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35 Functions of Matrix ArgumentProperties

§35.5 Bessel Functions of Matrix Argument

  1. §35.5(i) Definitions
  2. §35.5(ii) Properties
  3. §35.5(iii) Asymptotic Approximations

§35.5(i) Definitions

35.5.1 Aν(𝟎)=1Γm(ν+12(m+1)),
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+𝐗1))|𝐗|ν12(m+1)d𝐗,
ν, 𝐓𝛀.

§35.5(ii) Properties

35.5.5 𝟎<𝐗<𝐓Aν1(𝐒1𝐗)|𝐗|ν1Aν2(𝐒2(𝐓𝐗))|𝐓𝐗|ν2d𝐗=|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|νBν(𝐓),
ν, 𝐓𝛀.
35.5.7 𝛀Aν1(𝐓𝐗)Bν2(𝐒𝐗)|𝐗|ν1d𝐗=1Aν1+ν2(𝟎)|𝐒|ν2|𝐓+𝐒|(ν1+ν2+12(m+1)),
(ν1+ν2)>1; 𝐒,𝐓𝛀.
35.5.8 𝐎(m)etr(𝐒𝐇)d𝐇=A1/2(14𝐒𝐒T)A1/2(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).