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11: 12.9 Asymptotic Expansions for Large Variable
12.9.1 U ( a , z ) e 1 4 z 2 z a 1 2 s = 0 ( 1 ) s ( 1 2 + a ) 2 s s ! ( 2 z 2 ) s , | ph z | 3 4 π δ ( < 3 4 π ) ,
12.9.2 V ( a , z ) 2 π e 1 4 z 2 z a 1 2 s = 0 ( 1 2 a ) 2 s s ! ( 2 z 2 ) s , | ph z | 1 4 π δ ( < 1 4 π ) .
12.9.3 U ( a , z ) e 1 4 z 2 z a 1 2 s = 0 ( 1 ) s ( 1 2 + a ) 2 s s ! ( 2 z 2 ) s ± i 2 π Γ ( 1 2 + a ) e i π a e 1 4 z 2 z a 1 2 s = 0 ( 1 2 a ) 2 s s ! ( 2 z 2 ) s , 1 4 π + δ ± ph z 5 4 π δ ,
12.9.4 V ( a , z ) 2 π e 1 4 z 2 z a 1 2 s = 0 ( 1 2 a ) 2 s s ! ( 2 z 2 ) s ± i Γ ( 1 2 a ) e 1 4 z 2 z a 1 2 s = 0 ( 1 ) s ( 1 2 + a ) 2 s s ! ( 2 z 2 ) s , 1 4 π + δ ± ph z 3 4 π δ .
12: 8.20 Asymptotic Expansions of E p ( z )
13: 13.7 Asymptotic Expansions for Large Argument
13.7.2 𝐌 ( a , b , z ) e z z a b Γ ( a ) s = 0 ( 1 a ) s ( b a ) s s ! z s + e ± π i a z a Γ ( b a ) s = 0 ( a ) s ( a b + 1 ) s s ! ( z ) s , 1 2 π + δ ± ph z 3 2 π δ ,
14: 8.21 Generalized Sine and Cosine Integrals
When ph z = 0 (and when a 1 , 3 , 5 , , in the case of Si ( a , z ) , or a 0 , 2 , 4 , , in the case of Ci ( a , z ) ) the principal values of si ( a , z ) , ci ( a , z ) , Si ( a , z ) , and Ci ( a , z ) are defined by (8.21.1) and (8.21.2) with the incomplete gamma functions assuming their principal values (§8.2(i)). Elsewhere in the sector | ph z | π the principal values are defined by analytic continuation from ph z = 0 ; compare §4.2(i). … When | ph z | < π and a < 1 , … When | ph z | < 1 2 π , … When z with | ph z | π δ ( < π ), …
15: 11.11 Asymptotic Expansions of Anger–Weber Functions
11.11.11 𝐀 ν ( λ ν ) ( 2 π ν ) 1 / 2 e ν μ k = 0 ( 1 2 ) k b k ( λ ) ν k , ν , | ph ν | π 2 δ ,
16: 13.19 Asymptotic Expansions for Large Argument
13.19.2 M κ , μ ( z ) Γ ( 1 + 2 μ ) Γ ( 1 2 + μ κ ) e 1 2 z z κ s = 0 ( 1 2 μ + κ ) s ( 1 2 + μ + κ ) s s ! z s + Γ ( 1 + 2 μ ) Γ ( 1 2 + μ + κ ) e 1 2 z ± ( 1 2 + μ κ ) π i z κ s = 0 ( 1 2 + μ κ ) s ( 1 2 μ κ ) s s ! ( z ) s , 1 2 π + δ ± ph z 3 2 π δ ,
