ferreira zeros of K

arXiv:math/0607471v2 [math.CA] 14 Nov 2006Zeros of the Macdonald function of complex order Erasmo M. Ferreira Instituto de F´ ısica, Universidade Federal do Rio de Janeir o, Caixa Postal 68528, 21941-972 Rio de Janeiro, RJ, Brasil Javier Sesma Departamento de F´ ısica Te´ orica, Facultad de Ciencias, Un iversidad de Zaragoza, 50009 Zaragoza, Spain Abstract Thez-zeros of the modified Bessel function of the third kind Kν(z), also known as modified Hankel function or Macdonald function, are consi dered for arbitrary complex values of the order ν. Approximate expressions for the zeros, applicable in the cases of very small or very large |ν|, are given. The behaviour of the zeros for varying |ν|or arg ν, obtained numerically, is illustrated by means of some grap hics. Key words: Macdonald function, modified Bessel function of the third ki nd, Hankel function, zeros 1991 MSC: 33C10 Corresponding author: Javier Sesma address: Departamento de Fisica Teorica, Facultad de Ciencias, 50009 Zaragoza, Spain. phone: 34 - 976 761 265 fax: 34 - 976 761 264 e-mail: [email protected] 1 Introduction. The relevance of Bessel functions to the solution of a great v ariety of problems in Physics and Engineering is widely known and does not need t o be stressed. Preprint submitted to Elsevier 2 February 2008 In particular, zeros of the different kinds of those function s are closely related to energies of bound or resonant physical systems. For insta nce, the ν-zeros of the Hankel function Hν(z) for positive values of zdetermine the poles of the amplitude of scattering of various kinds of waves by spheres and cylinders [20]. Almost four decades ago we considered [10] the zeros of the mo dified Bessel function of the third kind, Kν(z), also known as modified Hankel function or Macdonald function. Information about the z-zeros of Kν(z) for positive order νwas found in the book by Watson [33]. However, our interest, m otivated by a quantum mechanical problem, was on z-zeros when νassumes purely imag- inary values. For that case it was already known [12,18] that Kν(z) presents an infinity of zeros, all of them located on the positive real s emiaxis of the complex z-plane. We showed that these positive zeros, labelled accor ding to decreasing values, form a sequence tending to be a geometric progression with ratio exp( −π/|ν|). We gave, also, asymptotic approximations to the position of the largest zeros for large values of |ν|. Further discussions of the properties of the zeros of Kν(z),νpure imaginary, have been carried out by Laforgia [21] and by Dunster [7]. Zeros of Kν(z) with complex νhave been less studied. In view of their application in the analysis of electromagneti c wave propagation in anisotropic inhomogeneous conducting media, Nalesso [2 6] has considered theν-zeros of Kν(tν) for fixed positive values of t. But, to our knowledge, a discussion of the z-zeros of Kν(z) for arbitrary complex νis still lacking. In recent years, stimulated by review papers by Lozier and Ol ver [23,24] and by the DLMF project, a considerable number of new algorithms for the com- putation of special functions have been published. Concern ing the Macdonald function, Refs. [9,13,14,15,16,17,29] are to be noticed. A s a general rule, algo- rithms lose accuracy in the vicinity of the zeros. Hence the i nterest of having a knowledge, at least approximate, of the location of those ze ros. On the other hand, Kν(z) presents only a finite number of z-zeros in the Riemann sheet −π <argz≤πifνis real, whereas an infinity of zeros occur if νis pure imaginary. It has therefore seemed to us interesting to stud y the evolution of thez-zeros of Kν(z) as the modulus and/or argument of νare continuously changed. In what follows, we consider only values of νin the quadrant 0≤argν≤π/2, andzis assumed to lie in the principal Riemann sheet, −π <argz≤π, for large values of ν. The symmetry relations K−ν(z) =Kν(z), K ν(z) =Kν(z), (1) allow to extend our results to values of νin all quadrants. On the other hand, 2 the relation [1, Eq. 9.6.4] Kν(z) =1 2πie1 2νπiH(1) ν/parenleftBig ze1 2πi/parenrightBig , (2) valid when KandH(1)are analytically continued to every Riemann sheet, proves that the pattern of z-zeros of the Hankel function H(1) ν(z) results from that of Kν(z) by rotation by an angle of + π/2 around the origin. We discuss in Section 2 the behaviour of the zeros for decreas ing values of |ν|. The case of large values of |ν|is considered in Section 3, where asymptotic approximations to the zeros are given. Numerical results fo r the zeros of Kν(z) with moderate |ν|are presented graphically in Section 4. Finally, the partic ular case of real νis commented in Section 5. 2 Zeros of Kν(z)for|ν| ≪1. The particular case of pure imaginary νrevealed [10] that, as |ν|goes to zero, all zeros of Kν(z) approach the origin. Therefore we start by assuming |z| ≪1 in the expression of Kν(z) as a sum of two convergent ascending series [1, Eqs. 9.6.2 and 9.6.10] Kν(z) =(π/2) sin(νπ)/parenleftBigg(z/2)−ν Γ(1−ν)∞/summationdisplay k=0(z2/4)k k!(1−ν)k−(z/2)ν Γ(1+ν)∞/summationdisplay k=0(z2/4)k k!(1+ν)k/parenrightBigg ,(3) valid for νdifferent from an integer. Obviously, the zeros of Kν(z) should satisfy the relation (z/2)2ν=Γ(1+ν) Γ(1−ν)∞/summationdisplay k=0(z2/4)k k!(1−ν)k ∞/summationdisplay k=0(z2/4)k k!(1+ν)k, (4) that, in the case of being |z| ≪1, can be approximated by (z/2)2ν=Γ(1+ν) Γ(1−ν)/parenleftBigg 1 +ν 2(1−ν2)z2+O(z4)/parenrightBigg ,|z| ≪1. (5) By taking logarithms, one obtains, as a first approximation, log|z|+iargz≃ −nπi ν+ log 2 +1 2νlog(Γ(1+ ν)/Γ(1−ν)) +z2 4(1−ν2),(6) 3 where ncould in principle take the values n= 0,±1,±2, . . ., although, accord- ing to that obtained in the case of νbeing pure imaginary, actually existing zeros correspond to n= 1,2,3, . . .. These values of nwill be used as a label to characterize each zero in the form zn. Ignoring, in the crudest approximation, the last term in th e right hand side of Eq. (6), one has log|zn|≃ℜ/parenleftbigg1 ν(−nπi+1 2log(Γ(1+ ν)/Γ(1−ν))/parenrightbigg + log 2 , (7) argzn≃ℑ/parenleftbigg1 ν(−nπi+1 2log(Γ(1+ ν)/Γ(1−ν))/parenrightbigg . (8) The behavior of the zeros znas|ν|decreases can be immediately obtained from these expressions by using the series expansion [1, Eq. 6.1.34] 1 Γ(1+ν)= 1 + γν+c3ν2+. . . (9) to obtain Γ(1+ν) Γ(1−ν)= 1−2γν+ 2γ2ν2+. . ., (10) where γrepresents the Euler constant, γ= 0.5772156649 . . .. By retaining terms up to O(|ν|−1), it turns out log|zn|≃−nπℑν |ν|2+ log 2 −γ, (11) argzn≃−nπℜν |ν|2. (12) These two equations allow us to obtain a picture of the behavi our of the zeros ofKν(z) as|ν|tends to zero. For instance, let us assume that |ν|decreases with argνfixed. If arg ν=π/2, the zeros znapproach the origin along the positive real semiaxis, as described in Ref. [10]. If 0 <argν < π/ 2,|zn|tends to zero whereas arg zndecreases without limit. In other words, the zeros approach the origin spiraling clockwise infinitely. According to Eq. (12 ), most of zeros (or even all of them, for sufficiently small |ν|) occur outside the principal Riemann sheet. For arg ν= 0, Eqs. (11) and (12) are no more valid because, as we will see later, the possibility exists of zngoing to infinity for certain values of ν. 4 The behaviour of the zeros in this case is much more complicat ed and will be described in Section 5. 3 Zeros of Kν(z)for|ν| ≫1. We have already mentioned that Eq. (2) allows one to obtain th e zeros of the Macdonald function from those of the Hankel function, an d viceversa. Previous research, by other authors [3,11,19,20,25,27], o n the ν-zeros of the Hankel function shows that, for large values of |z|, they occur at ν≃z. Asymptotic expansions of H(1)valid in the transition region should then be used to obtain approximate expressions of its z-zeros for |ν| ≫1. From Eqs. 9.3.23 and 9.3.24 of Ref. [1], by using Eqs. 9.1.3 and 10.4.9, a convenient expansion can be written, namely H(1) ν(ν+θν1/3)∼24/3 ν1/3e−iπ/3/parenleftBigg Ai(−21/3ei2π/3θ)∞/summationdisplay k=0fk(θ)ν−2k/3 +21/3 ν2/3ei2π/3Ai′(−21/3ei2π/3θ)∞/summationdisplay k=0gk(θ)ν−2k/3/parenrightBigg ,(13) where f0(θ) = 1, the other functions fk(θ) and gk(θ) being given in Eqs. 9.3.25 and 9.3.26 of the same reference. A more powerful asymptotic expansion of H(1)exists [1, Eq. 9.3.37], but its coefficients are much more comp licated and, although reversion of that expansion could be done by means o f a procedure due to Fabijonas and Olver [8], we find preferable, for the sak e of simplicity and transparency, to work with Eq. (13). A simple inspection of that expansion shows that, for large |ν|, it vanishes for values of θverifying θs=−2−1/3e−i2π/3(as+εs(ν)), s = 1,2,3, . . ., (14) where asrepresents each one of the zeros of the Airy function, given i n Ta- ble 10.13 of Ref. [1], and εs(ν) =O(ν−2/3) is to be determined. Trying an expansion of the form εs(ν) =∞/summationdisplay j=1αs,jν−2j/3, (15) and using Taylor expansions of the Airy function and of its de rivative around as, it is straightforward, with the aid of a computer algebra so ftware package, to check that the right hand side of Eq. (13) cancels out for 5 αs,1=−3 102−1/3e−i2π/3a2 s, αs,2=−1 70021/3ei2π/3(a3 s+ 10), αs,3=1 126000as(479a3 s−40), αs,4=1 161700002−1/3e−i2π/3a2 s(20231 a3 s+ 55100) ,etc. Approximate values of the z-zeros of H(1) ν(z) are then given by zs∼(ν+θsν1/3),|ν| ≫1, s = 1,2,3, . . .. (16) Reversion of this relation between zandνwould provide an approximation for the ν-zeros of H(1) ν(z) when |z|is large, νs∼z+ 2−1/3e−i2π/3asz1/3+1 6021/3ei2π/3a2 sz−1/3−1 700(a3 s+ 10) z−1 +1 11340002−1/3e−i2π/3as(281a3 s+ 10440) z−5/3 −1 261954000021/3ei2π/3a2 s(73769 a3 s+ 6624900) z−7/3,|z| ≫1,(17) to be compared with those given in Refs. [3,11,19,20,27]. Ac cording to Eq. (2), thez-zeros of K(1) ν(z) are given approximately by zs∼e−iπ/2(ν+θsν1/3),|ν| ≫1, s = 1,2,3, . . .. (18) The usefulness of Eqs. (17) and (18) is restricted to the first values of s= 1,2,3, . . ., due to the fact that |as|increases with s. This makes the the omit- ted terms in the expansion (17) to increase and invalidates t heir suppression. Besides this, |θs|, given by (14), increases with sand, for a given ν, may reach a value such that expansion (13) ceases from being useful. If z eros corresponding to a large sare to be considered, approximate values can be obtained fol low- ing a procedure, already used by Magnus and Kotin [25] and by C ochran [3], consisting in taking logarithms in Eq. (4), 2νlog(z/2) +i2nπ= log(Γ( ν))−log(−Γ(−ν)) + log/parenleftBigg∞/summationdisplay k=0(z2/4)k k!(1−ν)k/parenrightBigg −log/parenleftBigg∞/summationdisplay k=0(z2/4)k k!(1+ν)k/parenrightBigg , n= 0,±1,±2, . . ., (19) and approximating the logarithms of the gamma functions by t heir asymptotic expansions [1, Eq. 6.1.40]. One obtains in this way 6 log(z/2)∼logν−1−iπ/2−i(n−1 4)π/ν+∞/summationdisplay m=1B2m 2m(2m−1)ν2m +1 2ν/parenleftBigg log/parenleftBigg∞/summationdisplay k=0(z2/4)k k!(1−ν)k/parenrightBigg −log/parenleftBigg∞/summationdisplay k=0(z2/4)k k!(1+ν)k/parenrightBigg/parenrightBigg . (20) Assuming |z| ≪ |ν|and keeping the dominant terms, it comes out for the z-zeros of Kν(z) zn≃e−iπ/22νexp/parenleftBig −1−i(n−1 4)π/ν/parenrightBig . (21) Obviously, since we are considering values of νin the first quadrant, only large positive values of the integer nare compatible with the assumption |z| ≪ |ν|. Moreover, for a given value of |ν|, the approximation (21) looses accuracy as arg νgoes from π/2 towards 0 and becomes completely useless for real ν. A numerical comparison of the values obtained with Eqs. (18) and (21) for moderate values of sandnshows that both labels coincide. 4 Zeros of Kν(z)for intermediate values of |ν|. We have used a numerical procedure to explore the behaviour o f thez-zeros ofKν(z), as a function of ν, for moderate values of |ν|. In order to simplify the presentation of the results, two different situations ha ve been considered: (i) variable |ν|with fixed arg ν, and (ii) variable arg νkeeping |ν|fixed. The resulting trajectories of the first, second, and third zeros are represented in Figures 1, 2, and 3, respectively. For the evaluation of Kν(z), expression (3) has been used. More sophisticated algorithms can be found in the literature [2,9,13,14,15,16,17,29,30,31,32], but either they have b een conceived only for particular (real or pure imaginary) values of the variable o r the order, or they become unnecessarily complicated for our purpose of gi ving a merely qualitative description of the zeros. For the location of th ese, the Newton method has been used. Let us mention, however, other existin g techniques like, for instance, the auxiliary tables for the evaluation of the ν-zeros of Kν(z), forzreal or pure imaginary, proposed by Cochran and Hoffspiegel [ 4], the finite element approximation method, applied by Leung an d Ghaderpanah [22] to a very precise determination of the zeros of Kn(z), and a procedure, applied by Segura [28] to the computation of zeros of Bessel a nd other special functions, that uses fixed point iterations and does not requ ire the evaluation of the functions. 7 -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz Fig. 1. Trajectories of the first zero, z1, ofKν(z). The continuous lines represent the trajectories followed by z1asνvaries whereas arg νremains fixed and equal tomπ/20, with m= 0 (first trajectory from the left), 1 ,2,... ,10 (horizontal tra- jectory). The continuous lines turn into short-dashed ones as the zero leaves the Riemann sheet −π <argz≤π. The trajectory corresponding to real ν(m= 0) goes to (+ ∞,π/4), with arg z1→ −2π, forν= 1/2. The nearly circular dashed arcs represent the trajectories of z1for fixed integer |ν|and varying arg νgoing from 0 toπ/2. The trajectories corresponding to |ν|= 5 and 10 are explicitly indicated. A double precision FORTRAN code was used to compute the expre ssion Eν(z) =∞/summationdisplay k=0(z2/4)k k!(1−ν)k−(z/2)2νΓ(1−ν) Γ(1+ν)∞/summationdisplay k=0(z2/4)k k!(1+ν)k(22) whose zeros coincide with those of the right hand side of Eq. ( 3). The two series were summed term by term until the absolute value of the ratio of the last term to the partial sum turned out to be less than 10−18. A warning message was foreseen for the case of the number of summed terms becomi ng larger than 100, but this limit was never reached. Fortunately, zeros at large values of |z| occur for values of νeither of large modulus or in the neighbourhood of n+1 2, n= 0,1,2, . . .. In the first case, the series in (22) converge reasonably, fr om the numerical point of view; in the second case, analytical meth ods can be used to approximate the zeros. Typical runs of our procedure can be s een in Table 1, which shows intermediate results in the determination of th e first, second and third zeros of Kν(z), with |ν|= 21, arg ν= 7π/20. Initial approximate values 8 -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz -10-8-6-4-2024 -10 -5 0 5 10105 RezImz Fig. 2. Trajectories of the second zero, z2, ofKν(z). The same convention as in Fig. 1 has been adopted to represent the behaviour of z2. The trajectory for real νgoes to (+∞,3π/4), with arg z2→ −2π, forν= 3/2. of the zeros, for a given ν, were obtained by extrapolation of the trajectories followed by the zeros as νvaries. Besides the smallness of the values of the real and imaginary parts of Eν(z), their changes of sign in the successive steps of the Newton method make us to be confident about our results. Our figures show that for large |ν|, with arg ν∈(0, π/2), the zeros lie far from the origin in the half plane −π <argz <0, according to Eq. (18). When |ν|decreases, they approach the origin but, before reaching it , they cross the semiaxis arg z=−π, leaving in this way the principal Riemann sheet−π <argz≤π, for a certain |νcrit,s|whose value depends on arg νand on the order sof the zero zs. The smaller arg ν(between 0 and π/2) and the higher the label sof the zero, the larger the value of |νcrit,s|. (In the case of realν, it can be shown that νcrit,s= 2s−1/2.) Therefore, except for the case of pure imaginary ν, the number of zeros of Kν(z) in the principal Riemann sheet is finite: only those zeros with label s= 1,2, . . .such that |νcrit,s|<|ν|, for the corresponding value of arg ν, remain in that principal Riemann sheet. If|ν|decreases again, being arg ν/negationslash= 0, the zeros approach the origin spiraling clockwise infinitely, according to Eqs. (11) and (12). The ca se of real νdeserves a particular consideration. 9 -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz -10-8-6-4-2024 -10 -5 0 5 1015105 RezImz Fig. 3. Trajectories of the third zero, z3, ofKν(z). The lines represent the behaviour ofz3with the same convention as in Fig. 1. The trajectory for real νgoes to (+∞,5π/4), with arg z3→ −2π, forν= 5/2. 5 Zeros of Kν(z)for real ν. Let us now concentrate on the particular case of νbeing real. According to the second of Eqs. (1), the zeros of Kν(z) appear in complex conjugate pairs. We refer, in what follows, only to those zeros of negative arg ument. Let us also recall that, for half-integer ν,ν=n+1 2, the Macdonald function reduces to a factor, different from zero in the finite plane, times a polyn omial of degree n. Therefore, Kν(z) presents exactly n z-zeros in each one of the Riemann sheets. Of course, the locations of these zeros are the same i n all sheets. The aproximate expression (18) for the first zeros of Kν(z) (ν≫1) is valid also in this case. Equation (21), instead, is not applicable beca use the assumption |z| ≪ |ν|, used in its derivation, is not justified. Also, Eqs. (11) and (12) are no more valid due to the possibility of |z|becoming infinite for certain values ofν, a fact that invalidates Eq. (5), from which Eqs. (11) and (12 ) stem. According to Eq. (2), the z-zeros of Kν(z) are obtained from the z-zeros of H(1) ν(z) by a rotation of −π/2 in the complex z-plane. It is then immediate to deduce the behaviour of the zeros of Kν(z) for varying real νfrom a previous discussion [5,6] of the zeros of H(1) ν(z). For convenience of the reader, we recall 10 Table 1 Typical output of the Newton method applied to the location o f the first, second and third zeros of Kν(z), with |ν|= 21, arg ν= 7π/20. Besides the successive approximations to the zeros, the values of Eν(z), given by Eq. (22), are shown. ℜz1 ℑz1 ℜEν(z1) ℑEν(z1) 14.00000000 −8.60000000 −.127330E+01 −.817656E −01 14.02983390 −8.68222831 .154771E+00 .778107E −01 14.02406924 −8.67520518 .105223E −01 .979004E −03 14.02389450 −8.67463545 −.594713E −04 .105606E −04 14.02389461 −8.67463884 .429746E −07 −.440585E −08 ℜz2 ℑz2 ℜEν(z2) ℑEν(z2) 10.90000000 −8.00000000 −.151232E+01 −.835798E+00 10.98486986 −7.94986297 −.232404E+00 .536993E −01 11.00073780 −7.95926858 .207686E −01 −.296534E −01 10.99984823 −7.95696547 .191900E −03 −.214639E −03 10.99983888 −7.95694790 −.278481E −06 .649298E −06 10.99983889 −7.95694795 .698624E −11 −.156303E −10 ℜz3 ℑz3 ℜEν(z3) ℑEν(z3) 8.50000000 −7.00000000 .741241E+00 .345969E+01 8.60000000 −6.92107409 .978987E+00 .293843E+01 8.70000000 −7.02107409 .658986E+00 .247911E+01 8.80000000 −7.12107409 .598150E+00 .174818E+01 8.88169635 −7.22107409 .831925E+00 .871598E+00 8.80046094 −7.31734050 −.282066E+00 .243279E+00 8.82858965 −7.32026067 .229670E −01 .745812E −01 8.82891727 −7.32660986 .257068E −04 −.695341E −03 8.82889674 −7.32655881 −.673649E −06 −.726223E −05 8.82889659 −7.32655825 .622240E −10 .575485E −09 here, without demonstration, the main results of that discu ssion translated to those zeros in the lower half plane ℑz <0. To be specific, let us focus on the trajectory followed by a zero, zs, asνdecreases from + ∞to 0. For very large ν,zsis given approximately by Eq. (18). It lies far from the origin in the quadrant −π <argz <−π/2. As νdecreases, it moves upwards 11 and to the right, reaching the horizontal semiaxis arg z=−πforν= 2s− 1/2. Then, it makes a nearly circular quarter of a turn, crosses the vertical semiaxis arg z=−3π/2 forν=s−1/3, and goes nearly horizontally towards (+∞,(2s−1)π/4) asν→s−1/2 from above. The behaviour of zsforν < s−1/2 is not uniquely defined. In fact, the implicit function zs(ν) defined by the condition Kν(z) = 0 presents a logarithmic branch point at ν=s−1/2, and the values of zsforν < s−1/2 depend on the chosen branch. One can pass continuously from the valu es ofzsfor ν > s−1/2 to those for ν < s−1/2 if one avoids the branch point by the usual procedure of adding a small imaginary part to νas its real part crosses the value s−1/2. But the resulting values of zsforν < s−1/2 depend on the sign of that imaginary part. Since this ambiguity in the v alues of zshas been thoroughly discussed in Ref. [6], in the context of the z-zeros of H(1) ν(z), we do not consider necessary to pursue further. Besides those described above, Kν(z) presents an infinite set of zeros near and at the left of the vertical semiaxis arg z=−3π/2 for every integer ν= n[33, p. 513]. As νincreases, they approach that semiaxis, cross it when ν=n+ 1/3, and move nearly horizontally to the right, going to infinit y as ν→n+ 1/2, which is a logarithmic branch point for the position of the zeros as a function of ν, as it was shown in Refs. [5,6] for the zeros of H(1) ν(z). 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