Error function, complement

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 error function complement erfcy is defined by the relationship erfcy = 1 — erf y. The concentration at the reference plane c is... 

Here, erfcx is the error function complement of x and ierfc is its inverse. The physical properties are represented by a, the thermal diffusivity, which is equal to K/pCp, where k is the thermal conductivity, p is the density and Cp, the specific heat capacity at constant pressure. The surface temperature during this irradiation, Ts, at x = 0, is therefore... 

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

The quantitative treatment for i as a function of a varying T f was first solved analytically by Sevdk in 1948. The solution involves Laplace transformation and the error function complement expressions applied in Vol. I, Section (4.2.11). It is better to quote here the rather simpler equations that can be found if one takes the entire surface as available for the exchange of electrons, i.e., the easy case of 0 = 0. Then (Gileadi, 1993),22 with this assumption, the peak potential is related to the rate constant (Ay) for the interfacial reaction, to the Tafel constant b, and to the sweep rate s, by the equation ... 

Figure 3. Gel permeation data for linear randomly coiled polypeptides on various agarose resins, plotted according to the method of Ackers (9). M0 555 is plotted vs. the inverse error function complement of Kd (erfc 1 Kd). Lines drawn through the data points represent best fits obtained from linear least-squares analysis of the data. Numerical designation of each curve represents the percent agarose composition for the resin used. Filled triangles on the curve for the 6% resin, and the filled squares on the curve for the 10% resin are points determined using fluorescent proteins. Data for the labeled polypeptides were not included in the least-squares analysis.

The function erfc that appears here is the error function complement, which will be discussed in section 2.3.3.1 and is also shown in Table 2.4. This transcendental function rapidly approaches zero for increasingly large arguments. This provides us with a series for large values of , correspondingly small values of t, which converges very well. Introducing the dimensionless time... 

Table 2.5 Values of the error function complement erfc from (2.126) and the integrated error function ierfc in the form t/jt ierfc = e — yTr erfc ...

This function corresponds to the area under the normal error distribution curve from its maximum value to a value z. The error function has a value of 0 when z is zero, and a value of one when z is infinity. The error function complement is simply defined as... 

Erfc is the error function complement, defined as erfc x = 1 — erf x. On the computation of erfc, see the Appendix. To obtain the mean concentration within the upper layer of thickness l — h, the value of Mt (in grams cm 2) in Relationship 16 is divided by l — h. 

At the other extreme, when the diffusion layer grows much larger than ro (as at a UME), the concentration profile near the surface becomes independent of time and linear with 1/r. One can see this effect in (5.2.22), where error function complement approaches unity for (r - ro) << 2(DoO - In that case. 

Table B.14 Values of the error function, erf, and error function complement, erfc...

It is the area under the normalised Gaussian function (Fig. A.l) between the ordinates z and The error function complement erfc(2) = 1 — erf(z) represents the remainder of the area imder the Gaussian curve. Equation (A.16) can be written as... 

Here erfc is the error function complement, and Ti and T2 are parameters related to the redprocal of the jump rate for on-Iattice rotameric transitions. This expression follows from the continuous limit of a master equation approach applying to three-bond motions on a tetrahedral lattice. The major defidency of this expression is its infinite slope at x = 0, which is (di cally unrealistic. This feature arises fix>m the mathematical approximation of the process of discrete rotsuneric transitions by a continuous difiusion equation. Likewise, Jones and Stockmayer (JS) [48], Bendler and Yaris (BY) [49] and Hall and Helfand (HH) [50] have given condse expressions for the first OACF. These are semiempirical expressions and do not distinguish between the first and second OACFs described above. Accordingly, the superscripts 1 and 2, as well as the bond indices of the OACFs, will be omitted ip Ma(x) and Muft) in the following. 

The solution of the problem of diffusion from a semi-infinite medium into another semiinfinite medium, is generally expressed in terms of the error-function complement [7] ... 

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