13.19.3 W κ , μ ( z ) e 1 2 z z κ s = 0 ( 1 2 + μ κ ) s ( 1 2 μ κ ) s s ! ( z ) s , | ph z | 3 2 π δ .
17: 15.8 Transformations of Variable
15.8.8 𝐅 ( a , a + m c ; z ) = ( z ) a Γ ( a + m ) k = 0 m 1 ( a ) k ( m k 1 ) ! k ! Γ ( c a k ) z k + ( z ) a Γ ( a ) k = 0 ( a + m ) k k ! ( k + m ) ! Γ ( c a k m ) ( 1 ) k z k m ( ln ( z ) + ψ ( k + 1 ) + ψ ( k + m + 1 ) ψ ( a + k + m ) ψ ( c a k m ) ) , | z | > 1 , | ph ( z ) | < π ,
15.8.9 𝐅 ( a , a + m c ; z ) = ( 1 z ) a Γ ( a + m ) Γ ( c a ) k = 0 m 1 ( a ) k ( c a m ) k ( m k 1 ) ! k ! ( z 1 ) k + ( 1 ) m ( 1 z ) a m Γ ( a ) Γ ( c a m ) k = 0 ( a + m ) k ( c a ) k k ! ( k + m ) ! ( 1 z ) k ( ln ( 1 z ) + ψ ( k + 1 ) + ψ ( k + m + 1 ) ψ ( a + k + m ) ψ ( c a + k ) ) , | z 1 | > 1 , | ph ( 1 z ) | < π .
15.8.10 𝐅 ( a , b a + b + m ; z ) = 1 Γ ( a + m ) Γ ( b + m ) k = 0 m 1 ( a ) k ( b ) k ( m k 1 ) ! k ! ( z 1 ) k ( z 1 ) m Γ ( a ) Γ ( b ) k = 0 ( a + m ) k ( b + m ) k k ! ( k + m ) ! ( 1 z ) k ( ln ( 1 z ) ψ ( k + 1 ) ψ ( k + m + 1 ) + ψ ( a + k + m ) + ψ ( b + k + m ) ) , | z 1 | < 1 , | ph ( 1 z ) | < π ,
15.8.11 𝐅 ( a , b a + b + m ; z ) = z a Γ ( a + m ) k = 0 m 1 ( a ) k ( m k 1 ) ! k ! Γ ( b + m k ) ( 1 1 z ) k z a Γ ( a ) k = 0 ( a + m ) k k ! ( k + m ) ! Γ ( b k ) ( 1 ) k ( 1 1 z ) k + m ( ln ( 1 z z ) ψ ( k + 1 ) ψ ( k + m + 1 ) + ψ ( a + k + m ) + ψ ( b k ) ) , z > 1 2 , | ph z | < π , | ph ( 1 z ) | < π .
18: 5.11 Asymptotic Expansions
As z in the sector | ph z | π δ , … If z is complex, then the remainder terms are bounded in magnitude by sec 2 n ( 1 2 ph z ) for (5.11.1), and sec 2 n + 1 ( 1 2 ph z ) for (5.11.2), times the first neglected terms. … For this result and a similar bound for the sector 1 2 π ph z π see Boyd (1994). … If z in the sector | ph z | π δ , then … Lastly, and again if z in the sector | ph z | π δ , then …
19: 2.11 Remainder Terms; Stokes Phenomenon
In both the modulus and phase of the asymptotic variable z need to be taken into account. …Then numerical accuracy will disintegrate as the boundary rays ph z = α , ph z = β are approached. … In effect, (2.11.7) “corrects” (2.11.6) by introducing a term that is relatively exponentially small in the neighborhood of ph z = π , is increasingly significant as ph z passes from π to 3 2 π , and becomes the dominant contribution after ph z passes 3 2 π . … uniformly with respect to ph z in each case. The relevant Stokes lines are ph z = ± π for w 1 ( z ) , and ph z = 0 , 2 π for w 2 ( z ) . …
20: 4.45 Methods of Computation
For Shift-and-Add and CORDIC algorithms, see Muller (1997), Merrheim (1994), Schelin (1983). …
4.45.15 ln z = ln | z | + i ph z , π ph z π ,
See §1.9(i) for the precise relationship of ph z to the arctangent function. …