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UNIFORM REPRESENTATIONS OF THE INCOMPLETE BETA FUNCTION IN TERMS OF ELEMENTARY FUNCTIONS.

1. Introduction. We may find in the literature a large variety of convergent or asymptotic expansions of the special functions of mathematical physics that have the important property of being given in terms of elementary functions: direct or inverse powers of a certain complex variable z and, sometimes, other elementary functions of z. However, quite often, these expansions are not simultaneously valid for small and large values of | z |. Thus, it would be interesting to derive new convergent expansions of these functions in terms of elementary functions that hold uniformly in z in a large region of the complex plane that includes small and large values of |z|.

In [1,5], the authors derived new uniform convergent expansions of the incomplete gamma function [gamma](a, z) and the Bessel functions [J.sub.v](z) and [Y.sub.v](z) in terms of elementary functions of z that hold uniformly in unbounded regions of C containing the point z = 0. The starting point of the technique used in [1, 5] is an appropriate integral representation of these functions. The key point is the use of a Taylor expansion, at an appropriate point of the integration interval, of a certain factor of the integrand that is independent of the variable z. This fact, the independence of this factor with respect to z, translates into a convergent uniform expansion in a large region of the complex z-plane. The expansions given in [1, 5] are accompanied by error bounds and numerical experiments showing the accuracy of the approximations.

In this work, we continue that line of investigation considering the incomplete beta function [B.sub.z](a, b). This function is used extensively in statistics as the probability integral of the beta distribution and as a special case of the (negative) binomial distribution, Student's distribution, and the F-(variance-ratio)distribution [3]. Among its physical applications, we mention its use in Monte Carlo simulations in statistical mechanics [4] and in cosmology [2]. We consider [B.sub.z](a, b) as a function of the complex variable z and derive new convergent expansions that are uniformly valid in an unbounded region of the complex z-plane that contains the point z = 0. The starting point is the integral definition of the incomplete beta function [9, eq. 8.17.1],

(1.1) [mathematical expression not reproducible],

valid for Ra > 0 and z [member of] C \ [1, [infinity]). The incomplete beta function [B.sub.z](a, b) reduces to the ordinary beta function B(a, b) when z = 1 and, except for positive integer values of b, has a branch cut discontinuity in the complex z-plane running from 1 to [infinity]. When a or b are positive integers, the incomplete beta function is an elementary function of z.

For reasons that will become clear later, it is convenient to consider the integral (1.1) only for Rb [less than or equal to] 1. When Rb [greater than or equal to] 1, we consider instead the following integral representation of [B.sub.z](a,b) thatmay be obtained from (1.1) after the change of variable t [right arrow] 1 - t:

(1.2) [mathematical expression not reproducible],

valid for Ra > 0 and z [member of] C \ [1, [infinity]).

By using the recurrence relation [9, eq. 8.17.20],

[mathematical expression not reproducible],

we find that the function [B.sub.z](a,b) may be analytically continued in the complex variable a to the negative half plane Ra [less than or equal to] 0 with poles at the negative integers a = - 1, -2, -3,... And conversely, by using repeatedly this formula, we have that [B.sub.z](a, b), with Ra [less than or equal to] 0, may be written as a linear combination of elementary functions of its three variables and an incomplete beta function with Ra > 0. Therefore, without loss of generality, in the remaining of the paper we restrict ourselves to Ra > 0.

The power series expansion of the incomplete Beta function is given by [10]

(1.3) [mathematical expression not reproducible].

This expansion also converges absolutely when |z| = 1 if Rb > 0. It may be derived from the integral representation (1.1) by replacing the factor [(1 - zt).sup.b-1] by its Taylor series at the origin and interchanging the order of summation and integration. This Taylor series expansion converges for t [member of] [0,1], but the convergence is not uniform in |z|. Therefore, the expansion (1.3) is not uniform in |z| as the remainder is unbounded when |z| [right arrow] [infinity].

