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I wish to generate a Kronecker delta like function, δ(x−a) = 1 at x=a (or something similar) and = 0 elsewhere. But the problem is I have to do this using a combination of sin, cos, tan (and/or inverses), exponential and log. To be more specific, I am using a software which is like black-box to me but I know it can recognize the above mentioned terms and their combinations. Now, I want to get output as 1 only when I input 'a' as an integer, say 3.

Edit: Again, it is a 'software' so there is a possibility that it will round off very small numbers to 0 and this may give an advantage (given that expression is good enough).

Any help will be really appreciated.

Aman_X
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    The Dirac delta is a distribution, it doesn't take the value $1$ at $0$, you can't represent it as a combination of elementary functions (but you can represent it as a limit of a sequence of functions, for example $\lim_{n \to \infty} \frac{n}2 1_{|x|< 1/n}$ with the limit taken in the sense of distributions) – reuns Oct 30 '19 at 16:26
  • I understand your point and would like to add that my choice is not limited to Dirac delta and also the output does not have to be exactly 1. I can use any other function as well, as long as it gives me a unique and fixed value. – Aman_X Oct 30 '19 at 17:33
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    No idea of what you mean, please make it clear – reuns Oct 30 '19 at 18:02
  • Assuming you want the value to be $0$ for $x \ne a$ (you don't mention this anywhere), what you are describing is not the Dirac delta function. It could be considered as a form of the Kronecker delta, where the indices are allowed to vary over all real numbers. It is also the characteristic function of the set ${a}$. – Paul Sinclair Oct 30 '19 at 23:43
  • @PaulSinclair Yes, I think Kronecker delta may be the right term. I did try to edit my previous comment and question but it was locked because of some unknown reason, but I did tag my question with Kronecker-delta. – Aman_X Oct 31 '19 at 10:31

1 Answers1

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Sorry, but this can't be done (at least to perfect accuracy). Defining $$\chi_a(x) =\begin{cases}1, & x = a\\0,& x \ne a\end{cases}$$ This is a function which is discontinuous at $a$. Assuming that what you are allowed to use are

  • Constant functions $f(x) = c\ \forall x$
  • $f(x) = x$
  • $|x|$
  • $\cos x$
  • $\sin x$
  • $\tan x$
  • $\arccos x$
  • $\arcsin x$
  • $\arctan x$
  • $e^x$
  • $\log x$

And the operations of

  • addition
  • subtraction
  • multiplication
  • division
  • function composition

Unfortunately, no combination of functions using the operations can give you what you want. The problem is that every starting function is continuous everywhere it is defined. And every operation preserves continuity everywhere the result is defined (division by $0$ is undefined, even if the numerator is $0$). The lack of definition removes values from the domain, and there is no way with these functions of extending the domain to include them again. Since $\chi_a$ is defined everywhere, such restrictions cannot be allowed. So you must stick to where continuity is preserved.

However, if you have at least one starting function available that is defined but discontinuous at some point, then it is likely possible. So examine your function choices carefully and determine if they are all continuous. Also examine if there is some operation available not mentioned above. Any possibility of piecewise definition would of course make it easy.

Paul Sinclair
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  • There is another conditions (if it helps)...the maximum value integer 'a' can take is 8. – Aman_X Oct 31 '19 at 10:34
  • @Aman_X - Sorry, but that doesn't help. $a$ just moves the place where we'd like the discontinuity to be. But no matter the value of $a$, every function is either continuous or undefined at $a$. There is still no way of building a function that is defined, but discontinuous. – Paul Sinclair Oct 31 '19 at 17:17