Digital Library of Mathematical Functions
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35 Functions of Matrix ArgumentProperties

§35.5 Bessel Functions of Matrix Argument

Contents

§35.5(i) Definitions

35.5.1 Aν(0)=1Γm(ν+12(m+1)),
ν.
35.5.2 Aν(T)=Aν(0)k=0(-1)kk!|κ|=k1[ν+12(m+1)]κZκ(T),
ν, T𝒮.
35.5.3 Bν(T)=Ωetr(-(TX+X-1))|X|ν-12(m+1)X,
ν, TΩ.

§35.5(ii) Properties

35.5.4 Ωetr(-TX)|X|νAν(SX)X=etr(-ST-1)|T|-ν-12(m+1),
S𝒮, TΩ; (ν)>-1.
35.5.5 0<X<TAν1(S1X)|X|ν1Aν2(S2(T-X))|T-X|ν2X=|T|ν1+ν2+12(m+1)Aν1+ν2+12(m+1)((S1+S2)T),
νj, (νj)>-1, j=1,2; S1,S2𝒮; TΩ.
35.5.6 Bν(T)=|T|-νB-ν(T),
ν, TΩ.
35.5.7 ΩAν1(TX)B-ν2(SX)|X|ν1X=1Aν1+ν2(0)|S|ν2|T+S|-(ν1+ν2+12(m+1)),
(ν1+ν2)>-1; S,TΩ.
35.5.8 O(m)etr(SH)H=A-1/2(-14SST)A-1/2(0),
S arbitrary.

§35.5(iii) Asymptotic Approximations

For asymptotic approximations for Bessel functions of matrix argument, see Herz (1955) and Butler and Wood (2003).