Saint Rantic

The day before I stumbled upon the precise problem that I was searching for. I thought I knew calculus, but somehow it was not satisfying and complete. I learn by mapping my imagination with the math and then linking the to the facts that result. I could not simply get this mapping for calculus. I got the ‘gound-breaker’ the day before. If I find an answer to the problem then the Imagination, math and result linking would be done for calculus.

Let me get to the problem.

Consider a function

find a function f(x) such that

∫ f(x)dx = g(x)

I have not found the answer till now. What puzzles me is this.

Can there be a zero area output from an Integration Operation?

Till now I have found two ways in which this problem can be approached.

1) By reaching g(x) by making rect(E,-E) where E tends to 0;
2) By making a triangular signal peaking at x = 0 with value one, and making the width of it 0.

I am not able to differenciate these signals and then inegrate them in the limit. It simply requires the tools that I am not aware of. Or it requires some more imagination and gettting down to the basics.

This incident take me back to 2 to 3 years when i was struck with the same kind of an ‘end’.

It was the end of Signal Processing as I knew it to be. One of my idiot friends(who is in IISc now…. but other than that he does not qualify for a human being) quizzed my understanding about the meaning of an Impulse, or a Dirac Delta Function.

I used to assume that the value of an impulse is ‘very very’ high at the point of incidence of the impulse. Then my friend, who had a deeper understanding of ‘calculus’, gave me an explanation that an impulse is meaningful only under integration.

This was the trigger that I needed. I spent days and nights toiling with books in signal processing. That was the end of Signal Processing as i knew it to be. I unlearned everything that I learned. I learned that an impulse was defined in a limit and it was not defined at a ‘point’. I learned the definition of an impulse as a limit.

From then on understanding Fourier Transforms and other signal processing tools was just my imagination and its mapping with mathematics. I really learned a lot in the next two months at a pace that just envy right now. The remaining time in college was a ‘bliss’ or ’salvation’. I used to attack and gobble up every signal processing problem I could lay my hands on.

I wish I get the same ‘bliss’ by finding an answer to this problem.