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.
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
And the operations of
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.