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1: 10.72 Mathematical Applications
In regions in which (10.72.1) has a simple turning point z 0 , that is, f ( z ) and g ( z ) are analytic (or with weaker conditions if z = x is a real variable) and z 0 is a simple zero of f ( z ) , asymptotic expansions of the solutions w for large u can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order 1 3 9.6(i)). … In (10.72.1) assume f ( z ) = f ( z , α ) and g ( z ) = g ( z , α ) depend continuously on a real parameter α , f ( z , α ) has a simple zero z = z 0 ( α ) and a double pole z = 0 , except for a critical value α = a , where z 0 ( a ) = 0 . …
2: 5.2 Definitions
It is a meromorphic function with no zeros, and with simple poles of residue ( - 1 ) n / n ! at z = - n . 1 / Γ ( z ) is entire, with simple zeros at z = - n . …
3: 3.8 Nonlinear Equations
§3.8 Nonlinear Equations
If f ( z 0 ) = 0 and f ( z 0 ) 0 , then z 0 is a simple zero of f . … If ζ is a simple zero, then the iteration converges locally and quadratically. … If the wanted zero ξ is simple, then the method converges locally with order of convergence p = 1 2 ( 1 + 5 ) = 1.618 . … Then the sensitivity of a simple zero z to changes in α is given by …
4: 7.13 Zeros
erf z has a simple zero at z = 0 , and in the first quadrant of there is an infinite set of zeros z n = x n + i y n , n = 1 , 2 , 3 , , arranged in order of increasing absolute value. … At z = 0 , C ( z ) has a simple zero and S ( z ) has a triple zero. …
5: 1.10 Functions of a Complex Variable
When m = 1 the zero is simple. … If f ( z ) and g ( z ) are analytic on and inside a simple closed contour C , and | g ( z ) | < | f ( z ) | on C , then f ( z ) and f ( z ) + g ( z ) have the same number of zeros inside C . …
6: 2.4 Contour Integrals
In the commonest case the interior minimum t 0 of ( z p ( t ) ) is a simple zero of p ( t ) . … Suppose that on the integration path 𝒫 there are two simple zeros of p ( α , t ) / t that coincide for a certain value α ^ of α . …
7: 22.2 Definitions
Each is meromorphic in z for fixed k , with simple poles and simple zeros, and each is meromorphic in k for fixed z . …
8: Bibliography K
  • P. Kravanja, O. Ragos, M. N. Vrahatis, and F. A. Zafiropoulos (1998) ZEBEC: A mathematical software package for computing simple zeros of Bessel functions of real order and complex argument. Comput. Phys. Comm. 113 (2-3), pp. 220–238.
  • 9: 22.4 Periods, Poles, and Zeros
    Then: (a) In any lattice unit cell p q ( z , k ) has a simple zero at z = p and a simple pole at z = q . …
    10: 23.2 Definitions and Periodic Properties
    The function σ ( z ) is entire and odd, with simple zeros at the lattice points. …