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1: 3.8 Nonlinear Equations
§3.8 Nonlinear Equations
… ►§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 . … ►§3.8(vi) Conditioning of Zeros
… ►Corresponding numerical factors in this example for other zeros and other values of are obtained in Gautschi (1984, §4). …2: Bibliography M
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Asymptotic expansions for the zeros of certain special functions.
J. Comput. Appl. Math. 145 (2), pp. 261–267.
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The Theory of Analytic Functions: A Brief Course.
“Mir”, Moscow.
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Inequalities for the zeros of Bessel functions.
SIAM J. Math. Anal. 8 (1), pp. 166–170.
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Calculation of the modified Bessel functions of the second kind with complex argument.
Math. Comp. 20 (95), pp. 407–412.
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Analytic expressions for integrals of products of spherical Bessel functions.
J. Phys. A 24 (7), pp. 1435–1453.
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3: Bibliography S
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Coulomb functions analytic in the energy.
Comput. Phys. Comm. 25 (1), pp. 87–95.
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A global Newton method for the zeros of cylinder functions.
Numer. Algorithms 18 (3-4), pp. 259–276.
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Bounds on differences of adjacent zeros of Bessel functions and iterative relations between consecutive zeros.
Math. Comp. 70 (235), pp. 1205–1220.
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Uniform asymptotic forms of modified Mathieu functions.
Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.
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Numerical Methods Based on Sinc and Analytic Functions.
Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.
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4: Bibliography W
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The analyticity of Jacobian functions with respect to the parameter
.
Proc. Roy. Soc. London Ser A 459, pp. 2569–2574.
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The zeros of Euler’s psi function and its derivatives.
J. Math. Anal. Appl. 332 (1), pp. 607–616.
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The distribution of the zeros of Jacobian elliptic functions with respect to the parameter
.
Comput. Methods Funct. Theory 9 (2), pp. 579–591.
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The Nahm equations, finite-gap potentials and Lamé functions.
J. Phys. A 20 (10), pp. 2679–2683.
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On the zeros of a confluent hypergeometric function.
Proc. Amer. Math. Soc. 16 (2), pp. 281–283.
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5: Bibliography B
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Zeros of generalized Airy functions.
Mathematika 32 (1), pp. 104–117.
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A program for computing the Riemann zeta function for complex argument.
Comput. Phys. Comm. 20 (3), pp. 441–445.
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Coulomb functions (negative energies).
Comput. Phys. Comm. 20 (3), pp. 447–458.
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Bessel functions and modular relations of higher type and hyperbolic differential equations.
Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.
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An analytic continuation formula for the generalized hypergeometric function.
SIAM J. Math. Anal. 19 (5), pp. 1249–1251.
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6: Bibliography
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Complex Analysis: An Introduction of the Theory of Analytic Functions of One Complex Variable.
2nd edition, McGraw-Hill Book Co., New York.
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On the zeros of confluent hypergeometric functions. III. Characterization by means of nonlinear equations.
Lett. Nuovo Cimento (2) 29 (11), pp. 353–358.
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Repeated integrals and derivatives of Bessel functions.
SIAM J. Math. Anal. 20 (1), pp. 169–175.
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Hypergeometric Functions and Elliptic Integrals.
In Current Topics in Analytic Function Theory, H. M. Srivastava and S. Owa (Eds.),
pp. 48–85.
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Note on the trivial zeros of Dirichlet -functions.
Proc. Amer. Math. Soc. 94 (1), pp. 29–30.
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7: Bibliography K
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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library.
ACM Trans. Math. Software 20 (4), pp. 447–459.
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Methods of computing the Riemann zeta-function and some generalizations of it.
USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
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Complex zeros of an incomplete Riemann zeta function and of the incomplete gamma function.
Math. Comp. 24 (111), pp. 679–696.
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Complex zeros of two incomplete Riemann zeta functions.
Math. Comp. 26 (118), pp. 551–565.
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On the zeros of the incomplete gamma function.
Math. Comp. 26 (119), pp. 751–755.
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8: Bibliography P
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Interlacing of positive real zeros of Bessel functions.
J. Math. Anal. Appl. 375 (1), pp. 320–322.
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Complex zeros of the modified Bessel function
.
Math. Comp. 26 (120), pp. 949–953.
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Automatic computation of Bessel function integrals.
Comput. Phys. Comm. 25 (3), pp. 289–295.
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Chebyshev series approximations for the zeros of the Bessel functions.
J. Comput. Phys. 53 (1), pp. 188–192.
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Fast analytic formulas for the modified Bessel functions of imaginary order for spectral line broadening calculations.
J. Quantit. Spec. and Rad. Trans. 62 (4), pp. 389–395.
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9: Bibliography R
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Topics in Analytic Number Theory.
Springer-Verlag, New York.
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On the definition and properties of generalized - symbols.
J. Math. Phys. 20 (12), pp. 2398–2415.
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Erratum to:Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging.
Computing in Science and Engineering 23 (4), pp. 91.
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Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging.
Computing in Science and Engineering 23 (3), pp. 56–64.
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On the zeros of the hypergeometric function.
Math. Ann. 191 (1), pp. 53–58.
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10: Bibliography C
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Determination of -zeros of Hankel functions.
Comput. Phys. Comm. 32 (3), pp. 333–339.
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The zeros of Hankel functions as functions of their order.
Numer. Math. 7 (3), pp. 238–250.
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The analyticity of cross-product Bessel function zeros.
Proc. Cambridge Philos. Soc. 62, pp. 215–226.
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Validated computation of certain hypergeometric functions.
ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.
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On zeros of the hypergeometric function.
Serdica 7 (3), pp. 243–249.
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