Abstract
The Szegö kernel has many applications to problems in conformal mapping and satisfies the Kerzman-Stein integral equation. The Szegö kernel for an annulus can be expressed as a bilateral series and has a unique zero. In this paper, we show how to represent the Szegö kernel for an annulus as a basic bilateral series (also known as -bilateral series). This leads to an infinite product representation through the application of Ramanujan’s sum. The infinite product clearly exhibits the unique zero of the Szegö kernel for an annulus. Its connection with the basic gamma function and modified Jacobi theta function is also presented. The results are extended to the Szegö kernel for general annulus and weighted Szegö kernel. Numerical comparisons on computing the Szegö kernel for an annulus based on the Kerzman-Stein integral equation, the bilateral series, and the infinite product are also presented.
1. Introduction
The Ahlfors map is a branching -to-one map from an -connected region onto the unit disk. It is intimately tied to the Szegö kernel of an -connected region [1]. The boundary values of the Szegö kernel satisfy the Kerzman-Stein integral equation, which is a Fredholm integral equation of the second kind for a region with a smooth boundary [2]. The boundary values of the Alhfors map are completely determined from the boundary values of the Szegö kernel [1–3]. For an annulus region , the Szegö kernel can be expressed as a bilateral series from which the zero can be determined analytically [4]. The Kerzman-Stein integral equation has been solved using the Adomian decomposition method in [5] to give another bilateral series form for the Szegö kernel for that converges faster. There are various special functions in the form of bilateral and basic bilateral series [6–8]. For example, the bilateral basic hypergeometric series contain, as special cases, many interesting identities related to infinite products, theta functions, and Ramanujan's identities. It is therefore natural to ask if the bilateral series for the Szegö kernel for can be summed as special functions or an infinite product that exhibits clearly its zero.
In this paper, we show how to express the bilateral series for the Szegö kernel for as a basic bilateral series (also known as -bilateral series). Ramanujan’s sum is then applied to obtain the infinite product representation for the Szegö kernel for . The product clearly exhibits the zero of the Szegö kernel for , and its connection with the -gamma function and the modified Jacobi theta function is shown. Using the symmetry of Ramanujan’s sum, we show how to easily transform the bilateral series for the Szegö kernel for in [4] to the bilateral series in [5].
The plan of the paper is as follows: After the presentation of some preliminaries in Section 2, we derive the basic bilateral series and infinite product representations for the Szegö kernel for in Section 3. We then derive a closed form of the Szegö for in terms of -gamma function and the modified Jacobi theta function. In Section 4, we show how to extend the representations in Section 3 to the general annulus using the transformation formula for the Szegö kernel under conformal mappings. Similar -analysis for the weighted Szegö kernel for is presented in Section 5. In Section 6, we give numerical comparisons for computing the Szegö kernel for using bilateral series, infinite product, and integral equation formulations.
2. Preliminaries
Let be an annulus with and a point . The boundary of consists of two smooth Jordan curves with the outer curve oriented counterclockwise and the inner curve oriented clockwise. The positive direction of the contour is usually that for which the region is on the left as one traces the boundary.
Let be an orthonormal basis for the Hardy spaces . Since the Szegö kernel is the reproducing kernel for , it can be written as [4] with absolute and uniform convergence on compact subsets of . An orthogonal basis for is . Thus where is the arc length measure. Therefore, an orthonormal basis for is [3, 4]
Using (1) and (3), the series representation for the Szegö kernel for is given by [4]
Series (4) is a bilateral series. It has a zero at [4].
Another bilateral series representation for the Szegö kernel for is given by [5] (in an equivalent form) which is initially obtained by solving the Kerzman-Stein integral equation using the Adomian decomposition method. It is also shown in [5] how to derive (5) directly from (4) using geometric series. It is illustrated in [5] that series (5) converges faster than (4).
More generally, if is any doubly connected region with the smooth boundary , and is a biholomorphic map of onto , then the Szegö kernel for can be obtained via the transformation formula as [1] where is unknown but can be computed.
The Szegö kernel can also be computed without using conformal mapping. The boundary values of the Szegö kernel on satisfy the Kerzman-Stein integral equation [2, 4], where and is a parametrization of . The function is known as the Kerzman-Stein kernel, and it is continuous on the boundary of [9, 10]. In fact, the integral equation (7) is also valid for an -connected region.
Since bilateral series and basic bilateral series will be used throughout this paper, we recall some facts about -series notations and results.
Let and . The -shifted factorial is defined as [7]
This notation yields the shifted factorial as a special case through
If is written in place of , then (9) becomes
It can be shown that [7]
If , it is standard to write which is absolutely convergent for all finite values of , real or complex, when [6]. This yields
Observe that would have zero as a factor if . It would be zero also if but these are all outside the circle since [8].
The bilateral basic hypergeometric series in base with one numerator and one denominator parameters is defined by [6–8]
The series is convergent for and .
The classical Ramanujan’s summation is given by [7, 8]
The special case of Ramanujan’s summation yields [8] also known as Cauchy’s formula. Due to symmetry in and on the right-hand side of (17), it implies [8]
The -gamma function is defined as [7]
Another important special function that is used in this paper is the modified Jacobi theta function defined by [7] where and . For a more detailed discussion on -series and historical perspectives, see, for example, [6–8] and the references therein.
3. Szegö Kernel for an Annulus and Basic Bilateral Series
In this section, we express the bilateral series (4) as a basic bilateral series and derive the infinite product representation of the Szegö kernel for . It is given in the following theorem.
Theorem 1. Let be the annulus bounded by . For , , the Szegö kernel for can be represented by The zero of in is the zero of the factor , that is, .
Proof. From (4), we have
Letting and yields
Applying (12) and (15) gives
Note that the series above is convergent because and . Substituting and into (26) gives (21).
Applying Ramanujan’s sum (16) to (26), gives
But from (14), with , we have
Thus, (27) becomes
Substituting and into (30) gives (22).
The infinite product (22) would have poles if
which implies
But
Therefore, the poles are all outside .
The infinite product (22) would have zeros if
which implies
For the first case
which is outside . For the second case, observe that
which clearly has a zero inside when . Thus, the infinite product (22) for has only one zero inside at . This completes the proof.
We note that the series representation (21) for is valid only for , while the infinite product representation (22) for is meaningful for all except for the infinitely many poles at .
We next show that the Szegö kernel for can also be expressed in terms of the basic gamma function and modified Jacobi theta function. By applying (20) to (29) and substituting and , we have
Applying (19) with , observe that where satisfies . This equation may be written as which yields a solution
Thus, (38) becomes
This can be regarded as a closed-form expression for the Szegö kernel for .
In the following, we show how to easily transform series (4) to series (5) using (18). Letting and , (4) becomes where in the last step we have used (18). By replacing and , we get
Letting yields which is the same as (5).
4. Szegö Kernel for General Annulus
Consider the general annulus with boundary denoted by . The region reduces to if , , and
Theorem 2. Let , , and . The Szegö kernel for can be represented by the bilateral series as The zero of in is .
Proof. Observe that the function maps onto with .
Applying the transformation formula (6) yields
Applying (4) to (48) with and replaced by and , respectively, gives
which simplifies to (46).
Applying (5) to (48) instead of and replaced by and , respectively, gives
which simplifies to (47).
Using the fact that has a zero at for , the zero of for is which implies . This completes the proof.
Similarly, the infinite product representation of for can be obtained by applying (22) to (48) with and replaced by and , respectively.
5. The Weighted Szegö Kernel for an Annulus and Basic Bilateral Series
The weighted Szegö kernel is defined in [11] as
To adopt the notations used in this paper, we change to , to , and to in (51), which gives
Note that is exactly the kernel for discussed in Section 1. The zeros of the kernel are not discussed in [11] but have expressed interest on the effect of the weight on the location of its zeros. In the following theorem, we express the weighted Szegö kernel as a basic bilateral series and derive its associated infinite product representation as well as its zeros.
Theorem 3. Let be the annulus bounded by . For , , and , the weighted Szegö kernel for can be represented by The kernel has a zero in only if takes the form , . In both cases, the zero is .
