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derivatives with respect to order

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11: Bibliography B
  • Yu. A. Brychkov and K. O. Geddes (2005) On the derivatives of the Bessel and Struve functions with respect to the order. Integral Transforms Spec. Funct. 16 (3), pp. 187–198.
  • 12: Bibliography S
  • R. Szmytkowski (2009) On the derivative of the associated Legendre function of the first kind of integer degree with respect to its order (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Chem. 46 (1), pp. 231–260.
  • R. Szmytkowski (2011) On the derivative of the associated Legendre function of the first kind of integer order with respect to its degree (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Chem. 49 (7), pp. 1436–1477.
  • 13: 10.1 Special Notation
    For the spherical Bessel functions and modified spherical Bessel functions the order n is a nonnegative integer. For the other functions when the order ν is replaced by n , it can be any integer. For the Kelvin functions the order ν is always assumed to be real. … Abramowitz and Stegun (1964): j n ( z ) , y n ( z ) , h n ( 1 ) ( z ) , h n ( 2 ) ( z ) , for 𝗃 n ( z ) , 𝗒 n ( z ) , 𝗁 n ( 1 ) ( z ) , 𝗁 n ( 2 ) ( z ) , respectively, when n 0 . … For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).
    14: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
    §1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
    For T to be actually self adjoint it is necessary to also show that 𝒟 ( T * ) = 𝒟 ( T ) , as it is often the case that T and T * have different domains, see Friedman (1990, p 148) for a simple example of such differences involving the differential operator d d x .
    §1.18(iv) Formally Self-adjoint Linear Second Order Differential Operators
    The special form of (1.18.28) is especially useful for applications in physics, as the connection to non-relativistic quantum mechanics is immediate:  d 2 d x 2 being proportional to the kinetic energy operator for a single particle in one dimension, q ( x ) being proportional to the potential energy, often written as V ( x ) , of that same particle, and which is simply a multiplicative operator. The sum of the kinetic and potential energies give the quantum Hamiltonian, or energy operator; often also referred to as a Schrödinger operator. Other applications follow from the fact that is suitable for describing vibrations, especially standing waves, which arise in many parts of engineering and the physical sciences, see Birkhoff and Rota (1989, §§10.3 and 10.16). See §18.39(i). More generally, continuous spectra may occur in sets of disjoint finite intervals [ λ a , λ b ] ( 0 , ) , often called bands, when q ( x ) is periodic, see Ashcroft and Mermin (1976, Ch 8) and Kittel (1996, Ch 7). Should q ( x ) be bounded but random, leading to Anderson localization, the spectrum could range from being a dense point spectrum to being singular continuous, see Simon (1995), Avron and Simon (1982); a good general reference being Cycon et al. (2008, Ch. 9 and 10). For example, replacing 2 q cos ( 2 z ) of (28.2.1) by λ cos ( 2 π α n + θ ) , n gives an almost Mathieu equation which for appropriate α has such properties.
    15: 10.19 Asymptotic Expansions for Large Order
    §10.19 Asymptotic Expansions for Large Order
    §10.19(i) Asymptotic Forms
    §10.19(ii) Debye’s Expansions
    §10.19(iii) Transition Region
    See also §10.20(i).
    16: 28.1 Special Notation
    m , n

    integers.

    ν

    order of the Mathieu function or modified Mathieu function. (When ν is an integer it is often replaced by n .)

    primes

    unless indicated otherwise, derivatives with respect to the argument

    The radial functions Mc n ( j ) ( z , h ) and Ms n ( j ) ( z , h ) are denoted by Mc n ( j ) ( z , q ) and Ms n ( j ) ( z , q ) , respectively.
    17: 2.8 Differential Equations with a Parameter
    dots denoting differentiations with respect to ξ . Then … The expansions (2.8.11) and (2.8.12) are both uniform and differentiable with respect to ξ . … The expansions (2.8.15) and (2.8.16) are both uniform and differentiable with respect to ξ . … The expansions (2.8.25) and (2.8.26) are both uniform and differentiable with respect to ξ . …
    18: 10.72 Mathematical Applications
    These expansions are uniform with respect to z , including the turning point z 0 and its neighborhood, and the region of validity often includes cut neighborhoods (§1.10(vi)) of other singularities of the differential equation, especially irregular singularities. … In regions in which the function f ( z ) has a simple pole at z = z 0 and ( z z 0 ) 2 g ( z ) is analytic at z = z 0 (the case λ = 1 in §10.72(i)), asymptotic expansions of the solutions w of (10.72.1) for large u can be constructed in terms of Bessel functions and modified Bessel functions of order ± 1 + 4 ρ , where ρ is the limiting value of ( z z 0 ) 2 g ( z ) as z z 0 . These asymptotic expansions are uniform with respect to z , including cut neighborhoods of z 0 , and again the region of uniformity often includes cut neighborhoods of other singularities of the differential equation. … Then for large u asymptotic approximations of the solutions w can be constructed in terms of Bessel functions, or modified Bessel functions, of variable order (in fact the order depends on u and α ). These approximations are uniform with respect to both z and α , including z = z 0 ( a ) , the cut neighborhood of z = 0 , and α = a . …
    19: 18.36 Miscellaneous Polynomials
    These are OP’s on the interval ( 1 , 1 ) with respect to an orthogonality measure obtained by adding constant multiples of “Dirac delta weights” at 1 and 1 to the weight function for the Jacobi polynomials. … Sobolev OP’s are orthogonal with respect to an inner product involving derivatives. … These are polynomials in one variable that are orthogonal with respect to a number of different measures. … These are matrix-valued polynomials that are orthogonal with respect to a square matrix of measures on the real line. Classes of such polynomials have been found that generalize the classical OP’s in the sense that they satisfy second-order matrix differential equations with coefficients independent of the degree. …
    20: 9.9 Zeros
    They are denoted by a k , a k , b k , b k , respectively, arranged in ascending order of absolute value for k = 1 , 2 , . They lie in the sectors 1 3 π < ph z < 1 2 π and 1 2 π < ph z < 1 3 π , and are denoted by β k , β k , respectively, in the former sector, and by β k ¯ , β k ¯ , in the conjugate sector, again arranged in ascending order of absolute value (modulus) for k = 1 , 2 , . See §9.3(ii) for visualizations. …
    §9.9(ii) Relation to Modulus and Phase
    §9.9(iii) Derivatives With Respect to k
    For error bounds for the asymptotic expansions of a k , b k , a k , and b k see Pittaluga and Sacripante (1991), and a conjecture given in Fabijonas and Olver (1999). …