From the hypergeometric function representation of [B.sub.z](a, b) [9, eq. (8.17.7)],

[mathematical expression not reproducible],

and by combining the formulas [8, eq. (15.2.2)] and [8, eqs. (15.8.2) and (15.8.8)], we obtain, for 1 -a -b [??] N [union]{0} and |ph(-z)| < [pi] (ph(z) denotes the phase of the complex number z), the asymptotic expansion

(1.4) [mathematical expression not reproducible]

On the other hand, if 1-a - b [member of] N [union] {0}, |z| > 1, and |ph(-z)| < [pi], then we have

(1.5) [mathematical expression not reproducible]

where [psi] denotes the digamma function. The expansions (1.4) and (1.5) are asymptotic expansions of the incomplete beta function for large |z|, but the remainders are unbounded when |z| [right arrow] 0, and hence, these expansions are not uniform in |z| either. Other large parameter asymptotic approximations with certain uniformity properties with respect to the parameters can be found in [7, 11].

The expansions (1.3), (1.4), and (1.5) have the good property of being given in terms of elementary functions of z, but they have the inconvenience of not being uniform in |z| in unbounded regions of the complex plane that include the point z = 0. In this paper we show that it is possible to derive convergent expansions of [B.sub.z] (a, b) in terms of elementary functions that hold uniformly for z in an unbounded region of C that includes the point z = 0. As an illustration of the approximations that we are going to obtain (see Theorem 2.1 below), we derive, for example, the following one:

(1.6) [mathematical expression not reproducible],

with |e(z)| < 0.0089 in the negative half plane Rz [less than or equal to] 0. When z = 0, the right-hand side of (1.6) must be understood in the limit sense.

In order to derive these kinds of approximations, we use in this paper the technique proposed in [1, 5]: we consider a Taylor expansion of the factor ta-1 in (1.1) and of the factor [(1 - t).sup.a-1] in (1.2). The factor [t.sup.a-1] in (1.1) is not analytic at the origin unless a [member of] N (analogously, the factor [(1 - t).sup.a-1] in (1.2) is not analytic at t = 1). Following the arguments given in [5], we must consider the expansion of the factors ta-1 and [(1 - t).sup.a-1] at the middle point t = 1/2 of the integration interval (0,1) in the respective integrals (1.1) and (1.2) in such a way that we assure that the integration interval is contained in the disk of convergence of the Taylor series. This Taylor expansion is convergent for any t in the integration interval of (1.1) or (1.2), and obviously, it is independent of z. After the interchange of the order of summation and integration, the independence with respect to z translates into a remainder that can be bounded independently of z in a large unbounded region of the complex z-plane that contains the point z = 0 and that we specify in Theorems 2.1 and 3.1 below. In the following section we consider the integral representation (1.1) for Rb [less than or equal to] 1. In Section 3 we consider the integral representation (1.2) for Rb [greater than or equal to] 1. Section 4 contains some comments about the accuracy of the approximations. Throughout the paper we use the principal argument arg z [member of] (-[pi], [pi]].

2. A uniform convergent expansion of [B.sub.z](a, b) for Rb [less than or equal to] 1. In this section we consider the integral representation (1.1). We define the extended sector (see Figure 2.1):

(2.1) [mathematical expression not reproducible],

with arbitrary 0 < [theta] [less than or equal to] [pi] /2. We have the following theorem:

THEOREM 2.1. For Ra > 0, Rb [less than or equal to] 1, z [member of] [S.sub.[theta]], with 0 < [theta] [member of] [pi]/2, and n = 1, 2, 3,...,

(2.2) [mathematical expression not reproducible],

where [[beta].sub.k](z, b) are the elementary functions

(2.3) [mathematical expression not reproducible]

with [[delta].sub.k,j] the Kronecker delta: [[delta].sub.k,j] = 1 if k = j, [[delta].sub.k,j] = 0 if k [not equal to] j. In (2.3), the first term inside the brackets must be understood as zero if j = -b. For k = 1, 2, 3,... and b [not equal to]0, the coefficients [[beta].sub.k](z, b) satisfy the recurrence relation

(2.4) [mathematical expression not reproducible]

On the other hand, for k = 1,2,3,... and b = 0,

(2.5) [mathematical expression not reproducible]

When z = 0, the above expressions must be understood in the limit sense. In the extended sector [S.sub.[theta]], the remainder is bounded in the form

(2.6) [mathematical expression not reproducible].

