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Euler transformation

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1: 24.13 Integrals
§24.13(iii) Compendia
2: 3.9 Acceleration of Convergence
§3.9(ii) Euler’s Transformation of Series
Euler’s transformation is usually applied to alternating series. …
3: 2.11 Remainder Terms; Stokes Phenomenon
The transformations in §3.9 for summing slowly convergent series can also be very effective when applied to divergent asymptotic series. A simple example is provided by Euler’s transformation3.9(ii)) applied to the asymptotic expansion for the exponential integral (§6.12(i)): … Further improvements in accuracy can be realized by making a second application of the Euler transformation; see Olver (1997b, pp. 540–543). …
4: 2.5 Mellin Transform Methods
Since e t ( z ) = Γ ( z ) , by the Parseval formula (2.5.5), there are real numbers p 1 and p 2 such that c < p 1 < 1 , p 2 < min ( 1 , β 0 ) , and
2.5.40 I j ( x ) = 1 2 π i p j i p j + i x z Γ ( 1 z ) h j ( z ) d z , j = 1 , 2 .
2.5.41 I 1 ( x ) = h 1 ( 1 ) x 1 + 1 2 π i ρ i ρ + i x z Γ ( 1 z ) h 1 ( z ) d z ,
2.5.42 I 2 ( x ) = β 0 z 1 res [ x z Γ ( 1 z ) h 2 ( z ) ] + 1 2 π i ρ i ρ + i x z Γ ( 1 z ) h 2 ( z ) d z .
2.5.47 res z = 1 [ ζ z 1 Γ ( 1 z ) h 2 ( z ) ] = ( ln ζ γ ) h 1 ( 1 ) ,
5: 15.14 Integrals
15.14.1 0 x s 1 𝐅 ( a , b c ; x ) d x = Γ ( s ) Γ ( a s ) Γ ( b s ) Γ ( a ) Γ ( b ) Γ ( c s ) , min ( a , b ) > s > 0 .
6: 11.7 Integrals and Sums
11.7.10 0 t ν 1 𝐇 ν ( t ) d t = π 2 ν + 1 Γ ( ν + 1 ) , ν > 3 2 ,
11.7.11 0 t μ ν 1 𝐇 ν ( t ) d t = Γ ( 1 2 μ ) 2 μ ν 1 tan ( 1 2 π μ ) Γ ( ν 1 2 μ + 1 ) , | μ | < 1 , ν > μ 3 2 ,
11.7.12 0 t μ ν 𝐇 μ ( t ) 𝐇 ν ( t ) d t = π Γ ( μ + ν ) 2 μ + ν Γ ( μ + ν + 1 2 ) Γ ( μ + 1 2 ) Γ ( ν + 1 2 ) , ( μ + ν ) > 0 .
7: 13.23 Integrals
13.23.1 0 e z t t ν 1 M κ , μ ( t ) d t = Γ ( μ + ν + 1 2 ) ( z + 1 2 ) μ + ν + 1 2 F 1 2 ( 1 2 + μ κ , 1 2 + μ + ν 1 + 2 μ ; 1 z + 1 2 ) , μ + ν + 1 2 > 0 , z > 1 2 .
13.23.2 0 e z t t μ 1 2 M κ , μ ( t ) d t = Γ ( 2 μ + 1 ) ( z + 1 2 ) κ μ 1 2 ( z 1 2 ) κ μ 1 2 , μ > 1 2 , z > 1 2 ,
13.23.3 1 Γ ( 1 + 2 μ ) 0 e 1 2 t t ν 1 M κ , μ ( t ) d t = Γ ( μ + ν + 1 2 ) Γ ( κ ν ) Γ ( 1 2 + μ + κ ) Γ ( 1 2 + μ ν ) , 1 2 μ < ν < κ .
13.23.5 0 e 1 2 t t ν 1 W κ , μ ( t ) d t = Γ ( 1 2 + μ + ν ) Γ ( 1 2 μ + ν ) Γ ( κ ν ) Γ ( 1 2 + μ κ ) Γ ( 1 2 μ κ ) , | μ | 1 2 < ν < κ .
13.23.6 1 Γ ( 1 + 2 μ ) 2 π i ( 0 + ) e z t + 1 2 t 1 t κ M κ , μ ( t 1 ) d t = z κ 1 2 Γ ( 1 2 + μ κ ) I 2 μ ( 2 z ) , z > 0 .
8: Bibliography M
  • A. R. Miller and R. B. Paris (2011) Euler-type transformations for the generalized hypergeometric function F r + 1 r + 2 ( x ) . Z. Angew. Math. Phys. 62 (1), pp. 31–45.
  • 9: 13.4 Integral Representations
    13.4.16 𝐌 ( a , b , z ) = 1 2 π i Γ ( a ) i i Γ ( a + t ) Γ ( t ) Γ ( b + t ) z t d t , | ph z | < 1 2 π ,
    13.4.17 U ( a , b , z ) = z a 2 π i i i Γ ( a + t ) Γ ( 1 + a b + t ) Γ ( t ) Γ ( a ) Γ ( 1 + a b ) z t d t , | ph z | < 3 2 π ,
    13.4.18 U ( a , b , z ) = z 1 b e z 2 π i i i Γ ( b 1 + t ) Γ ( t ) Γ ( a + t ) z t d t , | ph z | < 1 2 π ,
    10: 13.16 Integral Representations
    13.16.10 1 Γ ( 1 + 2 μ ) M κ , μ ( e ± π i z ) = e 1 2 z ± ( 1 2 + μ ) π i 2 π i Γ ( 1 2 + μ κ ) i i Γ ( t κ ) Γ ( 1 2 + μ t ) Γ ( 1 2 + μ + t ) z t d t , | ph z | < 1 2 π ,
    13.16.11 W κ , μ ( z ) = e 1 2 z 2 π i i i Γ ( 1 2 + μ + t ) Γ ( 1 2 μ + t ) Γ ( κ t ) Γ ( 1 2 + μ κ ) Γ ( 1 2 μ κ ) z t d t , | ph z | < 3 2 π ,
    13.16.12 W κ , μ ( z ) = e 1 2 z 2 π i i i Γ ( 1 2 + μ + t ) Γ ( 1 2 μ + t ) Γ ( 1 κ + t ) z t d t , | ph z | < 1 2 π ,