Pdf in the paper, the authors establish some subadditive and completely monotonic properties of the tricomi confluent hypergeometric functions. Special functions are the general class of functions like sine, cosine, the exponential function. One can show that a function is hypergeometric if and only if in the representation fz x k. Chapter iv confluent hypergeometric functions sciencedirect. The confluent hypergeometric function springerlink. Confluent hypergeometric kummer u function matlab kummeru. It is a solution of a secondorder linear ordinary differential equation ode.
Although many common functions can be defined as a special case of this function, it is typically better to use functions that take advantage of features of the specific function for more efficient and accurate evaluation. The confluent hypergeometric function kind 1 distribution with the probability density function pdf proportional to occurs as the distribution of the ratio of independent gamma and beta variables. An extension of the beta function by introducing an extra parameter, which proved to be useful earlier, is applied here to extend the hypergeometric and confluent hypergeometric functions. The expansion of the con uent hypergeometric function on the. On the expansion of confluent hypergeometric functions in terms of bessel functions n. The results are presented in terms of hypergeometric functions and confluent hypergeometric functions. The confluent hypergeometric function with special emphasis. Confluent hypergeometric functions are a type of special function. In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular singularity.
The confluent hypergeometric function is useful in many problems in theoretical physics, in particular as the solution of the differential equation for the velocity distribution function of electrons in a high frequency gas discharge. Confluent hypergeometric functions kummers function. Thus, is a generalization of the integral addition theorem for the confluent hypergeometric function see, formula 15, p. Hypergeometric functions hypergeometricua,b,z transformations 2 formulas transformations and argument simplifications 1 formula products, sums, and powers of the direct function 1 formula transformations 2 formulas hypergeometricu. The confluent hypergeometric function is an analytic function in the complex. The confluent hypergeometric function and wkb solutions by toshinori takahashi abstract explicit formulae which describe relations between the confluent hypergeometric function and wkb solutions are found.
The subject of this book is the higher transcendental function known as the confluent hypergeometric function. Products, sums, and powers of the direct function 1 formula. Bivariate extended confluent hypergeometric function. It is of interest to note that the confluent hypergeometric function, m, yields the prior moment generating function of p from density 1. In the last two decades this function has taken on an ever increasing significance because of its use in the application of mathematics to physical and technical problems. We present a method for solving the classical linear ordinary differential equations of hypergeometric type 8, including bessels equation, legendres equation, and others with polynomial coe. The confluent hypergeometric functions have proved useful in many branches of physics. Many special functions are also expressible as special cases of 1f1, including the incomplete gamma function, bessel functions and laguerre polynomials. They have been used in problems involving both diffusion and sedil. In this article, a multivariate generalization of this distribution is defined and derived. Subadditive and completely monotonic properties of the tricomi confluent hypergeometric functions article pdf available august 2018 with 4 reads how we measure reads. Pdf subadditive and completely monotonic properties of the. Numerical methods for the computation of the confluent and gauss.
Introduction the purpose of this paper is to describe a relation between the confluent hypergeo. Hypergeometric functions reading problems introduction the hypergeometric function fa, b. The confluent hypergeometric function and wkb solutions. An asymptotic expansion of the confluent hypergeometric function ua,b,x for large positive 2a. We also have a barnestype integral representation for the confluent hypergeometric function. The motivation for computing hypergeometric functions will be discussed, with details given of some of the practical applications of these functions.
Expansion of the confluent hypergeometric function in series. The partition method for a power series expansion was first introduced in the derivation of an asymptotic expansion for the particular kummer or confluent hypergeometric function that arises in the response theory of the charged bose gas 1. Representation of the confluent hypergeometric function in series of bessel functions of integral order. The solution of eulers hypergeometric differential equation is called hypergeometric function or gaussian function introduced by gauss. Confluent hypergeometric function article about confluent. Information regarding the scope of tables, analytical. This is also known as the confluent hypergeometric function of the first kind. Pdf the various forms of the differential equation for the confluent.
If the real parts of z and a are positive values, then the integral representations of the kummer u function is as follows. Other special cases of confluent hypergeometric functions and appropriate references are noted and properties of these and bessel functions are listed in the. Certain integrals of generalized hypergeometric and confluent. A connection formula for the q confluent hypergeometric function 5 method becomes a powerful tool. Tables of the confluent hypergeometric function fn2, 12. Temme abstract for the confluent hypergeometric functions u a, b, z and m a, b, z asymptotic expansions are given for a,o. Note that we can find a new parameter in the resummation 2f 0a. Information regarding the scope of tables, analytical properties, and approximations are provided. The details of the hypergeometric function were developed in chapter 5.
Pdf recently, some generalizations of the generalized gamma, beta, gauss hypergeometric and confluent hypergeometric functions have been introduced in. The term confluent refers to the merging of singular points of families of differential equations. Heatley, transactions of royal society of canada, 1943, defines a function which is a confluent hypergeometric function in z2 multiplied by a power of z, and gives a short 5figure table. Solutions to the hypergeometric differential equation are built out of the hypergeometric series. Hypergeometric1f1a,b,z 750 formulasprimary definition 3 formulas specific values 533 formulas. Pdf notes on generalized hypergeometric and confluent. Identities for the gamma and hypergeometric functions.
Hypergeometric functions hypergeometricua,b,z transformations 2 formulas. Acknowledgment the tables of the confluent hypergeometric function have been computed by. Many problems in mathematical physics can be solved in terms of confluent hypergeometric functions, and for that reason it is useful to have collected together the basic formulae relating to these functions and an extensive set of numerical tables. The generalized hypergeometric function pf qis known to satisfy the following di erential. Multivariate generalization of the confluent hypergeometric. It is also known as kummers function of the first kind. Victor kowalenko, in the partition method for a power series expansion, 2017. The equation has two linearly independent solutions at each of the three regular singular points, and. In mathematics, the gaussian or ordinary hypergeometric function 2 f 1 a,b. On the expansion of confluent hypergeometric functions in. A new expansion of the confluent hypergeometric function.
Pdf subadditive and completely monotonic properties of. There is a different and unrelated kummers function bearing the same name. The confluent hypergeometric function and wkb solutions 143 we can also compute their borel sums by a streightforward calculation. In the case of real parameters, this predicts a complexvalued contribution from the subdominant expansion, when in fact it clearly must be realvalued. The expansion of the con uent hypergeometric function on.
Note that some other integral addition theorems of the confluent hypergeometric function are given in 15,16. Confluent hypergeometric function encyclopedia of mathematics. The confluent hypergeometric function with special. The confluent hypergeometric function is an entire analytic function in the entire complex plane. The parameters of the hypergeometric function are zeros and poles of this rational function. Report presenting tables of the confluent hypergeometric function and some related functions. Slater, confluent hypergeometric functions cambridge. Numerical methods for the computation of the confluent and. Analogous to the case for the confluent hypergeometric function, it is better numerically to compute f than to compute 2f1.
536 672 490 324 771 1105 1046 290 1216 897 1006 621 619 1500 1030 878 717 337 206 307 383 472 1224 102 421 1144 509 1344 958 480 1003 898