Copyright @ The Author, 2011. The publisher version of the article can be accessed at the link below.In this work we describe a simple method for finding approximate representations for special functions which are entire transcendental functions that can be represented by infinite products. This method replaces the infinite product by a finite polynomial and Gamma functions. This approximate representation is shown in the case of Bessel functions to be very accurate over a large range of parameter values. These approximate expressions can be useful for finding the roots of a transcendental equation and the Wiener-Hopf factorization of functions involving such Bessel functions.The method is shown to be potentially useful for other transcendental andWiener-Hopf problems, which involve other entire functions that have infinite product representations

Rawlins, AD

IMA Journal of Numerical Analysis

English

Brunel University Research Archive

IThe method of finite-product extraction and an application toWiener-Hopf theoryANTHONY D. RAWLINSDepartment of Mathematical Science, Brunel University ,                   Kingston Lane, Uxbridge, UB8 3PH, UK. In this work we describe a simple method for f nding approximate representations for special functionswhich are entire transcendental functions that can be represented by inf nite products. This method re-places the inf nite product by a f nite polynomial and Gamma functions. This approximate representationis shown in the case of Bessel functions to be very accurate over a large range of parameter values. Theseapproximate expressions can be useful for f nding the roots of a transcendental equation and the Wiener-Hopf factorization of functions involving such Bessel functions.The method is shown to be potentiallyuseful for other transcendental and Wiener-Hopf problems, which involve other entire functions that haveinf nite product representations.Keywords: Transcendental equations,roots,zeros,Bessel Functions,polynomial approximations, Wiener-Hopf factorization.1. IntroductionThe inspiration for this work arose from reading the article (Chapman (2010)) in which the authorsfound that by replacing trigonometric functions by a f nite product polynomial and a Gamma functionexpression they were able to get very good approximations for the roots of eigenvalue equations. Thereplacement was exact and followed from the well known identities between the Gamma function andtrigonometric functions. We propose here to replace any transcendental function that is an entire func-tion and that can be represented by an inf nite product by an approximate expression that involves apolynomial and a function of Gamma functions. Initially to describe this extension of the method off nite-product extraction we apply the method to derive an approximation for the Bessel function Jν(z).We use this approximation to f nd the roots of a transcendental equation that involves Bessel functionsand check the numerical accuracy of the approximation. In many physical applications one often needsto consider the roots of Bessel function equations of the formβ Jν(z)+ zαJ′µ(z) = 0, (1.1)where Jν(z)(Jµ(z)) is the Bessel function of the f rst kind of order ν(µ) and β , and α are parame-ters possibly complex. This is a fairly general equation that arises in numerous applications involvingcylindrical boundary value problems. For example special cases of this equation arise as an eigenvalueequations in acoustically lined ducts:β Jν(z)+ zJ′ν(z) = 0, (1.2)see Rawlins (2007) and references therein. An equation of the form :β Jν(z)+ J′ν(z) = 0. (1.3)                                                        © Institute of Mathematics and its Applications 2005; all rights reserved.2 of 12 A. D. Rawlinsarises for the vibration of a circular membrane with elastic boundary support Chen & Zhou (1993) andfor tsunami wave generation, see references in Synolakis (1988) We also show how we can carry out anapproximate Wiener-Hopf ( Lawrie & Abrahams (2007)) factorization for a Bessel function expression.We then outline a generalization of the method of f nite-Product extraction to a general class of entirefunction. These results will have potential applications for other transcendental equations and Wiener-Hopf factorization problems in the future.2. An approximation for Bessel functionsWe use the Bessel function representation given in Abramowitz (1964),(page 370, 9.5.10) and Watson(1944), (page 498)Jν(z) =( z2 )νΓ (ν + 1)∞∏n=1[1−(zjν,n)2]. (2.1)where jν,n is the n-th nonnegative zero of the Bessel function of order ν . We begin by splitting theinf nite product at the value n = m−1 givingJν(z) =( z2 )νΓ (ν + 1)m−1∏n=1[1−(zjν,n)2] ∞∏n=m[1−(zjν,n)2]. (2.2)Provided we make m large enough the roots in second inf nite product can be approximated by theirasymptotic form:jν,n = (n +ν2− 14)pi + O(1N), (2.3)as N = (n + ν2 − 14 )pi → ∞, see Abramowitz (1964),(page 371, 9.5.12) . This being so we can approxi-mate the Bessel function by the expression:Jν(z)⋍( z2 )νΓ (ν + 1)m−1∏n=1[1−(zjν,n)2] ∞∏n=m[1− z2((n + ν2 − 14)pi)2]. (2.4)which we can write asJν(z)⋍( z2 )νΓ (ν + 1)m−1∏n=1[1−(zjν,n)2] ∞∏k=1[1− (z/pi)2(k + m+ ν2 − 54 )2]. (2.5)By using an identity, that can be easily derived from the result Erdelyi (1953),(p5, 1.3(4)):∞∏k=1[1− ζ2(ak + b)2]=Γ 2(1 + b/a)Γ (1 + b+ζa )Γ (1 +b−ζa ). (2.6)we can write (2.6) as Jν(z)⋍ J˜ν(z) whereJ˜ν(z) =( z2 )νΓ (ν + 1)Γ 2(m+ ν2 − 14)Γ (m+ ν2 − 14 + zpi )Γ (m+ ν2 − 14 − zpi )m−1∏n=1[1−(zjν,n)2], (2.7)which can be regarded as a good approximation for (m+ ν2 − 14) ≫ 1. In an exactly analogous mannerfrom the results in Abramowitz (1964)(page 370, 9.5.11; and page 371, 9.15.13) it can be shown that forExtension of the method of Finite-product extraction 3 of 12ν > 0, and (m+ ν2 − 34 )≫ 1, then the derivative of the Bessel function of order ν can be approximatedby the expression J′ν(z)⋍ J˜′ν(z) where:J˜′ν (z) =( z2 )ν−12Γ (ν)Γ 2(m+ ν2 − 34 )Γ (m+ ν2 − 34 + zpi )Γ (m+ ν2 − 34 − zpi )m−1∏n=1[1−(zj′ν,n)2]. (2.8)As a demonstration of the accuracy of this approximation we display some typical graphs below. InFigures 1-7 the top plot is of Jν(z) and J˜ν(z) superimposed on each other over the range 0 < z < 10 forν = 0, and 0 < z < 100 for ν = 8. In the lower plots the difference Jν(z)− J˜ν (z) is plotted over the range0 < z < 100. The plots in Figure 1-3 are for ν = 0, and the plots in Figure 4-7 are for the larger valueof ν = 8.2 4 6 8 10-0.4-0.20.20.40.60.81.020 40 60 80 100-0.02-0.010.010.020.03FIG. 1. The error between approximate and exact expressions J0(z)− J˜0(z), for ν = 0,m = 1.4 of 12 A. D. Rawlins2 4 6 8 10-0.4-0.20.20.40.60.81.020 40 60 80 100-0.006-0.004-0.0020.0020.004FIG. 2. The error between approximate and exact expressions