For an integral over the whole real line, the transformation used is :

Laplace noted that though the terms themselves were small, when Integral over time they could become important.

The action is usually an integral over time.

Where is a normalizing constant such the integral over all possible and is equal to 1.

The trace (tr) becomes an integral over the configuration space.

The difference is in the third term, the integral over the source.

In general, an integral over a set E of a function f is written:

The gravitational potential energy could then be found as the integral over all the shells from the centre to its outer radius.

Since J is an integral over a full period, it is only a function of the energy.

The average crossing number is then defined as the integral over the unit sphere: