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11: Bibliography T
  • N. M. Temme (1994c) Steepest descent paths for integrals defining the modified Bessel functions of imaginary order. Methods Appl. Anal. 1 (1), pp. 14–24.
  • 12: Bibliography P
  • A. Poquérusse and S. Alexiou (1999) Fast analytic formulas for the modified Bessel functions of imaginary order for spectral line broadening calculations. J. Quantit. Spec. and Rad. Trans. 62 (4), pp. 389–395.
  • 13: Bibliography B
  • C. B. Balogh (1967) Asymptotic expansions of the modified Bessel function of the third kind of imaginary order. SIAM J. Appl. Math. 15, pp. 1315–1323.
  • S. Bielski (2013) Orthogonality relations for the associated Legendre functions of imaginary order. Integral Transforms Spec. Funct. 24 (4), pp. 331–337.
  • A. R. Booker, A. Strömbergsson, and H. Then (2013) Bounds and algorithms for the K -Bessel function of imaginary order. LMS J. Comput. Math. 16, pp. 78–108.
  • 14: Software Index
    15: 36.11 Leading-Order Asymptotics
    36.11.2 Ψ K ( 𝐱 ) = 2 π j = 1 j max ( 𝐱 ) exp ( i ( Φ K ( t j ( 𝐱 ) ; 𝐱 ) + 1 4 π ( 1 ) j + K + 1 ) ) | 2 Φ K ( t j ( 𝐱 ) ; 𝐱 ) t 2 | 1 / 2 ( 1 + o ( 1 ) ) .
    36.11.3 Ψ 2 ( 0 , y ) = { π / y ( exp ( 1 4 i π ) + o ( 1 ) ) , y + , π / | y | exp ( 1 4 i π ) ( 1 + i 2 exp ( 1 4 i y 2 ) + o ( 1 ) ) , y .
    36.11.5 Ψ 3 ( 0 , y , 0 ) = Ψ 3 ( 0 , y , 0 ) ¯ = exp ( 1 4 i π ) π / y ( 1 ( i / 3 ) exp ( 3 2 i ( 2 y / 5 ) 5 / 3 ) + o ( 1 ) ) , y + .
    36.11.7 Ψ ( E ) ( 0 , 0 , z ) = π z ( i + 3 exp ( 4 27 i z 3 ) + o ( 1 ) ) , z ± ,
    36.11.8 Ψ ( H ) ( 0 , 0 , z ) = 2 π z ( 1 i 3 exp ( 1 27 i z 3 ) + o ( 1 ) ) , z ± .
    16: 14.17 Integrals
    Orthogonality relations for the associated Legendre functions of imaginary order are given in Bielski (2013). …
    17: 14.20 Conical (or Mehler) Functions
    14.20.4 𝒲 { 𝖯 1 2 + i τ μ ( x ) , 𝖯 1 2 + i τ μ ( x ) } = 2 | Γ ( μ + 1 2 + i τ ) | 2 ( 1 x 2 ) .
    14.20.6 P 1 2 + i τ μ ( x ) = i e μ π i sinh ( τ π ) | Γ ( μ + 1 2 + i τ ) | 2 ( Q 1 2 + i τ μ ( x ) Q 1 2 i τ μ ( x ) ) , τ 0 .
    14.20.7 𝖰 ^ 1 2 + i τ μ ( x ) 1 2 Γ ( μ ) ( 2 1 x ) μ / 2 ,
    14.20.12 g ( x ) = 0 P 1 2 + i τ μ ( x ) f ( τ ) d τ .
    For the case of purely imaginary order and argument see Dunster (2013). …
    18: 10.7 Limiting Forms
    10.7.6 Y i ν ( z ) = i csch ( ν π ) Γ ( 1 i ν ) ( 1 2 z ) i ν i coth ( ν π ) Γ ( 1 + i ν ) ( 1 2 z ) i ν + e | ν ph z | o ( 1 ) , ν and ν 0 .
    19: 14.24 Analytic Continuation
    14.24.1 P ν μ ( z e s π i ) = e s ν π i P ν μ ( z ) + 2 i sin ( ( ν + 1 2 ) s π ) e s π i / 2 cos ( ν π ) Γ ( μ ν ) 𝑸 ν μ ( z ) ,
    14.24.2 𝑸 ν μ ( z e s π i ) = ( 1 ) s e s ν π i 𝑸 ν μ ( z ) ,
    14.24.4 𝑸 ν , s μ ( z ) = e s μ π i 𝑸 ν μ ( z ) π i sin ( s μ π ) sin ( μ π ) Γ ( ν μ + 1 ) P ν μ ( z ) ,
    20: 14.23 Values on the Cut
    14.23.5 𝖰 ν μ ( x ) = 1 2 Γ ( ν + μ + 1 ) ( e μ π i / 2 𝑸 ν μ ( x + i 0 ) + e μ π i / 2 𝑸 ν μ ( x i 0 ) ) ,
    14.23.6 𝖰 ν μ ( x ) = e μ π i / 2 Γ ( ν + μ + 1 ) 𝑸 ν μ ( x ± i 0 ) ± 1 2 π i e ± μ π i / 2 P ν μ ( x ± i 0 ) .