J0(z)− J˜0(z), for ν = 0,m = 3.2 4 6 8 10-0.4-0.20.20.40.60.81.020 40 60 80 100-0.0010-0.00050.00050.00100.0015FIG. 3. The error between approximate and exact expressions J0(z)− J˜0(z), for ν = 0,m = 5.Extension of the method of Finite-product extraction 5 of 1220 40 60 80 100-0.2-0.10.10.20.320 40 60 80 100-0.20-0.15-0.10-0.050.050.100.15FIG. 4. The error between approximate and exact expressions J8(z)− J˜8(z), for ν = 8,m = 5.20 40 60 80 100-0.2-0.10.10.20.320 40 60 80 100-0.050.050.100.15FIG. 5. The error between approximate and exact expressions J8(z)− J˜8(z), for ν = 8,m = 10.6 of 12 A. D. Rawlins20 40 60 80 100-0.2-0.10.10.20.320 40 60 80 100-0.10-0.050.05FIG. 6. The error between approximate and exact expressions J8(z)− J˜8(z), for ν = 8,m = 15.Extension of the method of Finite-product extraction 7 of 1220 40 60 80 100-0.2-0.10.10.20.320 40 60 80 100-0.04-0.020.02FIG. 7. The error between approximate and exact expressions J8(z)− J˜8(z), for ν = 8,m = 20.8 of 12 A. D. RawlinsFrom these graphs it can be seen that as we increase m, i.e increasing the degree of the polynomial,we improve the accuracy near z = 0. The large z behaviour is well approximated by the Gamma functionsfor large enough values of m. The approximation can be made good within prescribed bounds over theentire range of values of z by choosing a suitably large value of m. In practice this value of m issurprisingly small for small ν! The agreement is not so good for all z for larger values of ν , and this iscould be due to the fact that Bessel function approximations for large order are only good for z > ν2.3. Some numerical examples of the Bessel function approximationThese type of results were obtained for a wide range of complex values of the parameter β , for α = 0,1in equation (1). We f rst apply the approximations to the roots zn,n = 1,2,3..., of the transcendentalequation:J0(z)+ 2iJ1(z) = 0, (3.1)and its approximate version:J˜0(z)+ 2iJ˜1(z) = 0, (3.2)whose corresponding roots are denoted by z˜n,n = 1,2,3.... The same Newton’s method with Mathe-matica is used to calculate the roots of (3.1) and (3.2). The f rst three roots of the equation (3.1) withthe ordered magnitude (0 < |z1| < |z2| < |z3|,ℜ(zn) > 0) are given by z1 = 2.46633− 0.52998i,z2 =5.54951−0.545114i,z3 = 8.67279−0.547564i. For the approximate equation (3.2) we obtain the cor-responding approximate roots and error estimates:m 1 3 5z˜1 2.39773−0.536855i 2.46618−0.530725i 2.46631−0.530116iz1− z˜1 0.06860 + 0.00687i 0.00015 + 0.00074i 0.00002 + 0.00013im 1 3 5z˜1 5.50061−0.475500i 5.54866−0.550255i 5.54944−0.545877iz2− z˜2 0.03890−0.069600i 0.00085 + 0.005141i 0.00007 + 0.000763im 1 3 5z˜1 8.64009−0.466037i 8.64998−0.563053i 8.67259−0.549758iz3− z˜3 0.03270 + 0.081428i 0.02281 + 0.015489i 0.00020 + 0.002194iWe now apply the approximations to the roots zn,n = 1,2,3..., of the transcendental equation:zJ′0(z)+ 2iJ0(z) = 0, (3.3)and its approximate version:zJ˜′0(z)+ 2iJ˜0(z) = 0, (3.4)whose corresponding roots will be denoted by z˜n,n = 1,2,3.... For the f rst root of this equation, givenby z1 = 3.88737−0.566309i, we obtain the approximationsExtension of the method of Finite-product extraction 9 of 12m 1 3 5z˜1 1.67973−0.931529i 3.88757−0.564116i 3.88739−0.565936iz1− z˜1 2.20764 + 0.375220i −0.00020−0.002193i 0.00002−0.000373iResults of similar accuracy have been obtained for other parameter