Comprehensive Results for the Error Functions in the Complex Plane and Some of Their Consequences
HÜSEYİN IRMA, FATMA AHMED SALEM SALE
e, Scienc Faculty of s, Department of Mathematic
1
T.C. Çankırı Karatekin University, Uluyazı Campus,
TR-18100, Çankırı,
TURKEY
2Department of Mathematics, Faculty of Science,
Misurata University,
218-2478, Misurata,
LIBYA
Abstract: – In this extensive special note, various necessary information directly relating to the well-known
error functions considered in certain domains of the complex plane, which are both in the family of classical
special functions and important tools for nearly all sciences and technology, will be firstly introduced, and a
number of our main results (consisting of various analytic-geometric properties of those error functions) will be
also stated (and then proven) by an auxiliary theorem produced in recent studies. In addition, some special
consequences of those main results, which are also associated with certain different types of special functions,
be will pointed out to relevant researchers.
Key-Words: - Domains in the complex plane, the (complex) error functions, special functions, power series,
series expansions, analytic functions
1 Introduction
In this first section, various specific mathematical
structures, which are expressed by certain types
being of some integrals (or functions) and serving
very comprehensive fields of science and
technology, will be introduced. At the same time,
certain special information in relation to historical
dimensions, various possible applications, and
comprehensive references will be introduced to our
readers.
The first is directly related to the Gamma function,
which is generally denoted by and also is
defined by
The second is also related to the following
significant tools given by
which are concerned with the well-known
probability integral. In generally, it is also
encountered as the normal distribution (or Gaussian
distribution). For those, see the fundamental
works, [1], [2], [3], [4], [5]. In particular, the well-
known normal distribution describes its familiar
form given by
where and .
Moreover, both its complex form and each one
of its special forms can be considered for the
theory of complex function. This status will also
be taken into different importance for the literature.
In special, as its familiar special form, when
and the standard normal distribution,
which is called as
Received: November 28, 2022. Revised: July 16, 2023. Accepted: August 23, 2023. Published: September 25, 2023.
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can be easily received. With the help of the
function just above, Gauss considered the integral
in the form given by
which is one of the special forms of the function-
integral constituted as in (1).
In addition, for the other scientific fields, various
effective-comprehensive roles of the forms, given
in (2), continue to be undeniable. Specially, in light
of the information in (2), Gauss considered the
following-specific integral:
which also is appertaining to the error of normal
distribution.
In the light of our classical analysis information
and also by means of (2) and (6), it is known that
the assertion:
Of course, as is known, in the special forms given
in (2), the related independent variable can be a
real variable as well as a complex variable, which
are generally considered as and , respectively.
Naturally, it is clear that each of the mathematical
expressions given in (2) to (5) will be also centered
upon various roles for quite different dimensions in
terms of the theory of the real-complex functions.
In general, as pointed out before, various types
consisting of the different forms indicated in (2) are
widely used in both normal and limit distributions
in the theory of probability and statistics. See the
main works given in, [1], [5], [6], [7], [8], [9]. At
the same time, there are various types of those
special functions and their inverse functions
defined with the help of that mentioned integral. As
some examples, the error function, the generalized
error function, the complementary error function,
the imaginary error function, and also their inverse
functions, which are the inverse error function, the
inverse of the generalized error function, and the
inverse complementary error function, respectively,
are also identified by the help of that integral in (2).
Although the comprehensive roles are seen in very
wide areas, in particular, these functions and their
approximations are also used to predict various
results that are valid with high probability (or low
probability). In addition, they also appear in
solutions to the heat equation when the boundary
conditions are given by the Heaviside step function.
For both those and some, their applications and
their details, one may refer to the essential works,
[1], [2], [5], [6], [8], [9], [10], [11], [12], [13], in
the references.
In particular, we also note that, in this specific
note, various types of forms specified by the
integral with complex variables will be then
concentrated on the complex forms of those
functions as indicated just above. In general, each
of these special functions is thought of as a typical
probability integral and inverse. Especially in the
literature, although definitions of integral forms are
used for each of these special functions, their
definitions in serial expansion form, which can be
easily created with the help of classical analysis
information, are also frequently used. For these
complex special functions and their implications,
one may look over the earlier studies given in, [3],
[4], [14], [15], [16], [17], [18].
Let us first present some of the mentioned
definitions concerning those error functions and
then remind us of certain information relating to
their relationships possessing infinite series forms
in the next section.
2 The Power Series of some Basic
Complex Error Functions
In this special section, the basic complex error
functions will be introduced, which will have
important roles for various special (complex)
functions. The complex error function, the complex
complementary error function, and the imaginary
error function, which are designated by the help of
the form presented in (6), are denoted by
erf(z) , erfc(z) and erfi(z) (8)
and also defined by
and
respectively, where the mentioned parameter is a
member belonging to any domain of the complex
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plane , which consists of any path moving from
the point origin of the set to any point .
