A Special Note on the Error Functions Defined in Certain Domains
of the Complex Plane and Some of Their Implications
FATMA AHMED SALEM SALEM1,2, HÜSEYİN IRMAK1
1Department of Mathematics,
Faculty of Science,
Çankırı Karatekin University,
TR-18100, Uluyazı Campus, Çankırı,
TURKEY
2 Department of Mathematics,
Faculty of Science,
Misurata University,
218-2478, Misurata,
LIBYA
Abstract: - The fundamental object of this specific research is firstly to introduce certain requisite information
in relation to the error functions (with complex (or real) parameters), which possess various extensive roles and
responsibilities in applications of science, engineering, and technology, next to determine various consequences
of those complex functions and also to reveal (or point out) some of those possible consequences. Finally, to
present certain recommendations about the possible extent of the scope of this special research note for the
relevant researchers.
Key-Words: - Special functions, the complex error functions, -plane, analytic functions, complex series
expansions, complex inequalities, complex powers.
Received: January 29, 2023. Revised: May 23, 2023. Accepted: June 15, 2023. Published: July 10, 2023.
1 Introduction and Necessary Basic
Information
As it is well known from published literature,
special functions (together with integral
transformations identified by special functions)
undertake a wide range of roles associated with
various extensive subjects of mathematical analysis,
the theory of differential and integral equations,
approximation theory, and to many other fields of
pure and applied mathematics for nearly all
sciences. To see some of those relevant roles and
their scope, it will be sufficient to take a look at the
fundamental resources given in references in [1],
[3], [7], [9], [12], [13], [20], [28], [30], [32], [34],
[37].
Naturally, as centuries old, these fundamental
subjects are under intense development, for use,
especially in pure and applied mathematics,
statistics, physics, engineering, and computer
science. This also stimulates continuous interest for
relevant researchers in those fields. Shortly, the
main aim of those functions (and transformations
specified special functions) is to foster further
growth by providing a means for the scientific
publication of important research on all aspects of
those subjects. For those special functions,
transformations, and also some of their extensive
applications, one can also look over a great deal of
properties, extensive relations, elementary results,
various implications, and also theories pointed out,
[3], [7], [11], [13], [14], [17], [18], [21], [23], [25],
[28], [30], in the references of this paper.
In particular, the classical error functions are
well-known functions as some special types in the
families consisting of all special functions with
complex (or real) parameters. In the written
literature, we generally encounter these special
functions with real parameters. But, in this special-
scientific research, the error functions with complex
parameters will be considered for our possible-
special results. Specially, for the concerned
researchers, we offer the earlier papers given in [3],
[4], [5], [6], [18], [19], [26], [33], [38].
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As well as the basic theory in relation to error
functions (or analysis), these special functions are
also made allowances for probability, statistics,
applied mathematics, mathematical physics, and a
vast number of other theoretical (or practical)
applications. For instance, the Fresnel integrals,
which are derived with the help of those functions,
are very important functions and they are also used
in the theory of optics. For some of those special
forms, one may also examine [1], [7], [9], [11].
In particular, we point out here that the main
purpose of this scientific note is firstly to consider
these error functions, then to put forward some basic
theories in relation to those special functions, and
also to focus on some of their possible implications.
For elementary results associated with those error
functions and also various implications (or extra
applications) of their different forms, one may also
refer to the academic studies presented in [2], [8],
[12], [15], [16], [19], [22], [32], [33], [34], [35],
[36], [38], [39].
For this scientific research, as extra information,
there is a need to introduce various basic notations,
definitions, and also lemmas.
Let us now begin to inform our readers about that
special information.
First of them, as is known, by the familiar notations
given by
and ,
are the set of the real numbers and the set of the
complex numbers, respectively.
In addition, by the best-known notation , we
describe the open unit disk in the complex plane,
which is of the form given by
.
Let also
and
Second of all, the complex error function is
represented by the notation and also
identified by the integral given by
In the same time, in general, as the second definition
of the complex function just above, it is defined by
using the MacLaurin series expansion of the
function with the variable:
,
which is given by
where .
