Digital Library of Mathematical Functions
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10 Bessel FunctionsBessel and Hankel Functions

§10.18 Modulus and Phase Functions

Contents

§10.18(i) Definitions

For ν0 and x>0

10.18.1 Mν(x)θν(x) =Hν(1)(x),
10.18.2 Nν(x)ϕν(x) =Hν(1)(x),

where Mν(x) (>0), Nν(x) (>0), θν(x), and ϕν(x) are continuous real functions of ν and x, with the branches of θν(x) and ϕν(x) fixed by

10.18.3 θν(x) -12π,
ϕν(x) 12π,
x0+.

§10.18(ii) Basic Properties

10.18.4 Jν(x) =Mν(x)cosθν(x),
Yν(x) =Mν(x)sinθν(x),
10.18.5 Jν(x) =Nν(x)cosϕν(x),
Yν(x) =Nν(x)sinϕν(x),
10.18.6 Mν(x) =(Jν2(x)+Yν2(x))12,
Nν(x) =(Jν2(x)+Yν2(x))12,
10.18.7 θν(x) =Arctan(Yν(x)/Jν(x)),
ϕν(x) =Arctan(Yν(x)/Jν(x)).
10.18.8 Mν2(x)θν(x) =2πx,
Nν2(x)ϕν(x) =2(x2-ν2)πx3,
10.18.9 Nν2(x)=Mν2(x)+Mν2(x)θν2(x)=Mν2(x)+4(πxMν(x))2,
10.18.10 (x2-ν2)Mν(x)Mν(x)+x2Nν(x)Nν(x)+xNν2(x)=0.
10.18.11 tan(ϕν(x)-θν(x))=Mν(x)θν(x)Mν(x)=2πxMν(x)Mν(x),
10.18.12 Mν(x)Nν(x)sin(ϕν(x)-θν(x))=2πx.
10.18.13 x2Mν′′(x)+xMν(x)+(x2-ν2)Mν(x)=4π2Mν3(x),
10.18.14 w′′+(1+14-ν2x2)w=4π2w3,
w=x12Mν(x),
10.18.15 x3w′′′+x(4x2+1-4ν2)w+(4ν2-1)w=0,
w=xMν2(x).
10.18.16 θν2(x)+12θν′′′(x)θν(x)-34(θν′′(x)θν(x))2=1-ν2-14x2.

§10.18(iii) Asymptotic Expansions for Large Argument

As x, with ν fixed and μ=4ν2,

10.18.17 Mν2(x) 2πx(1+12μ-1(2x)2+1324(μ-1)(μ-9)(2x)4+135246(μ-1)(μ-9)(μ-25)(2x)6+),
10.18.18 θν(x) x-(12ν+14)π+μ-12(4x)+(μ-1)(μ-25)6(4x)3+(μ-1)(μ2-114μ+1073)5(4x)5+(μ-1)(5μ3-1535μ2+54703μ-3 75733)14(4x)7+.

Also,

10.18.19 Nν2(x)2πx(1-12μ-3(2x)2-124(μ-1)(μ-45)(2x)4-),

the general term in this expansion being

10.18.20 -(2k-3)!!(2k)!!(μ-1)(μ-9)(μ-(2k-3)2)(μ-(2k+1)(2k-1)2)(2x)2k,
k2,

and

10.18.21 ϕν(x)x-(12ν-14)π+μ+32(4x)+μ2+46μ-636(4x)3+μ3+185μ2-2053μ+18995(4x)5+.

The remainder after k terms in (10.18.17) does not exceed the (k+1)th term in absolute value and is of the same sign, provided that k>ν-12.