Integrals numerical evaluation

For the case where t8 = rp — tp, Equation 13 can be integrated directly to give ITto1al, the total amount of tertiary ion formed. For the other two cases, integration cannot be performed directly, and values of ITtotal were evaluated numerically on a KDF 9 computer, using a procedure for Simpson s rule. (Numerical evaluation of the directly integrable case provided a check on this procedure.) Ip and I8 are then given by... 

Exact analyses of experimental spectra from thick absorbers, however, have to be based on the transmission integral (2.26). This can be numerically evaluated following the procedures described by Cranshaw [10], Shenoy et al. [11] and others. 

The numerical evaluation of definite integrals can be carried out in several ways. However, in all cases it must be assumed that the function, as represented by a table of numerical values, or perhaps a known function, is well behaved. While this criterion is not specific, it suggests that the functions haying pathological problems, e.g. singularities, discontinuities,..may not survive under the treatment in question. 

Relationship 23 provides a method for evaluating the parameter "a" that is defined by Equation 2A. The cumulative molar concentration of polymeric species PT0T was numerically evaluated via integration of population density distributions. The contribution of network molecules to the zeroth moment of the distribution is negligible. Results are presented by Figure A and show that... 

Since we have seen that the integral cannot be evaluated analytically, there are several alternatives to analytic integration of equation 43-63 we can perform the integration numerically, we can investigate the behavior of equation 43-63 using a Monte-Carlo simulation, or we can expand equation 43-63 into a power series. 

All three integrals are evaluated numerically and tabulated. The pseudo first order ka2 is as nearly constant as the second order value > al ... 

The integral is evaluated numerically, the results are tabulated and appear to be nearly constant, with... 

The integrals are evaluated numerically for various combinations of the exponents a and b. 

The various integrals are evaluated numerically and the results are tabulated. 

The above equation is the basic equation to estimate D at every concentration by numerically evaluating the integral in the numerator and the derivative in the denominator from C versus p relation, obtainable from a measured profile and experimental duration. Because the duration f for a given experimental profile is a constant. Equation 3-58d can also be written as... 

So, STOs give "better" overall energies and properties that depend on the shape of the wavefunction near the nuclei (e.g., Fermi contact ESR hyperfine constants) but they are more difficult to use (two-electron integrals are more difficult to evaluate especially the 4-center variety which have to be integrated numerically). GTOs on the other hand are easier to use (more easily integrable) but improperly describe the wavefunction near the nuclear centers because of the so-called cusp condition (they have zero slope at R = 0, whereas Is STOs have non-zero slopes there). 

This integral was analytically simplified to a one-dimensional integral of a complete elliptic integral, which admits numerical evaluation with an arbitrary precision [20]... 

The latent-heat terms (3.112) become necessary whenever the integrand ACP undergoes discontinuous change at a phase transition, with accompanying release of hidden AH. [The latent heat contribution is automatically included if one understands J(ACV) dT as Lehesgue integration.] For numerical evaluation of the integral in (3.111), power series... 

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