Additionally, the complex-complementary error
function is symbolized by the notation and
described by the integral given by
Moreover, as its second definition of that function, it
is also defined by considering the MacLaurin series
expansion of the mentioned function with the
complex parameter , which is given by
where .
More particularly, by considering the well-known
result given by
and the relation between the definitions precisely
defined as in (1) and (2), the special relationships
are given by
can be easily received, where .
We privately note here that, for the complex error
functions and also various comprehensive relations
between those special functions, one can look over
the remarkable studies, which also possess different
type results, in the references given by [1], [3], [4],
[5], [6], [10], [17], [19], [26], [29], [38], [39].
2 Lemmas and Main Results and
Related Consequences
This section is related to our main results and some
of their possible consequences. Two lemmas will be
important for our research, which are also
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introduced just below. Those are required for setting
and also proving our main results, which will be
determined by certain types of complex error
functions introduced between (1)-(5). In particular,
as theoretical information, one may also check the
basic investigations presented in [17], [24], [27], for
the auxiliary theorems, which are Lemma 2.1 and
Lemma 2.2, respectively. In particular, we offer to
center on some of the earlier results given by [18],
for our readers.
Lemma 2.1. Let and . Then, the
assertion is given by
holds.
Lemma 2.2. Let and also let be an
analytic function in the domain and satisfy the
condition If there exists a point in
such that
and
then
where
Theorem 2.1. If any one of the cases of the
assertion given by
is satisfied, then the assertion given by
is satisfied, where and and the main
value of the power just above is taken cognizance of
its principal ones.
Proof. For the proof of the theorem above, we want
to use Lemma 2.2 together with Lemma 2.1 here.
Firstly, by considering the definition in (4), let us
now define a function being of the form given
by
where
It is quite obvious that both the function
just above is analytic in the complex domain and
it satisfies the condition as it has been
stated in Lemma 2.2. Moreover, the differentiating
of both sides of the statement given in (13) easily
gives us that
and
where , and each one of the main
values of the complex power located in Theorem 2.1
is made account of their principal ones.
Now we suppose that there exists a point like
satisfying the conditions indicated as in
(8), which consist of the mentioned hypotheses
in Lemma 2.2. Then, under the conditions
presented by (10), in the light of the mentioned
assertion given in (6) and the special information
given in (9), from (14), it follows the relationships
presented by
,
where and
Indeed, through the instrumentality of the
mentioned information given in (9) and (10),
from assertion (15), one then arrives at:
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where
,
and
.
Nonetheless, as centering on each one of the
mentioned cases, which express the hypotheses of
Theorem 2.1 constituted in (11), it is clearly seen
that each one of those cases determined as in (16)
is in contradiction with each one of those
inequalities given by (11), respectively.
Hence, this says us that there is no point like
belonging to , which satisfies the condition of the
mentioned theorem given by (8). This means that
the inequality given by (7) holds for all points
belonging to .
Therefore, the assertion constituted as in (13)
immediately requires the inequality given by (12),
which is the provision of Theorem 2.1. Hereby, this
completes the related proof of Theorem 2.1.
As it has been indicated in the part of the
introduction of the first section, there are many
relationships (or many special functions) in
associating with the (complex) error functions. For
those functions, as some examples, especially, one
may look over some of them given by the earlier
investigations given in [1], [3], [7], [9], [18], [20],
[21], [29], [31], [39].
In addition, for those many special relationships or
functions highlighted there, a number of special
conclusions (or suggestions) that may be relevant to
this particular study may be of interest to our
researchers. From this point of view, only one of
them has been presented here, which includes the
relation between those complex error functions
given by (5). Of course, it is possible to expose
others. Now we want to inform relevant researchers
about some of them.
As the first consequence of this extensive
investigation, all fundamental results (or each one
of their possible implications) can be reconstituted
as their equal forms by taking into consideration any
one of various complex series expansions of those
complex error functions which were presented in (2)
and (4).
The important roles of the related results
containing such series expansions can also be very
important, for example, in approximate calculations.
For some of them, one may check the different-type
results presented in [2], [15], [20], [21], [22], [23],
[36], [39].
