Notes on Various Transforms Identified by Some Special Functions
with Complex (or Real) Parameters and Some of Related Implications
1,2
, TOLGA HAN AÇIKGÖZ
1
IRMAK HÜSEYİN 1Department of Mathematics,
Faculty of Science,
Çankiri Karatekin University,
TR 18100, Çankiri,
TURKEY
1,2Ministry of National Education,
Provincial Directorate of National Education,
TR 06100, Ankara,
TURKEY
Abstract – The fundamental aim of this special research is first to introduce certain essential information in
regards to some special functions, which are the Gamma function and the Beta function and play a big role in
both (applied) mathematics and most engineering sciences, and then to present both a number of their familiar
properties and several relationships between them. Afterward, various possible-undeniable effects of those
special functions in the transformation theory, their special implications, and suggestions for the relevant
researchers will be also considered as special information.
Key-Words: - The Gamma function, the Beta function, power functions, fractional calculus, series expansions,
improper integrals, integral transformations, operators.
Received: May 25, 2022. Revised: July 27, 2023. Accepted: September 16, 2023. Published: October 18, 2023.
1 Introduction and Certain Special
Information
As it is well known, the Gamma function and the
Beta function play a large part in both the family of
special functions and the theory of integral
transformation(s). Apart from this, these two
functions are frequently encountered in their forms
with both real parameters and complex parameters
as various theoretical research and nearly all
applied sciences in the literature. As special
purpose applications, they are very important tools
for modeling situations comprising continuous
changes, with various applications to calculus,
differential equations, complex analysis, and
statistics. As some examples, for each one of them,
it can be checked over the scientific studies
presented in, [1], [2], [3], [4], [5], [6], [7], [8], [9],
[10], [11].
In addition, in cases where our classical
analysis information is not sufficient, as various
useful tools, the importance of those special
functions cannot be ever denied for science and
technology as their applications. Specially, as
natural results of unclassical examples, without
using any one of those related functions which is
the possible solutions of the following improper
integrals cannot be concluded:
and
where the concerned parameters and are in the
set with Quite
simply, as is frequently encountered in the fields of
mathematics and even many other sciences, it can
easily be seen that when the mentioned parameters
used in (1) and (2) are chosen appropriately,
denumerably infinite types of either the definite
type integrals and the convergent types of the
improper integrals will be also encountered. For
more (simple) examples, it even suffices for them
to simply focus on the (basic) classical analysis
books (cf., e.g., [12], [13], [14], [15]).
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As the most basic definitions of the Gamma
function and the Beta function, for the complex
parameter , as is known, the gamma function is
denoted by and also defined by the improper
integral given by
where .
At the same time, for the other, the beta function
(consisting of the complex parameters and ) is
also denoted by , which is called the Euler
integral of the first kind, and then defined by the
(improper) integral given by
where and
In terms of literature, the special functions,
defined by (3) and (4), possess several essential
properties. In particular, an extensive property of
the beta function is its close relation to the Gamma
function, which is given by
where and
Privately, when
,
the familiar relation given by (5) immediately
arrives at the assertion given by
In particular, for the details of those special
functions and other relevant identities, formulas,
properties, relations, and also some of the related
transforms to some of the other special functions,
one may concentrate on the fundamental works
accentuated in, [2], [3], [4], [10], [14], [16], [17],
[18], [19], [20], [21], [22], [23]. In addition, for
the relevant researchers, since this particular
investigation will be related to various forms of
special functions with complex parameters, for the
researchers who are interested in the theory of
complex functions and related areas of special
scientific interest, we think that the main works
given in the references in [7], [9], [12], [13], [15],
[22], [24], [25], [26], will be also important as extra
special information.
In the meantime, under the conditions that the
integrals given by (1) and (2) are either in any
forms of the definite integrals or in any convergent
forms of the improper integrals and by means of
the special definitions given by (3) and (4), the
main object of this extensive research is to
construct some of (general type) transformations in
certain domains of the complex plane and then to
make some possible inferences by taking into
account both the gamma function being of the form
given in (1) together with (3) and the beta function
having the form given in (2) along with (4). Now,
in light of the information between (1)-(6), let us
first create them in order and then represent some
of their possible implications as our elementary
results.
2 The Gamma Function and Some of
the Related Implications
The first implication is directly related to various
comprehensive applications of the gamma function
being of the form given in (3). For this special
function and its details, one may refer to the works
in, [1], [5], [7], [16], [18], [20]. The main
references for its characterization, when two
expedient functions like and are given, it
will be enough to consider any instrumental-
extensive forms of any function defined by
which concurrently identifies processes that change
exponentially in time (or related space).
Let us now begin to designate its details.
For those functions and
that satisfy the necessary-convergence criteria (for
improper integral just below), by the following
notation: ,
we firstly denote a general transformation of the
function created by (7) and then define it
as the improper integral given by
As a result of a simple focus, for all the scalars
and , and for all the functions and
, the following propositions can be easily
presented.
Remark 2.1. For the mentioned functions and
, the following are also asserted:
i)
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and
ii)
By considering the convergence of the
generalized integral constituted in (8) and in the
light of our classical analysis knowledge, a large
number of the familiar transformations can be
easily constructed by choosing various suitable
forms of those functions expressed by and For
example, by taking the function as the form given
by
the related transform, which is well recognized as
the Laplace transformation of the mentioned
function , is easily arrived at the familiar
definition being of
where and for all of the
values of belonging to the interval
We point out here that when considering the special
information between Laplace transform and the
relevant improper integral indicated as in (11), it
can be easily re-emphasized the familiar relations
related to
and
As various possible applications in relation to
the mentioned transformation are presented by the
definition given in (11) (or (8) together with (7)),
it can be also concentrated on a great number of its
elementary implications. We want to present only
some of those consisting of (real) parameters as
remarks just below.
Remark 2.2. Let
and .
As it is well known, when , the transform of
the function can be easily calculated by
making use of our classical analysis information.
But, when is a real number with of
course, it is impossible to calculate (or determine)
the transform of that function. At this time, with the
help of the gamma function, it can be also
determined. Then, the related-special assertions can
be easily represented as the equivalent forms given
by
where and
Naturally, from the special result just above, the
following assertions can easily be revealed:
and
Most particularly, as it is known, when , we
immediately get that
and
.
Remark 2.3. Let
and ,
where and Then, the Laplace
transformation of the constant function
can be easily determined as
We also get that
.
Remark 2.4. Let
and .
Then, by taking into account our classical analysis
information, it can be easily presented by the
special assertions given by
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where and
For the improper integral just above, by having
regard to the following-parametric-differential
changes:
,
the pending result:
is then obtained, where and
More specially, by choosing the values of as
in Remark 2.3, we then receive the
following assertions:
where
Of course, from Remark 2.3, it follows that the
convergence of each one of the improper integrals
being of the elementary forms:
and
also is obvious.
As an elementary-extensive application of the
mentioned forms specified by (7) and (8), in a
similar manner to the relevant transformation and
its special implications like Remarks 2.1-2.4, a
great deal of special transforms can be also
composed by choosing different types of those
functions and in for the related integral forms
constituted as in (8). In particular, for both
Laplace transform and some of the familiar others
transform, one may refer to the source materials in
the references of this work given in, [4], [9], [17],
[27], [28], [29], [30], [31], [32], [33], [34], [35],
[36], [37], [38], [39], as various differ-integral
transformations or numerous related different type
operators.
3 The Beta Function and Some of the
Related Implications
The second implication is directly associated with
numerous extensive applications of the beta
function possessing the form constituted in (4).
Specifically, for this special function and related
information, one looks over the works given in,
[1], [2], [4], [7], [16], [18], [19], [20], [21]. For its
construction, when two expedient functions like
and are given, it will suffice to
take into account any function like being
of the form given by
where and . As it is known, the
notation denotes the familiar set of the real
numbers. At the same time, of course, the special
parameter will also be any members belonging to
the set , which also is the set of complex numbers.
Especially, in terms of convergence and light of
those functions and that
should satisfy the necessary criteria, by
we also denote the second general
transformation of the function consisting of the
form indicated as in (7) and then define it as
the integral form given by
We especially note here that, by selecting those
parameters and differently, it can also receive
various integral representations consisting of many
rational forms of those improper integrals. In such
cases, by considering the convergence of those
integrals constituted by (13) along with (12), and
also making use of our classical analysis
knowledge, a great number of the familiar
transforms can be easily created in the light of
various different types of those mentioned
functions and At the same time, if the value of
the parameter is allowed as real or complex, then
that general integral transformation will lead to
various fields with extensive implications. Now let
us focus on some of its essential properties and/or
possible implications, and also constitute those as
remarks again.
