Sum is to Integral as Product is to (?).?

Question

I don't know if this operator is formally defined, but if it is, anyone know what it is?

Answer 1

Interesting question!

I think you could define such a thing, but it is not necessary, as to do a product of things arbitrarily close to 1, we could just take the log of them, and then do the integral and then put the result in the exponent.

To put it another way (I am being slightly informal and leaving off the index of the sum/product):

Product(f(x)) = e^(Sum(ln(f(x)))

Then we can say:

Sum is to Integral as Product is to e^(Integral(ln))

Not sure if that is what you were thinking of, but it will be interesting to see what others come up with.

Answer 2

It is not defined.

In fact, such an operator would be trivial, because any finite length of a curve contains an infinite number of points. By choosing a small enough length, it is possible to ensure that the segment is entirely either greater than or less than 1.

It is obvious that an infinite number of values multiplied together will go to infinity if all of the values are greater than one, and will go to zero if all of the values are less than one.

Answer 3

I have never seen anything like this defined. It doesn't seem possible for such an operation to be finite, just thinking about it naively. In what context is this problem arising?

Answer 4

There is a formally defined product operator, ∏, but it is analogous to ∑, i.e.
. . . . n
n! = ∏ k
. . . k=1

I've never seen an equivalent operator which uses differentials.

Answer 5

Multiply

Answer 6

I don't think this operation is defined.

Answer 7

Multiplication maybe?

Answer 8

I would say degree