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How can I find the derivative of a step function?
Hertz said:The derivative of the step function is the Dirac Delta function. I don't know how good of a proof this is but it's the best I could come up with haha:
julian said:[itex]= - {1 \over 2 \pi i} \lim_{\epsilon \rightarrow 0} \int_{-\infty}^\infty {-i \omega \over \omega + i \epsilon} e^{-i \omega \tau} d \omega[/itex]
[itex]= {1 \over 2 \pi} \int_{-\infty}^\infty e^{-i \omega \tau} d \omega = \delta (\tau)[/itex]
pwsnafu said:That is not how the Dirac delta is defined, and proving ##\delta = H'## uses a different definition of derivative. Indeed it should be obvious there is no piecewise function with the properties you outlined because integrating over a point is zero from the definition of the integral.
What are you using to justify the limit passage?
pwsnafu said:That is not how the Dirac delta is defined, and proving ##\delta = H'## uses a different definition of derivative. Indeed it should be obvious there is no piecewise function with the properties you outlined because integrating over a point is zero from the definition of the integral.
Hertz said:Anyways, consider the integral $$\int_{a}^{b}\delta(x)dx$$ where ##a<b<0## or ##b>a>0##. This integral equals zero right? <snip>
I see why integrating over a single point of finite value is guaranteed to equal zero, but integrating over a single point of infinite value is like a zero times infinity indeterminate case isn't it?
I just know that ##\delta = \frac{d}{dx}H##
pwsnafu said:...
pwsnafu said:In measure theory we define ##0 \cdot \infty## equal to zero, so the integral is zero from the definition of the integral.
This is why there is no real variable function with the properties of Dirac delta. Dirac is a generalized function, it does not take point values.
The Dirac delta function, also known as the impulse function, is a mathematical function used to model a point or spike at a specific location. It is defined as zero everywhere except at the origin, where it is infinite, and its integral over the entire real line is equal to one.
The Kronecker delta function is a discrete version of the Dirac delta function, where it takes the value of one when the input is zero and zero otherwise. Both functions are used to represent impulses or spikes, but the Dirac delta function is a continuous function while the Kronecker delta function is a discrete function.
The Dirac delta function is often used in physics to represent a point mass or charge at a specific location. It is also used to model idealized point sources of energy, such as a point source of light or heat.
In engineering and signal processing, the Dirac delta function is used to represent a very short or instantaneous signal. It is also used to model impulse responses of systems, such as filters and amplifiers.
The Dirac delta function has several important properties, including the sifting property, linearity, and time-shifting. It is also related to the unit step function and the Heaviside function, and it can be convolved with other functions to produce interesting results.