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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: 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 .
13.23.6 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 .
13.23.7 1 2 π i - ( 0 + ) e z t + 1 2 t - 1 t κ W κ , μ ( t - 1 ) d t = 2 z - κ - 1 2 Γ ( 1 2 + μ - κ ) Γ ( 1 2 - μ - κ ) K 2 μ ( 2 z ) , z > 0 .
7: 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 .
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 ,