For n [greater than or equal to] Ra - 1 > 0, the remainder term may also be bounded in the form

(2.7) [mathematical expression not reproducible].

The remainder term behaves as [R.sub.n](z,a,b) ~ [n.sup.-Ra] as n [right arrow] [infinity] uniformly in |z| in the extended sector [S.sub.[theta]].

Proof. Consider the truncated Taylor series expansion of the factor [t.sup.a-1] in the integrand of the integral definition (1.1) of [B.sub.z](a, b) at the middle point t = 1/2 of the integration interval,

(2.8) [mathematical expression not reproducible],

where [r.sub.n](t, a) is the Taylor remainder

(2.9) [mathematical expression not reproducible].

Instead of the point t =1/2 for the Taylor expansion of [t.sup.a-1], we could have chosen any other point t [member of] [1/2,1] such that the convergence disk of the Taylor series of [t.sup.a-1] contains the interval [0,1]. We have selected the point t = 1/2 because in a previous paper [1], where we analyzed a similar problem for the incomplete gamma function, we proved that the optimal choice was the middle point of the interval.

After suitable manipulations, we can write

(2.10) [mathematical expression not reproducible].

Replacing (2.8) in the integral representation of [B.sub.z](a, b) given in (1.1) and interchanging the order of summation and integration, we obtain (2.2) with

(2.11) [mathematical expression not reproducible]

and

(2.12) [mathematical expression not reproducible].

Expanding the first factor of the integrand in the second integral in terms of powers of u and integrating term-wise, we obtain (2.3). Integrating by parts in any of the integrals in (2.12), it is straightforward to see that, for k = 1,2,3,..., the functions [[beta].sub.k](z, b) satisfy the recurrence relations (2.4) and (2.5).

In order to derive the bound (2.6), we need a bound for the factor [(1 - zt).sup.b-1] uniformly valid for t [member of] [0,1]. It is straightforward to verify that, for t [member of] [0,1], we have the bound [mathematical expression not reproducible] with

(2.13) [mathematical expression not reproducible]

The regions of the complex z-plane considered in this formula are depicted in Figure 2.2. For z [member of] [S.sub.[theta]], with 0 < [theta] [less than or equal to] [pi]/2, we have that M(z, b) [less than or equal to] [[sin([theta])].sup.Rb-1]. This inequality may be proved by using the following geometrical arguments:

(i) at the points of the circle |z - 1/2| = 1/2, we have that |1 - z| = |sin(arg(z))|;

(ii) the closest points of the sector [theta] [less than or equal to] | arg(z) | < [pi]/2 to the point z = 1 are just the two points obtained from the intersection of the rays arg z = [+ or -][theta] with the circle |z - 1/2| = 1/2;

(iii) the closest points of the region [mathematical expression not reproducible] and |z - 1| [greater than or equal to] sin[theta]} to the point z = 1 are those of the portion of the circle |z - 1| = sin [theta] contained inside this region.

Now we use that [r.sub.n](t, a) is integrablein (0,1), thebound [mathematical expression not reproducible] for t [member of] [0,1], and insert (2.10) into (2.11). Thus, we obtain

[mathematical expression not reproducible].

From the integral representation of the hypergeometric function [8, eq. (15.6.1)], we find that, for t [member of] (0,1),

|[.sub.2][F.sub.1] (n + 1 - a, 1; n + 1; 1 - 2t)| [less than or equal to] [.sub.2][F.sub.1] (n + 1 - Ra, 1; n + 1; 1 - 2t).

Then,

(2.14) [mathematical expression not reproducible]

Using now the contiguous hypergeometric function [8, eq. (15.5.14)] with a = 1, b = n + 1 - Ra, c = n + 1, and z = -1, we find that

[mathematical expression not reproducible].

and by applying [8, eq. (15.5.11)] in the second hypergeometric function, we can write

2 [.sub.2][F.sub.1] (n + 1 - Ra, 2; n + 1; -1) = n + [(Ra + 1 - 2n).sub.2][F.sub.1] (n + 1 - Ra, 1; n + 1; -1).

Thus, inserting these formulas into (2.14), we get

[mathematical expression not reproducible].

From the integral representation of the hypergeometric function [8, eq. (15.6.1)], we have that

[mathematical expression not reproducible].

