Properties Of Special Functions Research Articles

Monotonic Properties of Analytic Functions

A theorem is proved which enables one to obtain monotonic properties of the real part, imaginary part, modules and phase of an arbitrary analytic function in the complex plane. The monotonic properties are established from the behavior of the analytic function and its derivative on the boundary of the domain in which the monotonic properties are required. As an applicationsome monotonic results are derived for the Bessel functions $J_\nu (z)$ and $H_\nu ^{(1)} (z)$, where z is complex and $\nu$ is real and positive. The theorem and a corollary can be used to obtain monotonic properties of many other special functions of mathematical physics.

Branching diagram and serber‐type spin functions. Algorithms for their construction and special properties

AbstractBranching diagram and Serber‐type spin functions are ordered according to their diagrams. The method of Graebenstetter et al. is slightly modified; it provides an effective algorithm for the construction of genealogical spin functions. A counterpart of the method is developed for Serber‐type spin functions. Special properties of branching diagram functions and Serber‐functions are established.

Some algorithms for prime testing using generalized Lehmer functions

Let N be an odd integer thought to be prime. The properties of special functions which are generalizations of the functions of Lehmer (Ann. of Math., v. 31, 1930, pp. 419-448) are used to develop algorithms that produce information concerning the possible prime divisors of N. It is shown how the factors of N ± 1 , N 2 + 1 , N 2 ± N + 1 N \pm 1,{N^2} + 1,{N^2} \pm N + 1 , together with the factor bounds on these numbers, may all be used to calculate lower bounds for the possible prime divisors of N. Frequently, these bounds are large enough that N may be shown to be prime. These tests were implemented on an IBM/370-158 computer and run on the pseudoprime divisors of the first 385 Fibonacci and Lucas numbers.

On Minimization of Fuzzy Functions

The study of fuzziness in combinational switching systems by means of a suitable fuzzy algebra is discussed. The insufficiency of the methods for simplification of fuzzy functions as presented by Lee and Chang [9] and by Siy and Chen [10] is discussed. A new technique for minimization of fuzzy functions is developed. Special properties of fuzzy functions are discussed and their relationships to two-valued logic are investigated.

Some Properties of Special Functions Derived from the Theory of Continuous Transformation Groups

The theory of continuous transformation groups is utilized in the study of some properties of special functions. On constructing the continuous transformation groups corresponding to a suitably defined infinitesimal transformation, a bilateral generating relation involving Laguerre polynomials $\{ L_n^{(\alpha )}(x)\}$ is obtained in $\S 2$. It is shown to be a generalisation of Brafman’s result. In the last section raising and lowering operators for $\{ P_n^{(\alpha ,\beta - n)}(x)\}$ and their commutator are introduced and on showing that they generate a 3-dimensional Lie algebra, the idea of c.t. groups is employed to establish a generating relation involving $\{ P_n^{(\alpha ,\beta - n)}(x)\}$ which is seen to yield a number of known results. Moreover, a bilateral generating relation involving $\{ P_n^{(\alpha ,\beta - n)}(x)\}$ is obtained; this is seen to be a generalisation of a well-known relation due to Weisner.

Some properties of special functions derived from the theory of continuous transformation groups

The theory of continuous transformation groups is utilized in the study of some properties of special functions. On constructing the continuous transformation groups corresponding to a suitably defined infinitesimal transformation, a bilateral generating relation involving Laguerre polynomials { L n ( α ) ( x ) } \{ L_n^{(\alpha )}(x)\} is obtained in § 2 \S 2 . It is shown to be a generalisation of Brafman’s result. In the last section raising and lowering operators for { P n ( α , β − n ) ( x ) } \{ P_n^{(\alpha ,\beta - n)}(x)\} and their commutator are introduced and on showing that they generate a 3-dimensional Lie algebra, the idea of c.t. groups is employed to establish a generating relation involving { P n ( α , β − n ) ( x ) } \{ P_n^{(\alpha ,\beta - n)}(x)\} which is seen to yield a number of known results. Moreover, a bilateral generating relation involving { P n ( α , β − n ) ( x ) } \{ P_n^{(\alpha ,\beta - n)}(x)\} is obtained; this is seen to be a generalisation of a well-known relation due to Weisner.

Function of enzymes in wood decaying fungi

1. Mycelial extracts and culture filtrates of two species of Trametes have been assayed for various enzymatic activities. The tests performed after 3 and 10 days of mycelial growth on 3% malt extract have revealed that the spectra of external enzymes secreted by both strains differ markedly from their intracellular equivalents. A great part of the external isozyme bands, shown in disc electrophoresis and isoelectric focusing, have no equivalent acting within the cell. 2. These findings are discussed on the basis of special functional properties which have to be attributed to external enzymes. It is concluded that a great number of them are not simply internal enzymes which happen to be active outside the cell but that they are synthesized or at least modified especially for extracellular action under these special conditions. 3. A comparison between the two species of Trametes has shown that both produce enzymes with identical function. This indicates their strong relationship, though the electrophoretic mobility of most of the enzymes is not comparable. The consequences of these findings for chemical taxonomy are discussed.

Methods for the Bethe-Salpeter Equation. I. Special Functions and Expansions in Spherical Harmonics

Elements of the structure of the Bethe-Salpeter equation are studied. Properties of useful special functions are obtained and free-particle solutions to the truncated expansion of the equation in four-dimensional spherical harmonics are derived in terms of known special functions. The validity of the truncation approximation is examined in terms of a convenient representation of the Green's function. In particular, it is shown that the method of truncating the differential Bethe-Salpeter equation cannot succeed for scattering. The development of alternative procedures is deferred to the paper following. As a by-product, a simple computational technique for the approximation of integrals by Gaussian quadrature is derived.

Runge-Lenz Vector and the Coulomb Green's Function

The fact that the Runge-Lenz vector is an extra constant of the motion for a charged particle moving in a Coulomb potential is found to account for the especially simple structure of the nonrelativistic coordinate space Coulomb Green's function. Also, the study of the consequences of this extra constant of the motion leads to a separation of variables in the differential equation for the coordinate space Coulomb Green's function, and hence to a new derivation of the closed-form expression for this Green's function which avoids the use of infinite series and the detailed properties of special functions.

Congruence properties of special elliptic functions

Congruence properties of special elliptic functions