Complex functions logarithms

Spreadsheets have program-specific sets of predetermined functions but they almost all include trigonometrical functions, angle functions, logarithms (p. 262) and random number functions. Functions are invaluable for transforming sets of data rapidly and can be used in formulae required for more complex analyses. Spreadsheets work with an order of preference of the operators in much the same way as a standard calculator and this must always be taken into account when operators are used in formulae. They also require a very precise syntax - the program should warn you if you break this ... 

In this chapter, we discuss symbolic mathematical operations, including algebraic operations on real scalar variables, algebraic operations on real vector variables, and algebraic operations on complex scalar variables. We introduce the concept of a mathematical function and discuss trigonometric functions, logarithms and the exponential function. 

In seeking the differential coefficient of a complex function containing products and powers of polynomials, the work is often facilitated by taking the logarithm of each member separately before differentiation. The compound process is called logarithmic differentiation. 

The protein-binding model of Franke yields a complex function between the logarithm of the biological activity and some suitable hydrophobic parameter consisting of a linear part that passes into a parabola if steric hindrance occurs (Franke and Schmidt, 1973). 

The evaluation of log A is thus carried out in terms of the complex scalar logarithmic function. For a complex number z, the logarithm is obtained from the polar representation... 

The important point is that v E,k) and A0( ,fe) are multivalued, with a logarithmic branch point at fe = s = 0, while the residual functions m(e, k, zo) and w(e, k) are single valued. The result, as discussed in more detail later, is that the value of the quantum number v depends on the chosen location of the branch cut and on which Riemann sheet is taken, bearing in mind that the branch of arctan( /fe) must be taken according to the appropriate quadrant of the complex k, e) plane. Thus cj) = arctan( /fe) + ti/2 increases smoothly from zero to In around a counterclockwise circle in the k, e) plane, starting at = 0 and < 0. 

Vincente-Perez, S. Durand, J. S. Alvarez, M. D. 1992. Limitations of complexes logarithmic diagrams as a function of the ligand concentration-diagrams of conditioned variable. An. Quim. 88(7-8) 683-688. 

Figure 4. Copper complexation by a pond fulvic acid at pH 8 as a function of the logarithm of [Cu2+]. On the x-axis, complex stability constants and kinetic formation rate constants are given by assuming that the Eigen-Wilkens mechanism is valid at all [M]b/[L]t. The shaded zone represents the range of concentrations that are most often found in natural waters. The + represent experimental data for the complexation of Cu by a soil-derived fulvic acid at various metakligand ratios. An average line, based on equations (26) and (30) is employed to fit the experimental data. Data are from Shuman et al. [2,184]...

Choosing the principal branch of the complex logarithm function, we can write (8.38) as... 

Fig. 15.3 Calculated percentage of bound ligand (pb = [PL]/[L]0, where [PL] is the concentration of the protein-ligand complex, and [L]0 is the total ligand concentration) as a function of total ligand concentration. The graph is in double logarithmic scale. The dissociation constants, K0, are 10 pM (squares), 100 pM (diamonds), and 1 pM (triangles). Protein concentrations are 1 pM (solid curves) and 10 pM (dashed curves).

Figure 6-3 Total solubility of lead(ll) (curve with circles) and solubilities of individual species (straight lines) as a function of the concentration of free iodide. To the left of the minimum, [Pb]tota, is governed by the solubility product for Pbl2(s). As [I ] is increased, [Pb]total decreases because of the common ion effect. At high values of [I ], Pbl2(s) redissolves because it reacts with I to form soluble complex ions, such as PblJ. Note logarithmic scales. The solution is made slightly acidic so that [PbOH+] is negligible.

Let us consider a projection of the complex many-dimensional motion (which variables are both concentrations and the correlation functions) onto the phase plane (iVa, iVb). It should be reminded that in its classical formulation the trajectory of the Lotka-Volterra model is a closed curve - Fig. 2.3. In Fig. 8.1 a change of the phase trajectories is presented for d = 3 when varying the diffusion parameter k. (For better understanding logarithms of concentrations are plotted there.)... 

---