The bound (2.6) follows from (2.15) and this last inequality.

When n [greater than or equal to] Ra - 1 > 0, we consider again the integral representation of the hypergeometric function [8, eq. (15.6.1)],

[mathematical expression not reproducible].

When t [member of] (0,1), we have [[1 - (1 - 2t)s].sup.Ra-n-1] < [(1 - s).sup.Ra-n-1], and thus,

[mathematical expression not reproducible].

Therefore, from (2.10) we have that

[mathematical expression not reproducible].

Formula (2.7) follows straightforwardly by inserting this bound into (2.11).

Finally, using the Stirling formula and [6, eq. (30)] in (2.6) or (2.7), we obtain that [mathematical expression not reproducible] as n [right arrow] [infinity]. Then, any of the bounds (2.6) or (2.7) show the uniform character of the expansion (2.2) in the extended sector [S.sub.[theta]]. []

Formula (1.6) follows from Theorem 2.1 with a = 5/2, b = 1/2, and n = 3. An error bound simpler than those given in (2.6) and (2.7) can be found when a is real. It is stated in the following proposition.

PROPOSITION 2.2. For a > 0, Rb [less than or equal to] 1, z [member of] [S.sub.[theta]], and n = 1, 2, 3,..., the error term [R.sub.n](z, a, b) in Theorem 2.1 may be bounded as

(2.16) [mathematical expression not reproducible].

Proof. Take p := [a], and define [alpha] := a - p. Then we have that, for k [greater than or equal to] p,

(2.17) [(1 - a).sub.k] = [(-1).sup.p][([alpha]).sub.p][(1 - [alpha]).sub.k-p].

Using this equality in (2.9), we find that

[mathematical expression not reproducible].

We insert this bound into (2.11), and by using that |[(1 - zt).sup.b-1]| [less than or equal to] [e.sup.[pi]|Tb|] [[sin([theta])].sup.Rb-1] for t [member of] [0,1] and (2.17), we find (2.16). []

Table 2.1 provides the first few terms of the approximation of [z.sup.-a][B.sub.z](a, b) given by the expansion (2.2) for Rb [less than or equal to] 1 and -b [??] N [union] {0}. These terms are rational functions of z and functions of [(1 - z).sup.b]. When -b [member of] N [union] {0}, the terms of the expansion (2.2) also contain the term log(1 - z).

In Figure 2.3 we plot the relative errors, on a logarithmic scale, of the approximation of [z.sup.-a][B.sub.z](a, b) given in Theorem 2.1 forn = 1, 2, 3,4, 5, z = [rho][e.sup.i[theta]], with [theta] = 0, [pi]/4, [pi]/2, -[pi]/3, [rho] [member of] [-100,1) or [rho] [member of] [-100,100], and different values of the parameters a and b. This serves as a numerical experiment investigating the rate of convergence provided by (2.2). We also observe the uniform character of the approximation in the region [S.sub.[theta]].

3. A uniform convergent expansion of [B.sub.z](a, b) for Rb [greater than or equal to] 1. In this section we consider the integral representation (1.2). For any 0 < r [less than or equal to] 1, consider the punctured complex plane at z = 1 with the interval [1, [infinity]) removed:

(3.1) [C.sub.r] := {z [member of] C;|z - 1| [greater than or equal to] r, | arg(1 - z)| < [pi]}.

We have the following theorem.

THEOREM 3.1. For Ra > 0, Rb [greater than or equal to] 1, z [member of] [C.sub.r], with 0 < r [less than or equal to] 1, and n = 1, 2, 3,...,

(3.2) [mathematical expression not reproducible],

where the functions [[beta].sub.k](z, b) are the elementary functions

(3.3) [mathematical expression not reproducible].

For k = 1, 2, 3,..., they satisfy the recurrence relation

(3.4) [mathematical expression not reproducible]

When z = 0, the above expressions must be understood in the limit sense. The remainder is bounded in the form

(3.5) [mathematical expression not reproducible].

For n [greater than or equal to] Ra - 1 > 0, the remainder term may also be bounded in the form

(3.6) [mathematical expression not reproducible].

