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Leibniz formula for derivatives

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11: 15.5 Derivatives and Contiguous Functions
§15.5 Derivatives and Contiguous Functions
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§15.5(i) Differentiation Formulas
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15.5.10 ( z ⁒ d d z ⁑ z ) n = z n ⁒ d n d z n ⁑ z n , n = 1 , 2 , 3 , .
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12: 25.4 Reflection Formulas
§25.4 Reflection Formulas
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25.4.1 ΞΆ ⁑ ( 1 s ) = 2 ⁒ ( 2 ⁒ Ο€ ) s ⁒ cos ⁑ ( 1 2 ⁒ Ο€ ⁒ s ) ⁒ Ξ“ ⁑ ( s ) ⁒ ΞΆ ⁑ ( s ) ,
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25.4.2 ΞΆ ⁑ ( s ) = 2 ⁒ ( 2 ⁒ Ο€ ) s 1 ⁒ sin ⁑ ( 1 2 ⁒ Ο€ ⁒ s ) ⁒ Ξ“ ⁑ ( 1 s ) ⁒ ΞΆ ⁑ ( 1 s ) .
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25.4.3 ξ ⁑ ( s ) = ξ ⁑ ( 1 s ) ,
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25.4.5 ( 1 ) k ⁒ ΞΆ ( k ) ⁑ ( 1 s ) = 2 ( 2 ⁒ Ο€ ) s ⁒ m = 0 k r = 0 m ( k m ) ⁒ ( m r ) ⁒ ( ⁑ ( c k m ) ⁒ cos ⁑ ( 1 2 ⁒ Ο€ ⁒ s ) + ⁑ ( c k m ) ⁒ sin ⁑ ( 1 2 ⁒ Ο€ ⁒ s ) ) ⁒ Ξ“ ( r ) ⁑ ( s ) ⁒ ΞΆ ( m r ) ⁑ ( s ) ,
13: 4.38 Inverse Hyperbolic Functions: Further Properties
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§4.38(ii) Derivatives
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4.38.9 d d z ⁑ arcsinh ⁑ z = ( 1 + z 2 ) 1 / 2 .
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4.38.11 d d z ⁑ arctanh ⁑ z = 1 1 z 2 .
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4.38.13 d d z ⁑ arcsech ⁑ z = 1 z ⁒ ( 1 z 2 ) 1 / 2 .
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§4.38(iii) Addition Formulas
14: 10.49 Explicit Formulas
§10.49 Explicit Formulas
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§10.49(i) Unmodified Functions
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§10.49(ii) Modified Functions
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§10.49(iii) Rayleigh’s Formulas
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§10.49(iv) Sums or Differences of Squares
15: 1.8 Fourier Series
β–Ίat every point at which f ⁑ ( x ) has both a left-hand derivative (that is, (1.4.4) applies when h 0 ) and a right-hand derivative (that is, (1.4.4) applies when h 0 + ). … … β–Ί
Parseval’s Formula
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Poisson’s Summation Formula
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1.8.16 n = e ( n + x ) 2 ⁒ Ο‰ = Ο€ Ο‰ ⁒ ( 1 + 2 ⁒ n = 1 e n 2 ⁒ Ο€ 2 / Ο‰ ⁒ cos ⁑ ( 2 ⁒ n ⁒ Ο€ ⁒ x ) ) , ⁑ Ο‰ > 0 .
16: 10.61 Definitions and Basic Properties
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10.61.3 x 2 ⁒ d 2 w d x 2 + x ⁒ d w d x ( i ⁒ x 2 + ν 2 ) ⁒ w = 0 , w = ber ν ⁑ x + i ⁒ bei ν ⁑ x , ber ν ⁑ x + i ⁒ bei ν ⁑ x ker ν ⁑ x + i ⁒ kei ν ⁑ x , ker ν ⁑ x + i ⁒ kei ν ⁑ x .
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10.61.4 x 4 ⁒ d 4 w d x 4 + 2 ⁒ x 3 ⁒ d 3 w d x 3 ( 1 + 2 ⁒ Ξ½ 2 ) ⁒ ( x 2 ⁒ d 2 w d x 2 x ⁒ d w d x ) + ( Ξ½ 4 4 ⁒ Ξ½ 2 + x 4 ) ⁒ w = 0 , w = ber ± Ξ½ ⁑ x , bei ± Ξ½ ⁑ x , ker ± Ξ½ ⁑ x , kei ± Ξ½ ⁑ x .
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§10.61(iii) Reflection Formulas for Arguments
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§10.61(iv) Reflection Formulas for Orders
17: 18.9 Recurrence Relations and Derivatives
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§18.9(iii) Derivatives
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Jacobi
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Ultraspherical
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Laguerre
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Hermite
18: 25.18 Methods of Computation
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§25.18(i) Function Values and Derivatives
β–ΊThe principal tools for computing ΞΆ ⁑ ( s ) are the expansion (25.2.9) for general values of s , and the Riemann–Siegel formula (25.10.3) (extended to higher terms) for ΞΆ ⁑ ( 1 2 + i ⁒ t ) . …Calculations relating to derivatives of ΞΆ ⁑ ( s ) and/or ΞΆ ⁑ ( s , a ) can be found in Apostol (1985a), Choudhury (1995), Miller and Adamchik (1998), and Yeremin et al. (1988). …
19: 3.5 Quadrature
β–ΊIf k in (3.5.4) is not arbitrarily large, and if odd-order derivatives of f are known at the end points a and b , then the composite trapezoidal rule can be improved by means of the Euler–Maclaurin formula2.10(i)). … β–Ίβ–Ί
Gauss–Legendre Formula
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Gauss–Laguerre Formula
β–Ίa complex Gauss quadrature formula is available. …
20: 16.3 Derivatives and Contiguous Functions
§16.3 Derivatives and Contiguous Functions
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§16.3(i) Differentiation Formulas
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16.3.5 ( z ⁒ d d z ⁑ z ) n = z n ⁒ d n d z n ⁑ z n , n = 1 , 2 , .