values of β in the equations (1.2)and (1.3), for example the equation studied in the works of Synolakis (1988) and Rawlins (1989).4. An example of an approximate Wiener-Hopf factorization for Bessel functionsThe Bessel function approximation also offers a simple way to give analytic representations for Wiener-Hopf factorizations of Bessel functions. For example suppose we require thatF(z) = F+(z)F−(z), (4.1)where F±(z) is analytic and non-zero in the respective half planes ℑz ≷ 0.Thus for example if we letF(z) = Jν(κ), where κ =√k2− z2 , with ℜκ > 0 in the cut z-plane, then from the result ( 2.7) byinspection we have the approximate factorizationF˜±(z) =(k± z)ν/2√2νΓ (ν + 1)m−1∏n=1[1± κjν,n] Γ (m+ ν2 − 14 )Γ (m+ ν2 − 14 ± κpi ). (4.2)5. Generalization for other entire transcendental functions with an Euler-Weierstrass productrepresentationIf F(z) is an integral (entire) function of z with simple zeros it can be represented in the formF(z) = F(0)ez F′(0)F(0)∞∏k=1(1− zzk)ezzk ,see Titchmarsh (1958), (p113). We write this in a more convenient alternative form where the zeroszp(−zn) are such that 0 6arg(zp) < pi ( −pi 6arg(−zn) < 0) and are ordered in an appropriate manner,with the asymptotic behaviour zn → an + b as n→ ∞ (zp → α p + β as p→ ∞). Then we can write thelast product representation asF(z) = F(0)ez F′(0)F(0)∞∏p=1(1− zzp)ezzp∞∏n=1(1 +zzn)e−zzn ,We now extract the f nite products and use the asymptotic form for the roots giving the approximationfor F(z)F˜(z) = F(0)ez F′(0)F(0)P−1∏p=1(1− zzp)ezzp∞∏p=P(1− zα p + β)ezα p+β×N−1∏n=1(1 +zzn)e−zzn∞∏n=N(1 +zan + b)e−zan+b (5.1)10 of 12 A. D. RawlinsorF˜(z) = F(0)ez F′(0)F(0)P−1∏p=1(1− zzp)ezzp∞∏s=0(1− zαs+(αP+ β ))ezαs+(αP+β)×N−1∏n=1(1 +zzn)e−zzn∞∏s=0(1 +zas+(aN + b))e− zas+(aN+b) . (5.2)We now need to eliminate the inf nite products. This is achieved by using properties of the related Psifunction and the Gamma function. Thus we note that sinceψ(x) = ddxlnΓ (x) = Γ′(x)Γ (x)=−C− 1x+∞∑n=1(1n− 1x + n),where C is Euler’s constant, theneyψ(x) = e−yC−yx∞∏n=1eyn− yx+n ,and sinceΓ (x)Γ (x + y)= eyC(1 +yx) ∞∏n=1(1 +yx + n)e−yn ,then eliminating the term involving the Euler constant C from the last two results we get Mellin’s result,Erdelyi (1953) (p6, 1.3(5))∞∏n=0(1 +yx + n)e−yx+n =Γ (x)Γ (x + y)eyψ(x).Now let y = z/a,x = b/a and we then get the identity∞∏n=0(1 +zan + b)e−zan+b =Γ ( ba )Γ ( z+ba )eza ψ( ba ). (5.3)We now use this last result to eliminate the inf nite products that appear in (5.2) giving the f nite productrepresentation for F(z) asF˜(z) = F(0)Γ ( ba )Γ (βα )Γ ( b+za )Γ (β−zα )ez[ F′(0)F(0) +ψ(N+b/a)/a−ψ(P+β/α)/α ]×P−1∏p=1(1− zzp)ezzpN−1∏n=1(1 +zzn)e−zzn . (5.4)This result simplif es considerably when F(z) is an even function of z, in which case it reduces toF˜(z) = F(0)(Γ ( ba ))2Γ ( b+za )Γ (b−za )N−1∏n=1(1−(zzn)2).The optimum choice of P and N in the above expressions will depend on the physical applications.For example in wave propagation in ducts P and N will be closely related to the number of propagatingExtension of the method of Finite-product extraction 11 of 12modes and the direction of propagation in the duct. Finally we can immediately carry out an approximateWiener-Hopf factorization by inspection of (5.4). Suppose we require thatF˜(z) = F˜+(z)F˜−(z), (5.5)where F˜±(z) is analytic and non-zero in the respective half planes ℑz≷ 0, then a factorization is givenbyF˜+(z) =√F(0)Γ ( ba )Γ ( b+za )ez2 [F′(0)F(0) +ψ(N+b/a)/a−ψ(P+β/α)/α ]+χ(z)×N−1∏n=1(1 +zzn)e−zzn .