The remainder term behaves as [R.sub.n](z, a, b) ~ [n.sup.-Ra] as n [right arrow] [infinity], uniformly for z [member of] [C.sub.r].

Proof. The proof is similar to that of Theorem 2.1 but considering the integral representation (1.2) instead of (1.1). That is, we must consider the Taylor expansion of the factor [(1 - t).sup.a-1] at t = 1/2 instead of the expansion of the factor [t.sup.a-1]. Moreover, we must replace z by z/(z - 1) in the factor [(1 - zt).sup.b-1]. We only give here a few significant details.

Replacing the truncated Taylor series expansion of [(1-t).sup.a-1] at t = 1/2 on the right-hand side of (1.2), we obtain (3.2) with

(3.7) [mathematical expression not reproducible]

and

[mathematical expression not reproducible].

Expanding the first factor of the integrand in the second integral in terms of powers of u and integrating term-wise, we obtain (3.3). Thus, we obtain (3.2) with [R.sub.n](z, a, b) given in (3.7). Now, in order to derive the bounds (3.5) and (3.6) instead of a bound for the factor [(1 - zt).sup.b-1] valid for every t [member of] [0,1], we need a bound for the factor (1 - z[(z - 1)-1t).sup.b-1] valid for every t [member of] [0,1]. It is given by |(1 - z[(z - 1)-.sup.1]t).sup.a-1]| [less than or equal to] [e.sup.[pi]|Tb] M(z, b) with

M(z, b) := max{1, [|1 - z|.sup.1-Rb]}.

It is clear that M(z, b) [less than or equal to] r[.sup.1-Rb]', for z [member of] [C.sub.r], and thus, instead of (2.6) and (2.7), we obtain (3.5) and (3.6). []

Numerical experiments show that the recurrence relations (2.4) and (3.4) are not stable for general values of the variable z and the parameter b. Therefore, for large values of k, it is more convenient to use the explicit expressions (2.3) or (3.3), respectively.

Simpler error bounds than (3.5) and (3.6) can be found when a is real. The proof is similar to the proof of Proposition 2.2, and we omit it.

PROPOSITION 3.2. For a > 0, Rb [greater than or equal to] 1, z [member of] [C.sub.r], with Cr defined in (3.1),for 0 < r [less than or equal to] 1 and n = 1,2, 3,..., the error term [R.sub.n](z, a, b) defined by (3.7) in Theorem 3.1 may be bounded in the form

[mathematical expression not reproducible].

Table 3.1 shows the first few terms of the approximation of [z.sup.-a][(1 - z).sup.1-b][B.sub.z](a,b) given by the expansion (3.2). These terms are rational functions of z and functions of [(1 - z).sup.b]. In Figure 3.1 we give the relative errors, on a logarithmic scale, of the approximation of [z.sup.-a][(1 - z).sup.1-b][B.sub.z](a,b) from Theorem 3.1 for n = 1,2,3,4,5 and z = [rho][e.sup.i[theta]] with [theta] = 0,[pi]/4, [pi]/2, -[pi]/3, [rho] [member of] [-100,1) or [rho] [member of] [-100,100], and different values of the parameters a and b. This serves as a numerical experiment confirming the rate of convergence provided by (3.2). We also observe the uniform character of the approximation in the region [C.sub.r].

4. Some remarks on the numerical experiments. The numerical experiments performed for large values of the parameters |a| and/or |b| show that the accuracy of the approximation (2.2) gets worse as the values of the parameters become larger (one or both). For the approximation (3.2), the relative errors grow for large |a| or large |a| and |b|, although the precision is acceptable for large |b|, except for values of z [member of] [0,2], approximately. So, both approaches get worse in general with increasing |a| and/or |b|. In order to avoid these numerical difficulties when one of these parameter is large, it is possible to apply the recurrence relations [9, eq. 8.17.16, eq. 8.17.17, eq. 8.17.20 or eq. 8.17.21] to replace them by smaller ones and then use the expansions (2.2) and (3.2). The new approximations obtained in this way remain expressions in terms of elementary functions. Of course, this approach can be applied as long as one is sure about the numerical stability of the above mentioned recursions.

