Naturally, our view is only one among so many others, that could differ sharply from ours. Laplace representation on the positive half-line. Integral representation of Laplace type on. Definitions of functions and as averages. Conversely, if is analytic, of order in some half-plane , with real and , then its inverse is , uniquely determined by:.
Two functions with same Laplace transform are identical. If is the density of X,. Equation 3 is valid under the condition that 2 exists as an analytic function of the complex variable s, for.
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The integral is independent of w. In this section we consider only series and their limits. We have the series representation of the exponential function, which is a special case of the hypergeometric series:. One generalization of this notion is associated with the hypergeometric series, where this ratio is a rational expression of n. Then we should have:. The hypergeometric series has the above expression. For , , we have Gauss hypergeometric series in 3 parameters.
Equation 4 reduces to the geometric series for , hence its name. If a or b is zero or a negative integer the series becomes a polynomial. The first work on hypergeometric function was made by Euler in , when he studied series 4 , as solution to Equation The whole field of Special Functions is characterized by integral representations of various kinds see e. Lebedev . We first recall the integral representation of the upper tail of the gamma distribution by a finite sum, well known in elementary statistics Hogg and Craig  , p.
Similarly, we have the integral representation of an infinite series. There are several advantages in dealing with an integral instead of a series, as already remarked by Carlson . Continuity and even analyticity are usually provided by the integral, hence leading to a deeper study of its properties and extensions, and also faster convergence on a digital computer. The hypergeometric series 4 , with its convergence region will be of limited interest if it cannot be extended to the whole complex plane.
The principle of analytic continuation in complex analysis will permit us to precisely do that operation. There are three integral representations of , of increasing complexity, that serve three different purposes, and propose three different ways in computing the values of a hypergeometric function:. For real, inside the unit disc we have if and if. If both double conditions are satisfied then. Outside the unit disc , either integral can be seen to converge for any value of z, except on , for or , respectively. Hence, the condition can be dropped and the series can be extended to a function analytic in the complex plane, with a cut along if Lebedev .
It serves to generalize the series outside the unit circle, by analytic continuation. This is the representation which is mostly used in statistics, where, frequently, the integral is encountered first, and hence the series can become redundant. But the terminology can become confusing. Also, these integrals are not defined for real positive values of z superior to 1, as the cut implies, but they converge for all complex values of a, b and c, and are analytic functions of these parameters for z fixed.
Figure 1 and Figure 2 illustrate the example. This is to be compared with the Mellin- Barnes representation in 2. Figure 1. Graphs of , and where They coincide. Figure 2. Graphs of a , b , and c , with , They coincide.
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This representation is useful when dealing with Laplace transform methods and moment generating functions, which is frequent in Statistics. However, , and later , is usually expressed in function of another hypergeometric function, with less parameters, and this fact is useful for a progressive definition of a family of functions. We have:. This hypergeometric function is an important function in its own right see Slater  , but due to space limitation we will not deal with it further.
On the other hand, the Laplace transform of is:. NOTE: Some results on this transform, and its inverse, are given on p. Complex analysis developed in the 19th century brought powerful tools such as the calculus of residues, and Mellin-Barnes formula gives a third representation, based on contour integration. The value of the integral is computed, not as a complex integral, but as the sum of the residues at poles of.
When they are simple we have:. Computing the residues at the simple poles of , , we have 4 equal to 7 a proof is given in 2. It can be shown, again, using 7 , that can be extended to a function analytic in the complex plane, with a cut along.
Mathai and Saxena  , p. Cases , with , , etc. Example 2: For , for example, we have on , and on by direct computation, and finally, on by analytic continuation, by taking as value of the value of within that interval. From 7 above, however, is not defined for since the formula in this case contains. This is a drawback from Mellin-Barnes representation. Mellin-Barnes integral formula has its origins in the work of Pincherle in see Mainardi and Pagnini  and this formula was developed later by Mellin and Barnes.
Athough 7 is very convenient to deal with when extensions of to forms which are more general are considered, 7 itself is seldom encountered in statistics. Let be Gauss hypergeometric function and the associated six functions, called contiguous functions: , ,. It can be shown that can be obtained as a linear combination of any two of these functions, with rational coefficients expressed in terms of and z.
There are hence 15 such relations, that can be generalized to , with and s being integers.
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Although is the direct generalization of Gauss we have, in general, the hypergeometric function in one scalar variable and parameters, defined as the series with expression:. For , it is absolutely convergent for if , conditionally convergent for if and divergent for if. For the general case, Olsson  proposed to express as an expression of progressively down to Gauss for , using 13 , the analytic continuation of which has been made.
gopideas.com/4892.php For an extensive study, and lists of properties and formulas of , we refer to Mathai and Saxena  , sect 5. As before, we have three types of integral representation:. This relation is not to be mistaken as the Laplace transform below. Considering and , using Laplace transforms, we have the couple:. The above expressions become Laplace and inverse Laplace transforms of when and respectively. Conversely, it can be shown that if are distinct of each other with differences different from integers, 8 is the sum of all residues of. Using the formula for residue value, we have:.
Since the poles are in infinite numbers, we can see that ,. NOTE: We have most common functions in mathematics represented by where a, b and c take simple values.
For example, we have:. A list of standard mathematical functions expressed as G-functions can be found in Mathai and Saxena  , sect. Conversely, section 2. For example, the command:. In an effort to generalize and make sense of the case , we define the H-function, using Mellin-Barnes formula, and consider the ratio of two products of gamma functions as integrand.
The Meijer function is a special case, when , , of. We notice that 15 is just one way to generalize the integrant in From 3 and 14 we can see that G and H-functions are Inverse Mellin Transforms of and that the Mellin-Barnes integral is now taken as the definition of the G-function, instead of a series, or a definite integral, as in preceding sections. But under some mild conditions on , the function can be expressed as a function and conversely:.
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For discussions on the G-function see Mathai and Saxena  , and on the H-function, see Springer  , which also treats some uses of these functions in Statistics, as well as some computational issues. We wish to mention the following points:. They can be seen in Mathai and Saxena .
Taking we have the Laplace transform of. Also, the relation permits the ana-. Generalizations of H-functions: We will not go beyond the H-function, but it is worth mentioning that generalized forms of H exist, e. It is defined by:. When we go from one variable to two variables there are different ways to sum the variables, reflected in different expressions for the coefficients given to , and hence, we have different functions.
In two variables, we have Appell hypergeometric functions, defined as follows:.
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