Proof. Observe that
Letting and , the above equation becomes
which is exactly the same form as (24). Applying the result (26) with , the above equation becomes
Series (57) is convergent because and . Substituting gives (41).
Applying the result (29) with to (57) yields
Replacing and applying (13) give (54).
In the proof of Theorem 1, we have shown that the factors have no zeros in . The factors would have zeros if
Since , we conclude that the kernel has no poles in for any . The factors would have zeros if
which implies
For the first case, observe that
To have a zero in , we must have the condition
which means
Hence, we must have . In this case, the zero of in is .
For the second case, observe that
To have a zero in , we must have the condition
which means
Hence, we must have . In this case, the zero of in is also . This completes the proof.
The weighted Szegö kernel can also be expressed in terms of the basic gamma function and the modified Jacobi theta function. By applying (20) to (58) with , we have
Observe that where satisfies . This equation may be written as which yields a solution
Observe also that where satisfies . This equation may be written as which yields a solution
Thus, (68) becomes
This can be regarded as a closed-form expression for the weighted Szegö kernel for an annulus . Observe that (75) reduces to (42) when .
6. Numerical Computation of the Szegö Kernel for an Annulus
In this section, we compare the speed of convergence of the three formulas for computing the Szegö kernel for based on the two bilateral series (4) and (5) and the infinite product (22).
To approximate (4) numerically, we calculate and and .
To approximate (5) numerically, we calculate and .
To approximate (22) numerically, we compute and and .
The approximations are then compared with the numerical solution of the Kerzman-Stein Equation (7). To solve (7), we used the Nyström method [5] with the trapezoidal rule with selected nodes on each boundary component and . The approximate solution is represented by where is the number of nodes. All the computations were done using MATHEMATICA 12.3. Four numerical examples are given for different values of and . The results for the error norms are presented for each example.
We consider an annulus bounded by with .
Example 1. We consider an annulus with and . The results for the error norms are presented in Tables 1–3.
Example 2. We consider an annulus with and . The results for the error norms are presented in Tables 4–6.
Example 3. We consider an annulus with and . The results for the error norms are presented in Tables 7–9.
Example 4. We consider an annulus with and . The results for the error norms are presented in Tables 10–12.
The numerical results presented in Tables 1–12 show that computations using the infinite product formula (22) converge faster than the bilateral series formulas (4) and (5).
7. Conclusion
This paper has shown that the bilateral series for the Szegö kernel for is a disguised bilateral basic hypergeometric series . Ramanujan’s sum for is then applied to obtain the infinite product representation for the Szegö kernel for . The product clearly exhibits the zero of the Szegö kernel for an . The Szegö kernel can also be expressed as a closed form in terms of the -gamma function and the modified Jacobi theta function. Similar -analysis has also been conducted for the Szegó kernel for general and for the weighted Szegö kernel for . The numerical comparisons have shown that the infinite product method converges faster than the bilateral series methods for computing the Szegö kernel for .
For future work, it is natural to devote further investigation on the infinite product representation for the Szegö kernel for doubly connected regions via the transformation formula (6) and Theorem 1. This however requires knowledge of conformal mapping of doubly connected regions to annulus [12–15]. For some ideas on numerical methods for computing the zero of the Szegö kernel for doubly connected regions, see [16]. Alternatively, perhaps some computational intelligence algorithms can also be considered to compute the zero, like the monarch butterfly optimization (MBO) [17], earthworm optimization algorithm (EWA) [18], elephant herding optimization (EHO) [19], moth search (MS) algorithm [20], slime mould algorithm (SMA) [21], and Harris hawks optimization (HHO) [22].
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The authors wish to thank the Universiti Teknologi Malaysia for supporting this work. This work was supported by the Ministry of Higher Education Malaysia under Fundamental Research Grant Scheme (FRGS/1/2019/STG06/UTM/02/20). This support is gratefully acknowledged. The first author would also like to acknowledge the Tertiary Education Trust Fund (TETFund) Nigeria for overseas scholarship award. The authors thank the referees for comments and suggestions which improved the paper.