Dirac Delta Function

The function being described on the right-hand side is the Heaviside step function, often written as H(x) or θ(x)

• Case 1 (x<0) The integral’s upper limit  x is negative. The integration range is from −∞ to x. This range does not include t=0 where the delta functon is non-zero. Therefore, you are integrating a function that is zero over the entire interval, and the result is 0.
• Case 2 (x>0): The integral’s upper limit x is positive. The integration range from  −∞ to xdoes include t=0. Because the entire “spike” of the delta function is included in the range, the integral evaluates to the total area of the delta function, which is 1.

Notice that .

Let’s take any function f(x) that is smooth everywhere except for a single jump discontinuity of height c at x=a.

We can represent this function f(x) as the sum of two parts:

1. A continuous smooth “base” function, let’s call it g(x).
2. A step function that adds the jump at the right place.

The formula is:

How do we find g(x)? The function g(x) is simply the “pre-jump” part of the function extended across the entire domain.

• For x < a, we have H(x-a)=0, so f(x) = g(x). This means g(x) is identical to f(x) before the jump.
• For x > a, we have H(x-a)=1, so f(x) = g(x) + c. This correctly describes the function after the jump.

Differentiating the Representation

Now that we have expressed f(x) in a form that is a sum of well-behaved functions (a continuous function and a Heaviside function), we can differentiate it using the sum rule:

Using the relationship , we get:

Now, what is g’(x)? Since g(x) is the continuous version of f(x), its derivative g’(x) is exactly the same as the derivative of f(x) away from the jump, i.e., for .

So, we can replace g’(x) with the notation f’(x)_{x a} to get the final formula:

Notice that cos(ßx) dx does not converge in the usual sense because the cosine function oscillates forever. To give it a meaningful value, we treat it as a distribution by examining its behavior in a limit.

It was previously shown that

The function has an area of π for any value of a. As a→0, the function becomes an infinitely thin, infinitely high spike at β=0, while its total area remains π. This is precisely the behavior of πδ(β).