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1: Simon Ruijsenaars
His main research interests cover integrable systems, special functions, analytic difference equations, classical and quantum mechanics, and the relations between these areas. …
2: 2.4 Contour Integrals
2.4.10 I ( z ) = a b e - z p ( t ) q ( t ) d t ,
Assume that p ( t ) and q ( t ) are analytic on an open domain T that contains 𝒫 , with the possible exceptions of t = a and t = b . …
  • (a)

    In a neighborhood of a

    2.4.11
    p ( t ) = p ( a ) + s = 0 p s ( t - a ) s + μ ,
    q ( t ) = s = 0 q s ( t - a ) s + λ - 1 ,

    with λ > 0 , μ > 0 , p 0 0 , and the branches of ( t - a ) λ and ( t - a ) μ continuous and constructed with ph ( t - a ) ω as t a along 𝒫 .

  • in which z is a large real or complex parameter, p ( α , t ) and q ( α , t ) are analytic functions of t and continuous in t and a second parameter α . …
    2.4.18 p ( α , t ) = 1 3 w 3 + a w 2 + b w + c ,
    3: 4.7 Derivatives and Differential Equations
    For a nonvanishing analytic function f ( z ) , the general solution of the differential equation
    4.7.5 d w d z = f ( z ) f ( z )
    4.7.6 w ( z ) = Ln ( f ( z ) ) +  constant .
    4.7.12 d w d z = f ( z ) w
    4.7.13 w = exp ( f ( z ) d z ) + constant .
    4: 5.2 Definitions
    5.2.1 Γ ( z ) = 0 e - t t z - 1 d t , z > 0 .
    5.2.2 ψ ( z ) = Γ ( z ) / Γ ( z ) , z 0 , - 1 , - 2 , .
    5: 3.8 Nonlinear Equations
    §3.8 Nonlinear Equations
    This is an iterative method for real twice-continuously differentiable, or complex analytic, functions: …
    §3.8(v) Zeros of Analytic Functions
    Newton’s rule is the most frequently used iterative process for accurate computation of real or complex zeros of analytic functions f ( z ) . …
    §3.8(vi) Conditioning of Zeros
    6: 4.14 Definitions and Periodicity
    4.14.7 cot z = cos z sin z = 1 tan z .
    7: 16.2 Definition and Analytic Properties
    §16.2(ii) Case p q
    §16.2(iii) Case p = q + 1
    Elsewhere the generalized hypergeometric function is a multivalued function that is analytic except for possible branch points at z = 0 , 1 , and . …
    §16.2(iv) Case p > q + 1
    §16.2(v) Behavior with Respect to Parameters
    8: 28.7 Analytic Continuation of Eigenvalues
    As functions of q , a n ( q ) and b n ( q ) can be continued analytically in the complex q -plane. … All the a 2 n ( q ) , n = 0 , 1 , 2 , , can be regarded as belonging to a complete analytic function (in the large). …
    28.7.4 n = 0 ( b 2 n + 2 ( q ) - ( 2 n + 2 ) 2 ) = 0 .
    9: 1.13 Differential Equations
    Assume that in the equation … u and z belong to domains U and D respectively, the coefficients f ( u , z ) and g ( u , z ) are continuous functions of both variables, and for each fixed u (fixed z ) the two functions are analytic in z (in u ). Suppose also that at (a fixed) z 0 D , w and w / z are analytic functions of u . Then at each z D , w , w / z and 2 w / z 2 are analytic functions of u . … The inhomogeneous (or nonhomogeneous) equation …
    10: 4.28 Definitions and Periodicity
    Periodicity and Zeros