Remark 3.1. Let and be any scalar. For the
functions and , the following statements are
also satisfied:
i)
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and
ii)
Remark 3.2. Let the functions and be of the
form given by
and
The special form of that mentioned transform can
be received as follows:
where each one of the values of all those
parameters located just above is also selected as
their sensible values for the (improper) integral
form given in (12).
Especially, as one of its essential forms, namely, in
Remark 3.2, by setting in (14), that integral
transform then arrives at the well-known Beta
function, which is one of its equivalent forms given
by
where the familiar notation is generally
called as the Beta function in the mathematical
literature. Naturally, due to and there,
it must be and (and, of
course, and when and
).
Shortly,
.
Remark 3.3. Let the functions and be of the
form given by
and
Then, the special form of that mentioned transform
is easily arrived at:
At this stage, for the integral given by (16), when
considering the change of variable:
,
and by means of the information given in (16), the
following relationships:
can be also received, where and are real
numbers greater than Of course, when those
parameters are any complex numbers, their real
parts have to be greater than -1.
Additionally, by taking cognizance of the
substantial relationship between the Gamma
function and the Beta function, which is the form in
(5) (or (6)), the result determined in (17) can be
equivalently represented as the form:
In addition, when the parameters and belong to
, the related assertions presented by (18)
equivalently yield that
In addition, only two of the most important roles of
the Gamma function and the Beta function appear
in both the integral of fractional arbitrary order and
the derivative of fractional arbitrary order, which
both are classical generalizations of the familiar
ordinary integration and the generalization of
differentiation to arbitrary non-integer order and
are designated for the independent variable with the
real , respectively.
Let us now present these two extensive
implications, which are closely related to the
mentioned information introduced in (3)-(5), within
(the scope of) the complex functions as remarks,
which are just below.
Remark 3.4. (Fractional Integral Operator):
The fractional integral of order is denoted, for a
function by
and also defined by
where and the related function is an
analytic function in any simply connected region of
-plane involving the origin, and the multiplicity
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of is raised by necessitating
to be a real number when For more
information in relation with fractional differ-
intgeral in the references.
Of course, the value of , which expresses the
fractional order, can also be any complex number.
Due to the hypotheses there, it is inevitable that the
real part of is greater than zero.
To illustrate the definition presented by this
remark as only one of the indicated implications,
let us consider the mentioned function as the form
given by
for some with (and, of course,
when
Then, in the light of the information between (3)
and (5), and also by making use of the variable-
differential changes given by
from (13), its fractional integral of real order
can be easily determined as the relations
given by
where
Remark 3.5. (Fractional Derivative Operator):
The fractional derivative of order is denoted, for
a function by
and, also described as
where and the function is constrained,
and the multiplicity of is also extinct as
in Remark 3.4. For the particulars of the definition
presented in (23), se the earlier studies in, [39],
[40], [41], [42], [43].
Naturally, as it is also emphasized in Remark
3.5, when the indicated number is any complex
number, of course, its real part must be greater
than zero, and also . For more
details in relation to the familiar definitions that
have been described in Remarks 3.4 and 3.5, one
may also refer to some of the essential-earlier
results in the investigation presented in, [44], [45],
[46], [47], [48].
As an extensive example, we want to consider the
function consisting of the complex-exponential
form given by (20) again. In parallel with the
information indicated between (20) and (21), and
by using the basic information between (2) and
(4), its fractional derivative(s) of (real) order
can be easily re-determined as the
elementary result consisting of the relationships
given by
where
Indeed, in a similar idea to the elementary result
just above and by considering the second case
given by (23) in Remark 3.5, its fractional
derivative(s) of the (real) order:
can be also re-determined as in the form given by
where
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4 Conclusion and Recommendations
In this special section, numerous possible
implications and various sensible suggestions in
relation to our essential results will be presented (or
emphasized) for our readers, who are also
interested in some (generalized) integral
transformations and a number of their interesting
applications.
