Complex argument

REAL Converts a complex argument to a real value... 

Usually we are only interested in mutual intensity suitably normalised to account for the magnitude of the helds, which is called the complex degree of coherence 712 (r). This quantity is complex valued with a magnitude between 0 and 1, and describes the degree of likeness of two e. m. waves at positions ri and C2 in space separated by a time difference r. A value of 0 represents complete decorrelation ( incoherence ) and a value of 1 represents complete eorrelation ( perfect coherence ) while the complex argument represents a difference in optical phase of the helds. Special cases are the complex degree of self coherence 7n(r) where a held is compared with itself at the same position but different times, and the complex coherence factor pi2 = 712(0) which refers to the case where a held is correlated at two posihons at the same time. 

J. Thompson, A.R. Barnett, COULCC A continued-fraction algorithm for Coulomb functions of complex order with complex arguments, Comput. Phys. Comm. 36 (1985) 363. 

More generally, transformation (217) can be considered as the Laplace transformation with complex argument z = e- - irj. 

Equation (16-2) leads by complex argument to the following tree-energy principle ... 

Regardless of which explanation is favored, we have no immediate grounds to exclude alternative nucleotides or amino acids in nonterran life if we encounter it. Somewhat complex arguments based on the chemistry of some of the alternative nucleotides were proposed in the committee s discussion to disfavor them relative to the standard four nucleotides, but it is not clear that equally compelling counterarguments would not be offered if human-like genetics exploited the nonstandard alternatives. 

We apply the RDT theory to calculate an estimate (rf) of the free energy profile. The central quantity in this analysis is a generalized characteristic function g z) = ((exp(izF)))- of complex argument z. Here (( )) indicates an average over the surrogate solvent equilibrium distribution function (defined in terms of the surrogate Hamiltonian... 

It is well known that complex-rotated basis-set expansions give very accurate values for complex energies [157]. But in order to apply FFS to obtain critical exponents, we observe that the scaling function F< x) in the scaling relation (60) has to be replaced by a complex function of a bound states. Then it is necessary to introduce new scaling functions and critical exponents. The convergence process with the number N of basis functions is not uniform, and therefore it is very difficult to make extrapolations from the numerical data. 

The Decision Representation Language (DRL, [808, 809]) is a notation for decision rationale. Its top-level element is the Decision Problem, equivalent to the Question in QOC. Alternatives and Goals in DRL correspond to Options and Criteria, respectively. The outstanding characteristic of DRL is that Claims, i.e., statements which may be judged or evaluated, can be linked in a way that enables to represent complex argumentations in a straightforward... 

Equation (76-2) leads, by a complex argument, to the following free-energy principle ... 

The reflection amplitudes for s and p polarization are used to determine the reflectivity (amplitude modulus squared), phase (complex argument), and ellipsometric variables A and reflection amplitude is expressed as r = r exp(i ), then 0 is the phase change, A = (j>p-(j>s and (p = arctan( 7-p / rs ) [86]. This treatment is used both for the static ellipsometric measurements of the thickness and refractive index and for modeling the dynamic problem. 

Again by a complex argument, the Debye-Hiickel theory gives the following equation for the mean ionic activity coefficient ... 

This requires a more complex argument than that used for determining and can be... 

The function Si (a ) is the sine integral [28, (5.2.1)]. The Barrett moment can be evaluated from incomplete gamma functions with complex arguments [60, (1.5.51.3)],... 

An alternate and formally very powerful approach to resonance extraction is complex scaling [7, 101. 102. 103, 104, 105, 106 and 107] whereby a new Hamiltonian is solved. In this Hamiltonian, the grid s multidimensional coordinate (e.g., jc) is multiplied by a complex constant a. The kinetic energy gains a constant complex factor d ldy (1/a )(5 /5jc )), while the potential needs to be evaluated at points with a complex argument V(a jc). In a typical calculation, one diagonalizes the resulting complex Hamiltonian for several complex values of a, and the complex resonances are... 

The time-independent states are characterised by the stationary state vector which is free of the time variable. The state vector can be represented by the (wave) function of the complex argument belonging to a Hilbert space. The wave function should be ... 

Usually, functions and P are complex functions which depend on complex arguments. 

Thus the method of separation of variables in Helmholtz equation allows us in such cases to present a solution in an explicit form. In problems with cylindrical interfaces, as will be shown later, a solution can be written in the form of an improper integral containing Bessel functions of complex argument. In media with horizontal interfaces a field is also expressed through improper integrals, but in these cases the integrand is much simpler. 

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