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10 Bessel FunctionsKelvin Functions

§10.68 Modulus and Phase Functions

  1. §10.68(i) Definitions
  2. §10.68(ii) Basic Properties
  3. §10.68(iii) Asymptotic Expansions for Large Argument
  4. §10.68(iv) Further Properties

§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 Mn(x) =Mn(x),
θn(x) =θn(x)nπ.

With arguments (x) suppressed,

10.68.8 berνx=12Mν+1cos(θν+114π)12Mν1cos(θν114π)=(ν/x)Mνcosθν+Mν+1cos(θν+114π)=(ν/x)MνcosθνMν1cos(θν114π),
10.68.9 beiνx=12Mν+1sin(θν+114π)12Mν1sin(θν114π)=(ν/x)Mνsinθν+Mν+1sin(θν+114π)=(ν/x)MνsinθνMν1sin(θν114π).
10.68.10 berx =M1cos(θ114π),
beix =M1sin(θ114π).
10.68.11 Mν=(ν/x)Mν+Mν+1cos(θν+1θν14π)=(ν/x)MνMν1cos(θν1θν14π),
10.68.12 θν=(Mν+1/Mν)sin(θν+1θν14π)=(Mν1/Mν)sin(θν1θν14π).
10.68.13 M0 =M1cos(θ1θ014π),
θ0 =(M1/M0)sin(θ1θ014π).
10.68.14 d(xMν2θν)/dx =xMν2,
x2Mν′′+xMνν2Mν =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 Nν(x) =Nν(x),
ϕν(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(1μ1821x+(μ1)22561x2(μ1)(μ2+14μ399)614421x3+O(1x4)),
10.68.17 lnMν(x) =x212ln(2πx)μ1821x(μ1)(μ25)38421x3(μ1)(μ13)1281x4+O(1x5),
10.68.18 θν(x) =x2+(12ν18)π+μ1821x+μ1161x2(μ1)(μ25)38421x3+O(1x5).
10.68.19 Nν(x) =ex/2(π2x)12(1+μ1821x+(μ1)22561x2+(μ1)(μ2+14μ399)614421x3+O(1x4)),
10.68.20 lnNν(x) =x2+12ln(π2x)+μ1821x+(μ1)(μ25)38421x3(μ1)(μ13)1281x4+O(1x5),
10.68.21 ϕν(x) =x2(12ν+18)πμ1821x+μ1161x2+(μ1)(μ25)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).