(5.6)F˜−(z) =√F(0)Γ ( βα )Γ (β−zα )ez2 [F′(0)F(0) +ψ(N+b/a)/a−ψ(P+β/α)/α ]−χ(z)×P−1∏p=1(1− zzp)ezzp .(5.7)where χ(z) is an arbitrary function that can be chosen to ensure that F˜+(z),F˜−(z) have suitable behaviouras |z| → ∞ in appropriate half planes of analyticity.6. ConclusionsWe have presented a new method of f nite-product extraction that offers scope for useful approximationswhich retains the analytic properties of the original function. As an example we have presented a simpleapproximation for Bessel functions that is very accurate along the real axis. It is a useful and accu-rate approximation for f nding complex solutions of general transcendental equations involving Besselfunctions. For example the method is valid mutatis-mutandis to the equation (1.1) with the derivativestroke removed. The approximation also offers simple Wiener-Hopf approximate factorizations thatconvey the analytic properties of the original function. Such factorizations in terms of Cauchy integralrepresentations would be non-trivial to numerically calculate because of convergence properties of theinf nite integral. The method used here has further potential applications in that it can be applied to anyspecial function that has an inf nite product representation whose asymptotic zeros for large argumentare known. It is hoped that in the future the present approximate Wiener-Hopf method will be used onexisting, and new applications, to show the accuracy and simplicity of the this method.REFERENCESABRAMOWITZ, M. & STEGUN,I. A. (1964) Handbook of Mathematical Functions. New York: Dover.CHAPMAN, C. J. & SOROKIN, S.V. (2010) The f nite-product method in the theory of waves and stability, Proc.R. Soc., A466 (219), 471–491.CHEN, G. & ZHOU, J. (1993) Vibration and damping in distributed systems: Vol. 2. New York: CRC Press.ERDELYI, A.,MAGNUS, M,OBERHETTINGER,F G. & TRICOMI,F. G. (1953) Higher transcendental functions.Vol. 1 New York: McGraw-Hill.12 of 12 A. D. RawlinsLAWRIE, JANE B. & ABRAHAMS,I.DAVID. (2007) A Brief historical perspective of the Wiener-Hopf Technique,J. Eng. Math. 59, 351–358.RAWLINS, A. D. (1989) Note on the roots of f (z) = J0(z)− iJ1(z)., Quart. Appl. Math., 47, 323–324.RAWLINS, A.D. (2007) Wave propagation in a bifurcated impedance-lined cylindrical waveguide, J. Eng. Math.,59, 419–435.SYNOLAKIS, C. M. (1988) On the roots of f (z) = J0(z)− iJ1(z)., Quart. Appl. Math., 46, 105–107.TITCHMARSH,E. C. (1958) The Theory of Functions. 2nd edition. Oxford: Oxford University Press.WATSON,G. N. (1944) The Theory of Bessel Functions. Cambridge: Cambridge University Press.

A Brief historical perspective of the Wiener-Hopf Technique,

Handbook of Mathematical Functions.

Higher transcendental functions.

Note on the roots of f(z)

On the roots of f(z)

The Theory of Bessel Functions. Cambridge:

The Theory of Functions. 2nd edition.

Vibration and damping in distributed systems:

Wave propagation in a bifurcated impedance-lined cylindrical waveguide,

The method of finite-product extraction and an application to Wiener-Hopf theory

A. D. Rawlins

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The method of finite-product extraction and an application to Wiener-Hopf theory

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