# Alternate Representation of Function

• Hertz
In summary, the conversation discusses attempts to find a single line representation for a piecewise function defined as 1 at x = 0 and 0 everywhere else. The first attempt using a Fourier cosine series was unsuccessful due to the symmetry of the function. The second attempt using a Gaussian bell curve and a Taylor series was more promising, but the convergence of the series is uncertain. The discussion also touches on the uniqueness of the Dirac delta function and the challenges of finding alternative definitions of the piecewise function.
Hertz
Hi all,

the function that I'm posting about is a piecewise function defined as follows:
$$\Delta(x) = \left\{ \begin{array}{ll} 1 & \quad x = 0 \\ 0 & \quad x \neq 0 \end{array} \right.$$

I decided to call it capital delta because of its similarity to the Dirac delta function. What I'm looking for is another way to represent this function such that it's a single line expression like a limit or a convergent series or something.

My first attempt at this was to represent it as a Fourier cosine series but I'm not sure that I did this 100% correctly. What I did was consider symmetry on the interval ##[-a, a]## where ##a>0## and consider that ##\Delta(x)=0## at ##x= \pm a##. I used this to find the frequencies of the cosines used in the series. Then, I used Fourier's trick to solve for the coefficient. After all was said and done, I ended up with a series that did not clearly converge to 0 for all ##x \neq 0##. It's been a while since I did a Fourier series so I wasn't too confident in my answer anyways. I'd be interested to see if someone else can make this work.

For another attempt, I realized the similarity to the Gaussian bell curve. So I defined the function as:
$$\Delta(x)=\lim_{\alpha \to \infty}e^{-\alpha x^2}$$
For ##|x|>0## this limit converges to zero, for ##x=0##, this limit converges to 1. This next step might be useless, but I'll include it anyways. The next thing I did was represent the function inside the limit as a taylor series. Again, I'm not sure of the convergence of this Taylor series (though I assume it converges everywhere) and I'm not sure how the outer limit interferes with the series or where it converges.
$$\Delta(x)=\lim_{\alpha \to \infty}\sum_{n=0}^{\infty}\frac{(-\alpha x^2)^n}{n!}=\lim_{\alpha \to \infty}\sum_{n=0}^{\infty}(-1)^n\frac{\alpha^nx^{2n}}{n!}$$

Anyways, the above attempt is the best I could really get at finding a 'single line representation' of ##\Delta##. Like I mentioned though, I'm not sure if the last attempt (with the series) even converges. Any comments about this problem or my methods or whatever are appreciated. Or if you have another attempt I would love to see it.

P.S.
Lastly I just want to mention that this seems eerily familiar to the Dirac delta function so it would indicate that you might want to consider generalized functions when dealing with it. I'm very new to generalized functions but from the bit that I do know I can say this:

The ##\Delta## function defined above, when multiplied with another function (i.e. ##\Delta(x)f(x)##) is equal to ##f(x)## at zero and 0 everywhere else. Assuming that ##f## is finite at zero, we can say that ##\int_{-\infty}^{\infty}\Delta(x)f(x)dx=0## right? But isn't there an infinite number of functions similar to ##\Delta## which integrate to zero when multiplied by another function? So does this mean the definition of ##\Delta## as a generalized function is not unique?

A distribution is a uniquely defined functional, but if it is induced by a measurable function f, then f is not unique as such. For example, if f induces a distrubution D by $D(g) = \int^{\infty}_{-\infty} fgdx$, then $f+1_A$ induces the same distrubution where $1_A$ is the unit function on a null set A.

The distribution induced by your function $\Delta$ is the null-distrubution. Any function which is zero almost everywhere induces the null-distrubution, including the dirac-$\delta$ function. In particular, the dirac-delta distribution (which takes any measurable function to its value in 0) is not induced by the dirac-delta function (which is zero everywhere but in the point 0, where it is $\infty$).

As to your attempt to find alternative definitions of $\Delta$, your equation is perfectly fine as the taylor series represents $e^{-\alpha x^2}$ everywhere.

Hertz said:
My first attempt at this was to represent it as a Fourier cosine series but I'm not sure that I did this 100% correctly. What I did was consider symmetry on the interval ##[-a, a]## where ##a>0## and consider that ##\Delta(x)=0## at ##x= \pm a##. I used this to find the frequencies of the cosines used in the series. Then, I used Fourier's trick to solve for the coefficient. After all was said and done, I ended up with a series that did not clearly converge to 0 for all ##x \neq 0##. It's been a while since I did a Fourier series so I wasn't too confident in my answer anyways. I'd be interested to see if someone else can make this work.
You won't be able to make this work. The integral defining the Fourier coefficients cannot distinguish your function from the zero function since they only differ at one point. If calculate the coefficients correctly, they will all be zero, so the Fourier series will converge to the zero function, not to your function.

## 1. What is an alternate representation of function?

An alternate representation of function is a way of expressing a mathematical relationship between two or more variables using a different format or set of symbols. This can include graphs, tables, equations, or even real-world scenarios.

## 2. Why are alternate representations of function important?

Alternate representations of function allow for a better understanding of the relationship between variables and can help with problem-solving and making predictions. They also allow for different perspectives and interpretations of the same mathematical concept.

## 3. How are alternate representations of function used in science?

In science, alternate representations of function are used to model and understand various natural phenomena. For example, graphs can be used to represent the relationship between temperature and pressure in a gas, while equations can be used to model the growth of a population over time.

## 4. Can alternate representations of function be used to solve problems?

Yes, alternate representations of function can be used to solve problems in various fields, including science, engineering, and economics. By representing a function in different ways, it may become easier to analyze and manipulate the relationship between variables to find a solution.

## 5. Is there a specific alternate representation of function that is considered the most effective?

No, the most effective alternate representation of function can vary depending on the context and the individual's understanding. It is important to be familiar with different representations and to choose the one that best suits the problem at hand.

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