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derivatives with respect to order

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1: 10.64 Integral Representations
2: 10.38 Derivatives with Respect to Order
§10.38 Derivatives with Respect to Order
3: 10.15 Derivatives with Respect to Order
§10.15 Derivatives with Respect to Order
4: 14.11 Derivatives with Respect to Degree or Order
§14.11 Derivatives with Respect to Degree or Order
14.11.2 ν 𝖰 ν μ ( x ) = 1 2 π 2 𝖯 ν μ ( x ) + π sin ( μ π ) sin ( ν π ) sin ( ( ν + μ ) π ) 𝖰 ν μ ( x ) 1 2 cot ( ( ν + μ ) π ) 𝖠 ν μ ( x ) + 1 2 csc ( ( ν + μ ) π ) 𝖠 ν μ ( x ) ,
5: 11.4 Basic Properties
11.4.27 d d z ( z ν 𝐇 ν ( z ) ) = z ν 𝐇 ν 1 ( z ) ,
11.4.28 d d z ( z ν 𝐇 ν ( z ) ) = 2 ν π Γ ( ν + 3 2 ) z ν 𝐇 ν + 1 ( z ) ,
11.4.31 ν m ( z ) = z m ν ( 1 z d d z ) m ( z ν ν ( z ) ) , m = 1 , 2 , 3 , ,
§11.4(vi) Derivatives with Respect to Order
For derivatives with respect to the order ν , see Apelblat (1989) and Brychkov and Geddes (2005). …
6: 10.40 Asymptotic Expansions for Large Argument
ν -Derivative
7: 14.29 Generalizations
14.29.1 ( 1 z 2 ) d 2 w d z 2 2 z d w d z + ( ν ( ν + 1 ) μ 1 2 2 ( 1 z ) μ 2 2 2 ( 1 + z ) ) w = 0
8: Bibliography
  • A. Apelblat (1989) Derivatives and integrals with respect to the order of the Struve functions 𝐇 ν ( x ) and 𝐋 ν ( x ) . J. Math. Anal. Appl. 137 (1), pp. 17–36.
  • A. Apelblat (1991) Integral representation of Kelvin functions and their derivatives with respect to the order. Z. Angew. Math. Phys. 42 (5), pp. 708–714.
  • 9: 10.21 Zeros
    §10.21(xiv) ν -Zeros
    10: 14.10 Recurrence Relations and Derivatives
    14.10.4 ( 1 x 2 ) d 𝖯 ν μ ( x ) d x = ( μ ν 1 ) 𝖯 ν + 1 μ ( x ) + ( ν + 1 ) x 𝖯 ν μ ( x ) ,
    14.10.5 ( 1 x 2 ) d 𝖯 ν μ ( x ) d x = ( ν + μ ) 𝖯 ν 1 μ ( x ) ν x 𝖯 ν μ ( x ) .