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Many physical process, and other processes that we like to think about, can be described using functions. In general, the function we seek is a complete description of the process, and we must construct this function from our incomplete knowledge of the process.

Differential Equations are the best example of this construction that I know of. Based on the rate of change for a process, you can determine a function that describes the process at all times. Because we know algebra and calculus, we can also find an explicit formula for the function in many cases. Another advantage of diff-eqs is that the solutions satisfy uniqueness in many cases.

I believe that mathematical physics needs more tools for uniquely specifying functions given incomplete, semi-unrelated information (such as the rate of change). I am aware that there is a generalized concept of a functional equation, which may involve derivatives, integrals, or inverses etc of an unknown function. I do not understand how these equations could naturally arise (other then just writing one down and then trying to solve it). This is my essential question: What other methods exist for uniquely specifying a function and how are these methods applied?

Here is a much more specific question: Suppose we have f(a) = 0 and f(b) = 0, and we have that the area under f from a to b is A, and we have that the arc length of f from a to b is C.

The question is, is f determined uniquely? Intuitively I say yes, but I am asking if it has been proven. Opinions and counterexamples are welcome.

If f is determined uniquely, can we construct an explicit formula for f? Perhaps this falls under the branch of integral equations (although I doubt it). I also thought it might be a problem in the calculus of variations involving constraints, but can't really say more then that.

Edit: I have added the condition that the function be strictly positive in the interval a to b, and that uniqueness be relaxed to mean "within a multiplicative constant", as in differential equations.

Differential Equations are the best example of this construction that I know of. Based on the rate of change for a process, you can determine a function that describes the process at all times. Because we know algebra and calculus, we can also find an explicit formula for the function in many cases. Another advantage of diff-eqs is that the solutions satisfy uniqueness in many cases.

I believe that mathematical physics needs more tools for uniquely specifying functions given incomplete, semi-unrelated information (such as the rate of change). I am aware that there is a generalized concept of a functional equation, which may involve derivatives, integrals, or inverses etc of an unknown function. I do not understand how these equations could naturally arise (other then just writing one down and then trying to solve it). This is my essential question: What other methods exist for uniquely specifying a function and how are these methods applied?

Here is a much more specific question: Suppose we have f(a) = 0 and f(b) = 0, and we have that the area under f from a to b is A, and we have that the arc length of f from a to b is C.

The question is, is f determined uniquely? Intuitively I say yes, but I am asking if it has been proven. Opinions and counterexamples are welcome.

If f is determined uniquely, can we construct an explicit formula for f? Perhaps this falls under the branch of integral equations (although I doubt it). I also thought it might be a problem in the calculus of variations involving constraints, but can't really say more then that.

Edit: I have added the condition that the function be strictly positive in the interval a to b, and that uniqueness be relaxed to mean "within a multiplicative constant", as in differential equations.

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