many applications. The function's name honors the British electrical engineer and mathematician Oliver Heaviside.

The Heaviside step function is fundamental in describing systems that have an instantaneous on or off state. For instance, in electrical circuits, it can represent the application of a voltage or current at a specific time. In control systems, it models the activation of a switch or a trigger. Its discontinuity at the origin makes it a useful tool for analyzing systems with sudden changes.

The function can be expressed using the Dirac delta function, which is its derivative. It also has applications in probability theory, where it can represent an indicator function for a non-negative random variable. The Heaviside step function is a cornerstone for understanding and modeling transient behavior in dynamic systems.