As a result of a simple focus on this special-
research note in the second and third sections, it is
seen that various special information has been
presented in which the two relevant special
functions play a very active role.
Moreover, in consideration of the special
information between (1) and (6), it can be also
pointed out that all the specific information passed
between (7) and (21) will have quite specific
consequences both in the context of the
transformations and in some of their various special
applications. Of course, in revealing (or
determining) those special implications, it will be
enough to select the relevant parameters and/or
functions in the relevant sections appropriately.
Especially, as more special information in
relation to applying the general form in (8), since
the Gamma function (with real (or complex)
parameters) and the related transformations takes
part in a vast number of applications in such
diverse areas as astrophysics, fluid dynamics,
quantum physics, mathematics, and statistics, it can
be also checked those related special studies given
in the references dealing with related themas in the
references of this special work.
At the same time, a great deal of
transformations can be also composed by choosing
various different types of those functions and
in for the integral forms given in (8). For both
Laplace transform and some of the others
transforms (or their applications), one may also
refer to some of the main works given in, [49],
[50], [51], [52], [53], [54], [55], [56]. Moreover,
as more special information, since the Gamma
function (with real (or complex) parameters) and
the related transformations takes part in a vast
number of applications in such diverse areas as
astrophysics, fluid dynamics, quantum physics,
mathematics, and statistics, it can be also look over
the earlier studies given by the reference in relation
with various applications in the other papers in the
references of this investigation.
Concurrently, in the light of the special
references just above and as more special
information relating to a variety of applications of
the Beta function with real (or complex)
parameters, we point specially that this special
function is a quite useful tool in both representing
and computing the scattering amplitude for Regge
trajectories. Moreover, it was the first known
scattering amplitude in string theory, which was
first conjectured by Gabriele Veneziano. It also
happens in the theory of the preferential attachment
process, a type of stochastic urn process. At the
same time, it is also an important tool for the Beta
distribution and the Beta prime distribution in the
theory of statistics. As it has been pointed out this
special function is closely connected to the Gamma
function and also plays important roles in the well-
known classical calculus.
After quite detailed information above, for the
relevant researchers, we can present some of those
extensive implications and/or examples. Let us now
create some of those special results and then make
certain extra suggestions.
The first implication is related to various
applications of the Laplace transform, which is
possible only to be determined with the help of the
gamma function. It also is just below.
Implication 4.1. Under the specific conditions
designated by the admissible values of the
parameters , and , let
and .
Then, by using the function given by (3), the
Laplace transform of the function (with real
variable) can be easily determined. For it, in the
light of uniform convergence of the relevant
function series and the information expressed in
Remark 2.2, and also by the help of the Maclaurin
series of the function (just above), which is
the improper integral of the function in (26), or,
equivalently, the Laplace transformation of the
mentioned function, can be firstly specified with
the help of those special functions as in the
following equivalent forms:
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where and As only one of its more
special results, when setting and
in both (26) and (27), Implication 4.1 immediately
gives us the following relationships given by
and
The second implication deals with various
applications of those differ-integral operators of
fractional order, for an elementary function with
the complex variable , which also is possible only
to be identified by combining the Gamma function
and the Beta function. Its creation is just below.
Implication 4.2. Under the conditions accentuated
in Remark 3.4 and by using the Gamma function
and the Beta function, the second implication can
be also determined. For it, it is enough to consider
the Maclaurin series of the function given by (26)
and make use of the similar-variable changes there.
Then, the fractional-order differ-integral of that
function given in (26):
and
are also determined, respectively.
As concluding remarks or various suggestions,
first of all, the specific conclusions of all the
comprehensive results we have achieved so far can
be easily obtained. Also, by using the specific
information given between (1) and (5), several
different-type integral transformations can be
reconstructed. For each of those transformations,
just as in the second part and after the main
conclusions made in the third part, similar
conclusions can easily be drawn to the
comprehensive conclusions mentioned in the fourth
part, that is, (26) to (32) and their more specific
conclusions. We bring the determination of each of
such determinations to the attention of relevant
researchers.
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DOI:10.37394/232025.2023.5.12
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Engineering World
DOI:10.37394/232025.2023.5.12
Hüseyi
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E-ISSN: 2692-5079
118
Volume 5, 2023
