Integrated error function

The argument of the error function integral is related to the gaussian distribution, also frequently called the normal error distribution, which is given by... 

The error function integral has to be evaluated numerically and can be found tabulated in texts on diffusion or conduction, or in mathematical tables. An abbreviated listing appears in Table 4.2, and some important properties of the function are summarized in Table 4.3. 

Contrary to the impression that one might have from a traditional course in introductory calculus, well-behaved functions that cannot be integrated in closed form are not rare mathematical curiosities. Examples are the Gaussian or standard error function and the related function that gives the distribution of molecular or atomic speeds in spherical polar coordinates. The famous blackbody radiation cuiwe, which inspired Planck s quantum hypothesis, is not integrable in closed form over an arbitiar y inteiwal. 

Here erfc( ) is the error function complement, a mathematical function of the argument u given by 1 - erf( ), where erf( ) is the error or Euler-Laplace integral. This integral in turn is defined by the expression... 

The variable y in the expression under the integral sign is an auxiliary variable the value of the integral depends only on the limits of integration (i.e., on the value of u). The numerical values of the error function vary from zero for m = 0 to an upper limit of unity for m —(this value is practically attained already for u 2). Plots of functions erf(n) and erfc(n) are shown in Fig. 11.2. 

The integral in Eq. (10) is the usual definition of the error function. A closely related function is the complementary error function... 

This may be reduced to an elementary function and error functions for b > o and an elementary function and Dawson s Integral for b < o (13,15). Calculations are easily performed since... 

Ccs is the constant concentration of the diffusing species at the surface, c0 is the uniform concentration of the diffusing species already present in the solid before the experiment and cx is the concentration of the diffusing species at a position x from the surface after time t has elapsed and D is the (constant) diffusion coefficient of the diffusing species. The function erf [x/2(Dr)1/2] is called the error function. The error function is closely related to the area under the normal distribution curve and differs from it only by scaling. It can be expressed as an integral or by the infinite series erf(x) — 2/y/ [x — yry + ]. The comp-... 

The integral is defined in terms of the error function given by... 

The integral in equation 10.31 cannot be evaluated analytically but it can be written in terms of the error function erf(rj) defined as... 

Note that s is a dummy variable the value of the integral depends only on the value of the upper limit. Tables of the error function are available and values can be calculated from power series [Dwight (1961), Kreyszig (1988)]. The error function has the properties erf(0) = 0 and erf(°°) = 1. Equation 10.31 can be written in terms of the error function as... 

However, the calculations are tedious and we shall not report them here in the final stage, we are left with a complicated one-variable integral involving in the integrand various combinations of error functions which are well known in plasma theory.10 One has then to evaluate this last integral by some numerical procedure. 

The integral of the error function complement ierfc is defined as... 

After converting the error function to the Fq function, both of the potential energy integrals have the same general form... 

Indefinite Integration. KACSYMA can handle integrals involving rational functions and combinations of rational, algebraic functions, and the elementary transcendental functions. It also has knowledge about error functions and some of the higher transcendental functions. 

The error function erf x and its complementary function erfc x appear in the solution of dilferential equations describing dilfusive processes (see, for instance, section 11.6.3). They are defined by the integrals... 

Gautschi W. (1964). Error function and Fresnel integrals. In Handbook of Mathematical Functions, M. Abramowitz and I. A. Stegun, eds. National Bureau of Standards, Washington. 

The integral can be related to the error function by partial integration, which gives... 

If D is constant, an experimental diffusion profile can be fit to the analytical solution (such as an error function) to obtain D. If it depends on concentration and the functional dependence is known. Equation 3-9 can be solved numerically, and the numerical solution may be fit to obtain D (e.g., Zhang et al., 1991a Zhang and Behrens, 2000). However, if D depends on concentration but the functional dependence is not known a priori, other methods do not work, and Boltzmann transformation provides a powerful way (and the only way) to obtain D at every concentration along the diffusion profile if the diffusion medium is infinite or semi-infinite. Starting from Equation 3-58a, integrate the above from Po to 00, leading to... 

The one-dimensional diffusion equation in isotropic medium for a binary system with a constant diffusivity is the most treated diffusion equation. In infinite and semi-infinite media with simple initial and boundary conditions, the diffusion equation is solved using the Boltzmann transformation and the solution is often an error function, such as Equation 3-44. In infinite and semi-infinite media with complicated initial and boundary conditions, the solution may be obtained using the superposition principle by integration, such as Equation 3-48a and solutions in Appendix 3. In a finite medium, the solution is often obtained by the separation of variables using Fourier series. 

Next we turn to the inference of cooling history. The length of the concentration profile in each phase is a rough indication of (jDdf) = (Dot), where Do is calculated using Tq estimated from the thermometry calculation. If can be estimated, then x, Xc and cooling rate q may be estimated. However, because the interface concentration varies with time (due to the dependence of the equilibrium constants between the two phases, and a, on temperature), the concentration profile in each phase is not a simple error function, and often may not have an analytical solution. Suppose the surface concentration is a linear function of time, the diffusion profile would be an integrated error function i erfc[x/(4/Ddf) ] (Appendix A3.2.3b). Then the mid-concentration distance would occur at... 

Integrated error functions are repeated integrations of the complementary error function. Define... 

Integrated error functions can be expressed in terms of error functions. For example, integrating by part, we can find... 

This probability integral (8.36) can be written using the error function of mathematical physics, denoted... 

---