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1: 21.10 Methods of Computation
§21.10(i) General Riemann Theta Functions
§21.10(ii) Riemann Theta Functions Associated with a Riemann Surface
  • Deconinck and van Hoeij (2001). Here a plane algebraic curve representation of the Riemann surface is used.

  • 2: 2.10 Sums and Sequences
    §2.10(iii) Asymptotic Expansions of Entire Functions
    The asymptotic behavior of entire functions defined by Maclaurin series can be approached by converting the sum into a contour integral by use of the residue theorem and applying the methods of §§2.4 and 2.5. …
    3: 7.2 Definitions
    erf z , erfc z , and w ( z ) are entire functions of z , as is F ( z ) in the next subsection. … ( z ) , C ( z ) , and S ( z ) are entire functions of z , as are f ( z ) and g ( z ) in the next subsection. …
    4: 1.9 Calculus of a Complex Variable
    Analyticity
    A function analytic at every point of is said to be entire. …
    Liouville’s Theorem
    Any bounded entire function is a constant. …
    5: 28.7 Analytic Continuation of Eigenvalues
    Therefore w I ( 1 2 π ; a , q ) is irreducible, in the sense that it cannot be decomposed into a product of entire functions that contain its zeros; see Meixner et al. (1980, p. 88). …
    6: 14.24 Analytic Continuation
    For fixed z , other than ± 1 or , each branch of P ν - μ ( z ) and Q ν μ ( z ) is an entire function of each parameter ν and μ . …
    7: 16.2 Definition and Analytic Properties
    When p q the series (16.2.1) converges for all finite values of z and defines an entire function. … When p q + 1 and z is fixed and not a branch point, any branch of F q p ( a ; b ; z ) is an entire function of each of the parameters a 1 , , a p , b 1 , , b q .
    8: 6.2 Definitions and Interrelations
    Si ( z ) is an odd entire function. … Cin ( z ) is an even entire function. …
    9: 28.29 Definitions and Basic Properties
    This is the characteristic equation of (28.29.1), and cos ( π ν ) is an entire function of λ . … It is an entire function of λ . …
    10: 12.2 Differential Equations
    All solutions are entire functions of z and entire functions of a or ν . …