Series Summation and Identities

The reciprocals of the terms of the arithmetic-progression series are called harmonic progression. No general summation formulas are available for this series. 

This is simply a special case of Taylor s series when h is set to zero. Exponential Series 

Taylor Series The Taylor series for a function of two variables, expanded about the point (x0, y o), is 

References Ablowitz, M. J., and A. S. Fokas, Complex Variables Introduction and Applications, Cambridge University Press, New York (2003) Asmar, N., and G. C. Jones, Applied Complex Analysis with Partial Differential Equations, Prentice-Hall, Upper Saddle River, N.J. (2002) Brown, J. W., and R. V Churchill, ComplexVariables and Applications, 7th ed., McGraw-Hill, New York (2003) Kaplan, W., Advanced Calculus, 5th ed., Addison-Wesley, Redwood City, Calif. (2003) Kwok, Y. K., Applied Complex Variables for Scientists and Engineers, Cambridge University Press, New York (2002) McGehee, O. C., An Introduction to Complex Analysis, Wiley, New York (2000) Priestley, H. A., Introduction to Complex Analysis, Oxford University Press, New York (2003). 

Numbers of the form z = x + iy, where x and y are real, i2 = -1, are called complex numbers. The numbers z = x + iy are representable in the plane as shown in Fig. 3-45. The following definitions and terminology are used  

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For the following relationships the sign = means approximately equal to, when X is small. These equations are derived by using a Taylor s series (see Series Summation and Identities ). 

Using atomic densities calculated from tabulated atomic wave functions, the summation was found [214] to produce results equivalent to the most elaborate molecular Hartree-Fock calculations for a series of small molecules, at a fraction of the computing expense. Surface areas and volumes computed by the two methods were found virtually identical. The promolecule calculation therefore has an obvious advantage in the exploration of surface electron densities, surface areas and molecular volumes of macromolecules for the analysis of molecular recognition. 

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