Ordinarily the exponential function is trivial to integrate: simply multiply it by the differential of its argument and the integration leaves the function unchanged. But in this case the differential multiplying the function is not the differential of the function痴 argument, so we will have to do something clever. Fortunately for us, other mathematicians have already done that something clever.

We begin by augmenting the number line (the x-axis) with a y-axis, thereby expanding the number line into an x-y plane. We now have our exponential defined on a two-dimensional manifold. Extend the lower limit of the integral from zero to negative infinity and let and also Then we have

(Eq地 1)

We now have an exponential function of circular symmetry relative to the
origin of our x-y plane integrated over an infinitely wide square region on that
plane. Because the value of that exponential function declines rapidly toward
zero with increasing distance from the origin, we get a result negligibly
different from the square integration if we integrate the function over an
infinitely wide disc on the x-y plane, so we can translate I^{2}
into polar coordinates, which means that we have

(Eq地s 2)

Thus we have Equation 1 as

(Eq地 3)

From that we obtain By symmetry we have then

(Eq地 4)

Reference: Reif, F., "Fundamentals of Statistical and Thermal Physics", McGraw-Hill Book Company, New York, 1965 (LCCCN 63-22730), Pg 606 - 607.

Example Application:

Gaussian Probability (The Maxwell Distribution)

We want to calculate the distribution of the velocities of particles in an ideal monatomic gas (such as helium or neon), given the absolute temperature of the gas and the number of atoms that constitute it. We assume that the gas is sufficiently dilute that the particles interact rarely; that is, we assume that the mean free path each particle follows between collisions is long relative to the size of the particle. We can thus regard the gas as a heat reservoir for the particle, which means that the collisions that the particle enacts do not change the temperature of the gas. Analysis of the gas under those assumptions gives us, then, the distribution that James Clerk Maxwell derived in 1860.

We have a spatial density n of particles, each carrying mass m at a speed v in a gas of absolute temperature T. For the phase space density, the number of particles per cubic meter per meter per second cubed, we have

(Eq地 A-1)

Now we want to calculate the density of particles moving in the x-direction
only. To obtain the desired figure we must integrate the above expression over
the v_{y} and v_{z} coordinates in phase space. We thus have

(Eq地 A-2)

subject to the normalization criterion

(Eq地 A-3)

When we carry out the integrations in Equation A-2 we get

(Eq地 A-4)

That then gives us the means to describe the properties of an ideal gas.

efefefefef