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1: 14.8 Behavior at Singularities
§14.8 Behavior at Singularities
14.8.16 Q - n - ( 1 / 2 ) μ ( x ) π 1 / 2 Γ ( μ + n + 1 2 ) n ! Γ ( μ - n + 1 2 ) ( 2 x ) n + ( 1 / 2 ) , n = 1 , 2 , 3 , , μ - n + 1 2 0 , - 1 , - 2 , .
2: 14.21 Definitions and Basic Properties
§14.21(iii) Properties
This includes, for example, the Wronskian relations (14.2.7)–(14.2.11); hypergeometric representations (14.3.6)–(14.3.10) and (14.3.15)–(14.3.20); results for integer orders (14.6.3)–(14.6.5), (14.6.7), (14.6.8), (14.7.6), (14.7.7), and (14.7.11)–(14.7.16); behavior at singularities (14.8.7)–(14.8.16); connection formulas (14.9.11)–(14.9.16); recurrence relations (14.10.3)–(14.10.7). …
3: 14.20 Conical (or Mehler) Functions
§14.20(iii) Behavior as x 1
4: 19.12 Asymptotic Approximations
With ψ ( x ) denoting the digamma function (§5.2(i)) in this subsection, the asymptotic behavior of K ( k ) and E ( k ) near the singularity at k = 1 is given by the following convergent series: …
5: 2.10 Sums and Sequences
For extensions of the Euler–Maclaurin formula to functions f ( x ) with singularities at x = a or x = n (or both) see Sidi (2004). … We seek the behavior as x + . …
  • (b´)

    On the circle | z | = r , the function f ( z ) - g ( z ) has a finite number of singularities, and at each singularity z j , say,

    2.10.30 f ( z ) - g ( z ) = O ( ( z - z j ) σ j - 1 ) , z z j ,

    where σ j is a positive constant.

  • The singularities of f ( z ) on the unit circle are branch points at z = e ± i α . To match the limiting behavior of f ( z ) at these points we set …
    6: 8.12 Uniform Asymptotic Expansions for Large Parameter
    The right-hand sides of equations (8.12.9), (8.12.10) have removable singularities at η = 0 , and the Maclaurin series expansion of c k ( η ) is given by … For the asymptotic behavior of c k ( η ) as k see Dunster et al. (1998) and Olde Daalhuis (1998c). … A different type of uniform expansion with coefficients that do not possess a removable singularity at z = a is given by …
    7: 2.7 Differential Equations
    All solutions are analytic at an ordinary point, and their Taylor-series expansions are found by equating coefficients. … If both ( z - z 0 ) f ( z ) and ( z - z 0 ) 2 g ( z ) are analytic at z 0 , then z 0 is a regular singularity (or singularity of the first kind). … The most common type of irregular singularity for special functions has rank 1 and is located at infinity. … The transformed differential equation either has a regular singularity at t = , or its characteristic equation has unequal roots. … For irregular singularities of nonclassifiable rank, a powerful tool for finding the asymptotic behavior of solutions, complete with error bounds, is as follows: …
    8: 1.8 Fourier Series
    Lebesgue Constants
    (1.8.10) continues to apply if either a or b or both are infinite and/or f ( x ) has finitely many singularities in ( a , b ) , provided that the integral converges uniformly (§1.5(iv)) at a , b , and the singularities for all sufficiently large λ . … Let f ( x ) be an absolutely integrable function of period 2 π , and continuous except at a finite number of points in any bounded interval. …at 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 + ). The convergence is non-uniform, however, at points where f ( x - ) f ( x + ) ; see §6.16(i). …
    9: Bibliography B
  • M. V. Berry (1981) Singularities in Waves and Rays. In Les Houches Lecture Series Session XXXV, R. Balian, M. Kléman, and J.-P. Poirier (Eds.), Vol. 35, pp. 453–543.
  • N. Bleistein (1966) Uniform asymptotic expansions of integrals with stationary point near algebraic singularity. Comm. Pure Appl. Math. 19, pp. 353–370.
  • N. Bleistein (1967) Uniform asymptotic expansions of integrals with many nearby stationary points and algebraic singularities. J. Math. Mech. 17, pp. 533–559.
  • R. Bo and R. Wong (1996) Asymptotic behavior of the Pollaczek polynomials and their zeros. Stud. Appl. Math. 96, pp. 307–338.
  • W. Bühring (1987b) The behavior at unit argument of the hypergeometric function F 2 3 . SIAM J. Math. Anal. 18 (5), pp. 1227–1234.
  • 10: 2.3 Integrals of a Real Variable
    Other types of singular behavior in the integrand can be treated in an analogous manner. … Without loss of generality, we assume that this minimum is at the left endpoint a . … For the more general integral (2.3.19) we assume, without loss of generality, that the stationary point (if any) is at the left endpoint. … For extensions to oscillatory integrals with logarithmic singularities see Wong and Lin (1978). … it is free from singularity at t = α . …