About the Project
10 Bessel FunctionsKelvin Functions

§10.68 Modulus and Phase Functions


§10.68(i) Definitions

10.68.1 Mν(x)eiθν(x)=berνx+ibeiνx,
10.68.2 Nν(x)eiϕν(x)=kerνx+ikeiνx,

where Mν(x)(>0), Nν(x)(>0), θν(x), and ϕν(x) are continuous real functions of x and ν, with the branches of θν(x) and ϕν(x) chosen to satisfy (10.68.18) and (10.68.21) as x. (See also §10.68(iv).)

§10.68(ii) Basic Properties

10.68.3 berνx =Mν(x)cosθν(x),
beiνx =Mν(x)sinθν(x),
10.68.4 kerνx =Nν(x)cosϕν(x),
keiνx =Nν(x)sinϕν(x).
10.68.5 Mν(x) =(berν2x+beiν2x)1/2,
Nν(x) =(kerν2x+keiν2x)1/2,
10.68.6 θν(x) =Arctan(beiνx/berνx),
ϕν(x) =Arctan(keiνx/kerνx).
10.68.7 M-n(x) =Mn(x),
θ-n(x) =θn(x)-nπ.

With arguments (x) suppressed,

10.68.8 berνx=12Mν+1cos(θν+1-14π)-12Mν-1cos(θν-1-14π)=(ν/x)Mνcosθν+Mν+1cos(θν+1-14π)=-(ν/x)Mνcosθν-Mν-1cos(θν-1-14π),
10.68.9 beiνx=12Mν+1sin(θν+1-14π)-12Mν-1sin(θν-1-14π)=(ν/x)Mνsinθν+Mν+1sin(θν+1-14π)=-(ν/x)Mνsinθν-Mν-1sin(θν-1-14π).
10.68.10 berx =M1cos(θ1-14π),
beix =M1sin(θ1-14π).
10.68.11 Mν=(ν/x)Mν+Mν+1cos(θν+1-θν-14π)=-(ν/x)Mν-Mν-1cos(θν-1-θν-14π),
10.68.12 θν=(Mν+1/Mν)sin(θν+1-θν-14π)=-(Mν-1/Mν)sin(θν-1-θν-14π).
10.68.13 M0 =M1cos(θ1-θ0-14π),
θ0 =(M1/M0)sin(θ1-θ0-14π).
10.68.14 d(xMν2θν)/dx =xMν2,
x2Mν′′+xMν-ν2Mν =x2Mνθν2.

Equations (10.68.8)–(10.68.14) also hold with the symbols ber, bei, M, and θ replaced throughout by ker, kei, N, and ϕ, respectively. In place of (10.68.7),

10.68.15 N-ν(x) =Nν(x),
ϕ-ν(x) =ϕν(x)+νπ.

§10.68(iii) Asymptotic Expansions for Large Argument

When ν is fixed, μ=4ν2, and x

10.68.16 Mν(x) =ex/2(2πx)12(1-μ-1821x+(μ-1)22561x2-(μ-1)(μ2+14μ-399)614421x3+O(1x4)),
10.68.17 lnMν(x) =x2-12ln(2πx)-μ-1821x-(μ-1)(μ-25)38421x3-(μ-1)(μ-13)1281x4+O(1x5),
10.68.18 θν(x) =x2+(12ν-18)π+μ-1821x+μ-1161x2-(μ-1)(μ-25)38421x3+O(1x5).
10.68.19 Nν(x) =e-x/2(π2x)12(1+μ-1821x+(μ-1)22561x2+(μ-1)(μ2+14μ-399)614421x3+O(1x4)),
10.68.20 lnNν(x) =-x2+12ln(π2x)+μ-1821x+(μ-1)(μ-25)38421x3-(μ-1)(μ-13)1281x4+O(1x5),
10.68.21 ϕν(x) =-x2-(12ν+18)π-μ-1821x+μ-1161x2+(μ-1)(μ-25)38421x3+O(1x5).

§10.68(iv) Further Properties

Additional properties of the modulus and phase functions are given in Young and Kirk (1964, pp. xi–xv). However, care needs to be exercised with the branches of the phases. Thus this reference gives ϕ1(0)=54π (Eq. (6.10)), and limx(ϕ1(x)+(x/2))=-58π (Eqs. (10.20) and (Eqs. (10.26b)). However, numerical tabulations show that if the second of these equations applies and ϕ1(x) is continuous, then ϕ1(0)=-34π; compare Abramowitz and Stegun (1964, p. 433).