There are a great number of relationships along
with the error functions introduced as in (8)-(11)
and also various special functions relating to those
functions. For each of them and their details, the
works given in, [1], [3], [4], [19], [20], [21], [22],
[23], in the references can be also presented as
main sources. It is only pertinent to remember the
following special information between the error
functions consisting of their complex forms given
by (9) and (10).
As one of a large number of relationships
between those special functions, by taking into
account the elementary result presented in (7) and
one of those relations between the complex forms
of the error functions given by 5
can be then received for all in .
In addition, some of the extra specific results,
which are directly interested in the mentioned
fundamental information, are also presented just
below. For further information, it can be checked
the references given in, [7], [21], [24], [25], [26],
[27], [28], [29], [30], [31].
Remarks 1. For the essential error functions, the
following are satisfied:
-
-
-
-
-
-
-
-
-
-
-
-
-
and so on.
The accuracy of each of the fundamental
assertions just above can be easily demonstrated by
making use of their mentioned definitions of the
mentioned complex functions. However, to shed
light on the special result constituted as in (11), we
would like to present some extra information for
the second property.
As a more special example, for the value of
at infinity (, by the definition given by
(9) it will be in the form below:
.
By considering the basic mathematical changes
given by
,
the following improper integral:
is then received. At the same time, with the help of
the well-known result given by
and the Gamma function defined by (1), can be also
expressed as the following-special result:
As we have emphasized before, in the light of
our classical analysis knowledge, we can easily
determine the complex series expansions of those
error functions between (9)-(11), which play very
important roles in many research areas. Moreover,
these series expansions are also considered as their
second definitions of the complex function given in
(8) in the literature. We want to present those
remarks as follows.
Remark 2. The complex error function defined by
the form in (9) possesses the series expansion given
by
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where
Remark 3. The complex complementary error
function defined by the form in (10) is of the series
expansion presented by
where
Remark 4. The imaginary error function defined
by the complex form in (11) has the series
expansion given by
where
As some of the possible main consequences of
this scientific study, when focusing on the complex
error functions introduced by the forms in (8) and
their series expansions, all right, various new-
classic results appertaining to those complex-power
series determined in (14)-(16) can be also
produced. For some of them, the auxiliary theorem
given below will play a very important role in both
their creations and their proofs. Although the
earlier paper given in, [32], is the main reference
for the related auxiliary theorem, and the extra
works (or studies), [8], [11], [33], can be also
presented as some extra references. In the same
time, in particular, for the theory of complex
functions, one may also center on, [22], [34], [35],
[36], [37], the related references of this present
paper.
Lemma 1. Let and also let a complex
function be both analytic in the domain and
satisfy the condition If there exists a
point in such that
and
Then
3 The Main Results and Some of
Their Implications
In this section, which is the main section of our
investigation, the proofs of the first three theorems,
which will be presented, can be easily composed
with the help of fairly basic mathematical
knowledge. The proof of the other three theorems
can be accomplished by using the relations between
the hypotheses and provisions of Lemma 1 in a
logical and coherent way.
Let us now present the relevant theorems in
relation to those complex error functions and then
center on their proofs.
Theorem 1. For the complex error function being
of the form given by (14), the following assertion
holds:
Theorem 2. For the complementary error function
having the form given by (15), the following
assertion holds:
Theorem 3. For the imaginary error function
possessing the form given by (16), the following
assertion is true:
Proofs of Theorems 1-3: For some and all
by taking cognizance of the
modulus of the power series of those complex
functions given by (14)-(16) and also making use
of the incontrovertible equalities given by
and
,
their pending proofs can be easily completed. We
think that they are easy. Therefore, their details are
omitted here.
Theorem 4. For the complex error function being
of the form presented by (14), if the statement
given by
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is true, then the assertion given by
is also true, where and, of course, the value
of the complex power given in (23) is considered as
its principal ones.
Proof. For the proof of Theorem 1, there is a need
to consider Lemma 1. For it, when considering the
mentioned function being of the complex
form given by
It is easily seen that both is an analytic
function in and it also satisfies the indicated
condition even although the critical
point is a removable point for the
function
By virtue of differentiating the function in (25)
with respect to the complex variable , the
following:
or, equivalently,
can be then propounded for some
Now, suppose that there exists a point
satisfying the conditions given by (17) and (18) as
it is pointed out in Lemma 1. Then, in the light of
this special information and by making use of the
assertion given in (26) together with (25), the
following equality:
can be easily represented for all belonging to the
complex domain
Since the function with the complex variable
and the concerned point are suitable for all
conditions of Lemma 1, therefore, with the help of
the information indicated in (17) and (18), the
equation given by (27) gives us
where
As a result of various mathematical operations, it is
easily seen the inequality determined in (28) is a
contradiction with the inequality presented in (23),
which is the hypothesis of Theorem 1. Therefore,
the desired proof is completed.
Theorem 5. For the error function being of the
complex form given as in (15), if the inequality
given by
is supplied, then the assertion is given by
is also supplied for and the value of the
complex power given in (29) is considered as its
principal one.
Theorem 6. For the error function being of the
complex form given as in (16), if the inequality
given by
is provided, then the assertion is given by
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is also provided for some in the set and the
value of the complex power presented in (30) is
considered as its principal one.
