Because of the form of the velocity and speed probability distribution functions in kinetic molecular theory,
(1a,b)we find ourselves having to do integrals of the form,
(2)where, in our cases, x could be vx, vy, vz, or just v, and λ = m/2kT, and n ≥ 0 is an even or odd integer.
When n is an even integer or zero these integrals are referred to as "Gaussian" integrals, or better, integrals of Gaussian functions. The prototype of all Gaussian integrals is
(3)The first thing we will do is show you how you can determine the value of this integral for yourself. Begin by squaring the integral on the left hand side of Equation 3,
(4)The variable x in Equation 4 is called a variable of integration or a "dummy variable." It really doesn't matter what symbol you make it since that symbol is "integrated out." On the right hand side of Equation 4 let's change the variable in the second integral to y and combine the two integrals into one double integral,
(5a,b,c,d)Notice that in Equation 5d we are integrating exp(− (x2 + y2)) over the whole x-y plane. There is no reason why we couldn't just as well do this integration in plane polar coordinates, r and θ . In plane polar coordinates dxdy becomes rdθ dr and x2 + y2 becomes r2. The integral becomes
7a,b,c,dwhere we have made use of the fact that the integral on the right of 7c is equal to unity. That is,
(8)We need only take the square root of Equation 7d to see that,
(3)Note, however, that there is no parameter, λ , in equation 3. To actually do the integral in Equation 2 (with n = 0) let's change variables. This time let λx2 = u2, then x√λ= u, and so dx = du/√λ . The integral becomes (note, for the limits of integration, that when x = ± ∞ then u = ±∞ ),
Suppose now the integer, n, in Equation 2 is not zero. If n is even we could go through a procedure similar to the one we have just carried out to find the value of the integral, but there is an easier and quicker way to obtain this integral. Consider that
(10)This result may seem strange because we are not used to taking derivatives of variables inside integrals. However, it is a correct result since the integral on the left of Equation 10 is truly a function of λ and not a function of x because x is the integration variable and is "integrated out." Since we already know the integral on the right of Equation 10 we can write,
(11a,b,c)We can extend this procedure to obtain the integrals with any even positive power of x in the integrand,
(12a,b)where it is understood that n must be an even positive integer.
"Pseudo" Gaussian Integrals
We now come to the case where n in Equation 2 is an odd positive integer. That is, in the integral,
(2)n is an odd positive integer.
Strictly speaking, these integrals are not Gaussian integrals because they can be calculated using standard methods. (One thing is clear, with n an odd integer the limits can not be from −∞ to + ∞ because then the integrand would be an odd function of x and the integral would vanish.) The prototype of these integrals is,
(13)where, in this case, n = 1.
This integral can be calculated, as was done in Equations 6 and 7, by changing variables. Let us set u = λx2. Then du = 2λxdx or xdx = du/2λ . We can set the limits of the new integral by noting that when x = 0 then u = 0, and when x = ∞ then u = ∞ . Then
(14a,b,c,d)Where, as before, we are assuming that everyone knows (or can easily show) that,
(8)What about the rest of the integrals with odd n? These integrals can be easily obtained in the same manner that we obtained the integrals in Equations 11 and 12. For example,
(15a, b, c)
In the same manner it is easily seen that, for the general case of odd n, we have,
The Error Function, erf(x)
Integrals of the form,
(2)where the limits are not −∞ , 0, or +∞ can not be given in terms of elementary functions. However, all such integrals can be written in terms of one particular integral,
(17)We can see from arguments given above that, if x goes to ∞ , the value of the expression 17 goes to (√π)/2. The error function, erf(x), is defined such that erf(∞) is equal to unity. That is,
(18)erf(x) can be thought of as just another transcendental function like sin(x) and exp(x). For example, to twelve significant figures, erf(0.5) = 0.520499877813....
Other integrals can be written in terms of this one basic integral. For example,
(19a,b)In this integral we have made the variable change λu2 = y2.
More complicated integrals are, of course, more complicated. However we can still use the techniques we used above to find integrals with even powers of u in the integrand, For example,
(20a,b,c,d)We can take derivatives of erf(x),
(21)Second and higher derivatives are then easy to take.
There are tables of erf(x) in many collections
of mathematical tables (for example the NBS Handbook of Mathematical Functions).
The error function is also a built-in function in such symbol manipulation
programs, such as Derive and Mathematica, and it is an add-in in Microsoft
Other Indefinite Integrals of the Gaussian Function
Other indefinite integrals of the Gaussian function can be written in terms of erf(x). For example,
Copyright 2004, W. R. Salzman
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