Moreover, as an elementary-special example, by
considering the information presented by Lemma
2.1 and also making use of the basic form of the
series expansion given by (4), one of the
implications, namely, Theorem 2.1 can be then
reconstituted as in the form given by theorem just
below.
Theorem 2.2. If any one of the cases of the
assertion given by
is occurred, then the assertion is given by
is occurred, where , and the main value
of the complex power (just above) are allowed as its
principal ones.
As the second consequence of this comprehensive
research, in consideration of those relationships
between the error functions which are introduced
by the forms given in (1)-(5), a large number of
related consequences can be also represented (or
determined) for relevant researchers.
In specially, for some of them, by considering
one of the well-known inequalities which are given
by
and
or, by using the special relation specified by (5),
namely, which is:
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some of the possible implications of those theorems
just above can be next represented for some
belonging to the set Since the desired proof of
each of them can be easily created, they are omitted
here.
Theorem 2.3. If any one of the cases of the
statement presented by
is true, then the assertion given by
is true, where , and the value of the
complex power is taken into account as its principal
ones.
Finally, as more special consequences of our main
results or certain recommendations for the relevant
researchers, by selecting the possible values of the
parameters used in all our results, it also possesses
the admissible potential to uncover (or reconstitute)
a vast number of those special results. Here we also
want to present only two of those special
implications.
In Theorem 2.3, with the help of the information
relating to the complex exponential presented by the
form in (6) and also by selecting the value of the
mentioned parameter as in where
the parameter belongs to the mentioned set , as
one of the other implications, the following
Proposition 2.1 can be firstly reobtained as one of
the special consequences in relation with those
complex error functions.
Proposition 2.1. If any of the cases of the assertion
given by
holds true, then the assertion given by
,
holds true, where and the main value of the
power just above are made allowances for its
principal ones.
As has been indicated before, in the light of the
mentioned information in connection with the well-
known inequalities constituted as in (17) and (18)
and by considering the sensible-real values of the
mentioned parameter it can be also re-determined
various comprehensive-special implications of our
essential results above. For an extra exmaple, by
selecting the value of the mentioned parameter as
in Proposition 2.1, the following-two-more-
special consequences can be then represented as one
of the various possible implications of this special
investigation.
Proposition 2.2. If any one of the implications
given by
and
is provided, then the implication is given by
is provided, where
In addition, in view of the mentioned information
pointed out in (17) and (18) and by making use of
the series expansion of the complex-error function
presented in (4) for the function with the complex
variable , which has been constituted as in
Proposition 2.2, the following special consequence
can be last reconstituted.
Proposition 2.3. If any one of the inequalities
given by
and
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holds, then the inequality given by
also holds, where Of course, for the complex
powers just above, the values of their values are
considered as their principal ones.
3 Concluding Remarks
In this special section, some reminders, special
information, and extra suggestions will be
mentioned. Since this comprehensive scientific
research note is directly concerned with the error
functions (with complex parameters), certain
necessary information in relation to those special
functions has been first introduced in the first
chapter and some theorems together with two main
lemmas have been then presented in the second
chapter.
In addition, it has been concentrated on some of the
meaningful implications of our fundamental results
and various recommendations, which will be related
to theoretical and applied applications of those error
functions together with various special functions.
Shortly, we think that all of our main results and
their possible consequences will be interesting for
our readers.
More particularly, as one of the extra suggestions
for interested researchers, we would also like to
remind those about the definition of one type of the
other error functions, which is known as the
imaginary error function in the literature and its
series expansion in the complex form.
The imaginary error function with complex
variable is denoted by
and it is also defined by
,
and, by using this complex function just above, its
series expansion series can be also determined by
the form given by
,
where
As a final word of this special section, we would
also like to emphasize that this special function
erfi(z) with complex variable z introduced above
can be re-evaluated within the scope of our main
results. At the same time, each one of its relevant
potential-specific results, which can be obtained (or
revealed), will also lead to various specific
conclusions, as we did in the second chapter. Of
course, for each of them, it will be necessary to
realize that various special efforts should be made
by the researchers who are interested in those
relevant scientific fields.
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final findings and solutions.
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Conflict of Interest
The authors have no conflict of interest to declare.
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