Proofs of Theorems 5 and 6. For each one of
those proofs, with similar thought, it will be enough
to follow all of the similar steps used in the proof
of Theorem 4. For them, when one reconsiders the
mentioned-analytic function as the complex
forms designed by
and
respectively, where , and then applies the
same ways, as we did in the proof of Theorem 4, to
those two theorems, the pending proofs of
Theorems 5 and 6 can be easily completed. The
related details of those theorems are also omitted
again.
4 Conclusions and Recommendations
As is known, in the first part of this special
research, some special mathematical forms have
been mentioned, various information has been
given about the (complex) error functions in the
second part, and in the third part, some extensive
theories (together with their possible implications)
in relation to those complex error functions have
been put forward and some of them have been also
demonstrated. In this section relating to the
conclusion and recommendations, in this note, we
would like to elaborate on the scope of our
scientific research with various information for our
readers. In such cases, we have to focus on
extensive research or information.
As we have emphasized in the previous
sections, error functions are special functions that
will appeal to almost all fields of science, which are
both with real variables and with complex
variables. As it is known, we have only
concentrated on three basic types of all error
functions encountered in the literature. We have
stated that each of them has many basic features
unique to them, and we have also presented some
of them. In specially, the diverse studies given in,
[29], [30], [33], can be reviewed for other special
properties as well as for other type error functions.
In addition, the relationships related to the three
related error functions and their other types with
other special functions are quite extensive.
It even differs considerably in its theoretical
applications together with its inverses in special
functions, serial forms, transformation theory,
approximation theory, and measurement theory.
Each result, which may be unusual, can have quite
different meanings in terms of error functions with
complex parameters. For those, the special
information given in, [1], [3], [4], [5], [10], [14],
[17], [18], [22], [23], [26], [28], [33], can be also
examined.
From various aspects, since our study includes
various theoretical results, for researchers, it would
be appropriate to mention only some special
implications of the related theories that we
obtained. Some of those possible suggestions can
be listed below.
1. Special relationships that we have not
highlighted with Remarks 1 can be taken into
account in the theories that we have obtained
before or in any specific results regarding them.
For example, considering the fundamental relation
given by (13) for Theorem 5 and Theorem 6, it is
easily seen that the related theorems are equivalent
to each other.
However, given the special relations between
the various special functions of error functions,
each of my theorems presented will also require
various special theories (or consequences) that may
be related to those special functions.
However, the mentioned special relations
between various special functions and error
functions, each of our main results will also require
various special theories (or consequences), which
are related to those special functions. As extra
information, those relationships concerning some
special functions can be checked in the paper in,
[1], [21], [29]. Moreover, as a simple example in
relation to the well-known incomplete Gamma
function, the special relations are given by
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can be also re-evaluated in the scope mentioned
just above, where .
2. As is known, Lemma 1 has been used for the
proofs of all of our theorems. Naturally, it is
essential to satisfy both the condition and the
differentiability condition for all points in the set U
for the function, which have been used in the
relevant lemma, Lemma 1.
Therefore, provided that these fundamental
conditions are taken into account, the relevant
complex function used in the proofs of all theorems
can be defined in various different types and all of
our main results can be also reconstructed
according to these new functions. For instance, by
considering the series expansion belonging to the
complex error function given in (15), the
corresponding function can be then selected as
follows:
which explicitly satisfies the mentioned conditions
in the lemma, where the variable belongs to of
the complex plane
3. By taking cognizance of various inequalities
frequently used in classical analysis, various
special results can be reconstituted with the help of
the mentioned theories (or their special results). As
an example, the following proposition can be easily
derived with the help of Theorem 1.
Proposition 1. For the complex error function
being of the form presented by (14), if any one of
the statements given by
and
is satisfied, then the inequality given by (24) is also
satisfied.
4. Lastly, some complex forms of various types
of real functions that are frequently used in the
literature can be also used. As novel research, by
using those possible forms of all those complex
functions that can be created with different logic,
together with our main results, various logical
results can be also reproduced. For example, the
complex forms of the normal distribution function
in (3) (or all their possible special cases), which
have important roles in statistics, can be
reconsidered. In such cases, various new results can
be also reproduced with the help of those special
complex functions with our main results. For
instance, by considering the special function
defined as in (4), the following special integral,
which was used by K. Gauss for this study, given
by
can be then reused for our main results. In this case,
for the indicated complex function, when it is
evaluated in the information of Theorem 1, only
two of the pending propositions can be easily
reconstructed with the help of the complex
functions given by
or
where
Especially, noting here that, clearly, if these
special-complex functions:
and
are taken into account together with the complex
function in the mentioned lemma, i.e., Lemma
1, those complex functions (just above), which are
also suitable for all the hypotheses of the lemma,
can be also re-arranged for each one of our
fundamental results constituted as in the third
section.
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Policy)
The authors contributed to the present research, at
all stages from the formulation of the problem to
the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
This research received no external funding.
Conflict of Interest
The authors have no conflict of interest to declare.
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