I am asking this question out of both curiosity and frustration. There are many problems in computer science which require you to perform operations on a finite set of numbers. It always bothers me that there is no way of mapping this discrete problem onto a continuous one and using the power of calculus to solve it, finally extracting out the answer of the original discrete problem.
Is this not possible? If it is, is there a branch of mathematics which deals with precisely this? Are we confined to solving such problems only by thinking of clever algorithmic techniques?
Thank you for taking the time to answer, and as always, apologies if the questions is silly.
Best Answer
The branch of math which connects calculus and combinatorics is called the theory of generating functions, or more evocatively analytic combinatorics. This theory is often used to find asymptotics of combinatorial sequences, such as might occur in the analysis of the running time of some algorithm. Here is a free textbook about it.
The branch of math which connects calculus and number theory is called analytic number theory.