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11: 8.7 Series Expansions
For an expansion for γ ( a , i x ) in series of Bessel functions J n ( x ) that converges rapidly when a > 0 and x ( 0 ) is small or moderate in magnitude see Barakat (1961).
12: 14.1 Special Notation
x , y , τ real variables.
δ arbitrary small positive constant.
The main functions treated in this chapter are the Legendre functions 𝖯 ν ( x ) , 𝖰 ν ( x ) , P ν ( z ) , Q ν ( z ) ; Ferrers functions 𝖯 ν μ ( x ) , 𝖰 ν μ ( x ) (also known as the Legendre functions on the cut); associated Legendre functions P ν μ ( z ) , Q ν μ ( z ) , 𝑸 ν μ ( z ) ; conical functions 𝖯 1 2 + i τ μ ( x ) , 𝖰 1 2 + i τ μ ( x ) , 𝖰 ^ 1 2 + i τ μ ( x ) , P 1 2 + i τ μ ( x ) , Q 1 2 + i τ μ ( x ) (also known as Mehler functions). … Among other notations commonly used in the literature Erdélyi et al. (1953a) and Olver (1997b) denote 𝖯 ν μ ( x ) and 𝖰 ν μ ( x ) by P ν μ ( x ) and Q ν μ ( x ) , respectively. Magnus et al. (1966) denotes 𝖯 ν μ ( x ) , 𝖰 ν μ ( x ) , P ν μ ( z ) , and Q ν μ ( z ) by P ν μ ( x ) , Q ν μ ( x ) , 𝔓 ν μ ( z ) , and 𝔔 ν μ ( z ) , respectively. Hobson (1931) denotes both 𝖯 ν μ ( x ) and P ν μ ( x ) by P ν μ ( x ) ; similarly for 𝖰 ν μ ( x ) and Q ν μ ( x ) .
13: 2.5 Mellin Transform Methods
The sum in (2.5.6) is taken over all poles of x z f ( 1 z ) h ( z ) in the strip d < z < c , and it provides the asymptotic expansion of I ( x ) for small values of x . …
2.5.43 h ( ζ ) = h 1 ( 1 ) + β 0 z 1 res [ ζ z 1 Γ ( 1 z ) h 2 ( z ) ] + 1 < z < l res [ ζ z 1 Γ ( 1 z ) h ( z ) ] + 1 2 π i l δ i l δ + i ζ z 1 Γ ( 1 z ) h ( z ) d z ,
14: 6.18 Methods of Computation
For small or moderate values of x and | z | , the expansion in power series (§6.6) or in series of spherical Bessel functions (§6.10(ii)) can be used. …
15: 10.74 Methods of Computation
The power-series expansions given in §§10.2 and 10.8, together with the connection formulas of §10.4, can be used to compute the Bessel and Hankel functions when the argument x or z is sufficiently small in absolute value. …
16: 18.25 Wilson Class: Definitions
18.25.9 y = 0 N p n ( y ( y + γ + δ + 1 ) ) p m ( y ( y + γ + δ + 1 ) ) γ + δ + 1 + 2 y γ + δ + 1 + y ω y = h n δ n , m .
17: 3.2 Linear Algebra
The sensitivity of the solution vector 𝐱 in (3.2.1) to small perturbations in the matrix 𝐀 and the vector 𝐛 is measured by the condition number
18: 19.36 Methods of Computation
Similarly, §19.26(ii) eases the computation of functions such as R F ( x , y , z ) when x ( > 0 ) is small compared with min ( y , z ) . …
19: 36.12 Uniform Approximation of Integrals
For example, the diffraction catastrophe Ψ 2 ( x , y ; k ) defined by (36.2.10), and corresponding to the Pearcey integral (36.2.14), can be approximated by the Airy function Ψ 1 ( ξ ( x , y ; k ) ) when k is large, provided that x and y are not small. …
20: 13.1 Special Notation
m integer.
x , y real variables.
δ arbitrary small positive constant.
Γ ( x ) gamma function (§5.2(i)).
ψ ( x ) Γ ( x ) / Γ ( x ) .