Error function and

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. 

Table 7.2 gives tabulated values of the error function and related functions in the solution of other semi-infinite conduction problems. For example, the more general boundary condition analogous to that of Equation (7.27), including a surface heat loss,... 

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. 

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. 

Figure 1-11 Concentration profile for (a) crystal growth controlled by interface reaction (the concentration profile is flat and does not change with time), (b) diffusive crystal growth with t2 = 4fi and = 4t2 (the profile is an error function and propagates according to (c) convective crystal growth (the profile is an exponential function and does not change with time), and (d) crystal growth controlled by both interface reaction and diffusion (both the interface concentration and the length of the profile vary).

Note that neither initial nor boundary conditions have been applied yet. The above equation is the general solution for infinite and semi-infinite diffusion medium obtained from Boltzmann transformation. The parameters a and b can be determined by initial and boundary conditions as long as initial and boundary conditions are consistent with the assumption that C depends only on q (or ). Readers who are not familiar with the error function and related functions are encouraged to study Appendix 2 to gain a basic understanding. 

Note that Equation 4-99 means that the solution is an error function with respect to the lab-fixed reference frame (x = x—2aVSf). In the interface-fixed reference frame, the solution appears like an error function, and its shape is often error function shape, but the diffusion distance is not simply especially... 

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... 

Table A2-1 Values of error function and related functions...

Although some spreadsheet programs provide values of error function and related functions, there may be limitations on the value of the independent variables. For example, in a spreadsheet program, erf(x) values are provided only for 0 < X < 10. Then one may use the following to obtain erf(x) for any real x ... 

Recall that for one-dimensional diffusion in infinite medium, the concentration profile is an error function and the mid-concentration point is the interface. The above profile is also roughly an error function (e.g., fitting the profile by an error function would give D accurate to within 0.1% if (4Df) la < 0.5), but the mid-concentration point is not fixed at Tq = a rather it moves toward the center as To = a 2Dtla. The evolution of concentration profile is shown in Figure A3.3.4. 

Figure A2-1 Error function and complementary error function 566...

The error function is also useful for solving chemical diffusion problems (Chapter 13) and thermal conduction problems and the details of the error function and the table of the error fiinction are given in Chapter 13. 

Fig. 9.4.3 Normal-probability plot (a) and lognormal probability plot (b) for In particles shown in Figure 9.4.2. The ordinate stands for the cumulative percent of particles, with diameters smaller than d on the abscissa. This follows an error function and should give a straight line if the plot obeys the correspondent distribution, as seen in case (b). (From Ref. 4.)...

Tables are also available (e.g., see Beyer 1987) for the area under the normal curve between 7 (7 = 0) and one value of tt (i.e., they apply to one tail only). This area is known as the error function and is often symbolized as erf (t) ...

However, no simple function gives the area between arbitrary limits. This integral is so fundamental that its value, numerically integrated by computers to high accuracy, is given its own name—it is called the error function and is discussed in Chapter 4. 

Values of the error function and its first derivative (the Gauss distribution) are tabulated1. 

The function Erf (a ) is called the probability integral or the error function, and standard tables for it have been prepared. Erf (a ) represents the chance of finding an error of less than x in absolute magnitude. We have Erf(O) = 0, and Erf(oo) = 1. Erf(x) will have the value when x =... 

---