On the other hand, numerical experiments indicate that the error bounds (2.6), (2.7), (3.5), and (3.6) oscillate between values of two and three times the maximum error in the corresponding approximation, depending on the parameters: the bounds are better for small values of the parameters a and b.

Acknowledgments. This research was supported by Ministerio de Economia, Industria y Competitividad, Gobierno de Espana, project MTM2017-83490-P, Gobierno de Aragon and European Social Fund (group E24_17R).

REFERENCES

[1] B. BUJANDA, J. L. LOPEZ, AND P. J. PAGOLA, Convergent expansions of the incomplete gamma functions in terms of elementary functions, Anal. Appl. (Singap.), 16 (2018), pp. 435-448.

[2] A. J. S. HAMILTON, Formulae for growth factors in expanding universes containing matter and a cosmological constant, Monthly Notices Roy. Astronom. Soc., 322 (2001), pp. 419-425.

[3] N. L. JOHNSON, S. KOTZ, AND N. BALAKRISHNAN, Continuous Univariate Distributions. Vol. 2., 2nd ed., Wiley, New York, 1995.

[4] D. A. KOFKE, Comment on "the incomplete beta function law for parallel tempering sampling of classical canonical systems" [J. Chem. Phys. 120 (2004) 4119], J. Chem. Phys., 121 (2004), p. 1167.

[5] J. L. LOPEZ, Convergent expansions of the Bessel functions in terms of elementary functions, Adv. Comput. Math., 44 (2018), pp. 277-294.

[6] J. L. LOPEZ, P. J. PAGOLA, AND E. PEREZ SINUSIA, A simplification of Laplace's method: applications to the gamma function and the Gauss hypergeometric function, J. Approx. Theory, 161 (2009), pp. 280-291.

[7] G. NEMES AND A. B. OLDE DAALHUIS, Uniform asymptotic expansion for the incomplete beta function, SIGMA Symmetry Integrability Geom. Methods Appl., 12 (2016), Art. 101 (5 pages).

[8] A. B. OLDE DAALHUIS, Hypergeometric function, in NIST Handbook of Mathematical Functions, F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, eds., Cambridge University Press, Cambridge, 2010, pp. 383-402.

[9] R. B. PARIS, Incomplete gamma functions, in NIST Handbook of Mathematical Functions, F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, eds., Cambridge University Press, Cambridge, 2010, ch. 8, pp. 173-192.

[10] K. PEARSON, Tables of Incomplete Beta Functions, 2nd ed., Cambridge University Press, Cambridge, 1968.

[11] N. M. TEMME, Uniform asymptotic expansions of the incomplete gamma functions and the incomplete beta function, Math. Comp., 29 (1975), pp. 1109-1114.

CHELO FERREIRA ([dagger]), JOSE L. LOPEZ ([dagger][dagger]), AND ESTER PEREZ SINUSIA ([dagger])

(*) Received March 22, 2018. Accepted October 16, 2018. Published online on December 4, 2018. Recommended by F. Marcellan.

([dagger]) Departamento de Matematica Aplicada, IUMA, Universidad de Zaragoza, 50009 Zaragoza, Spain ({cferrei,ester.perez}@unizar.es).

([dagger][dagger]) Departamento de Estadistica, Informaticay Matematicas, Universidad Publicade Navarra, 31006 Pamplona, Spain (jl.lopez@unavarra.es).

DOI: 10.1553/etna_vol48s450
TABLE 2.1 The first few terms in the expansion (2.2) of [z.sup.-a]
[B.sub.z](a, b) when -b [??] N [union] {0}.

n  [(1 - a).sub.n][[beta].sub.n](z, b)/n!

0  [mathematical expression not reproducible]
1  [mathematical expression not reproducible]
2  [mathematical expression not reproducible]

TABLE 3.1 The first few terms in the expansion (3.2) of [z.sup.-a]
[(1 - z).sup.1-b][B.sub.z](a, b).

n  [(-1).sup.n][(1 - a).sub.n][[beta].sub.n](z,b)/n!

0  [mathematical expression not reproducible]
1  [mathematical expression not reproducible]
2  [mathematical expression not reproducible]
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Author:Ferreira, Chelo; Lopez, Jose L.; Sinusia, Ester Perez
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Date:Jan 1, 2018
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