in a domain
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1—10 of 42 matching pages
1: Bibliography Y
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The calculation of the Riemann zeta function in the complex domain.
USSR Comput. Math. and Math. Phys. 25 (2), pp. 111–119.
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Computation of the derivatives of the Riemann zeta-function in the complex domain.
USSR Comput. Math. and Math. Phys. 28 (4), pp. 115–124.
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2: 3.7 Ordinary Differential Equations
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►where , , and are analytic functions in a domain
.
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►Assume that we wish to integrate (3.7.1) along a finite path from to
in a domain
.
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3: 1.10 Functions of a Complex Variable
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►Let be analytic in a domain
.
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►Suppose the subarc , is contained in a domain
, .
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►If is analytic in a domain
, and for all , then is a constant in
.
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►Assume that for each , is an analytic function of
in
, and also that is a continuous function of both variables.
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►Suppose , , a domain.
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4: 1.9 Calculus of a Complex Variable
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►A domain
, say, is an open set in
that is connected, that is, any two points can be joined by a polygonal arc (a finite chain of straight-line segments) lying in the set.
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►A function is analytic in a domain
if it is analytic at each point of .
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►Suppose is analytic in a domain
and are two arcs in
passing through .
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►Suppose the series , where is continuous, converges uniformly on every compact set of a domain
, that is, every closed and bounded set in
.
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5: 1.13 Differential Equations
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►A domain in the complex plane is simply-connected if it has no “holes”; more precisely, if its complement in the extended plane is connected.
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►where , a simply-connected domain, and , are analytic in
, has an infinite number of analytic solutions in
.
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►
1.13.6
,
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6: Bibliography W
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Uniform asymptotic expansion of the Jacobi polynomials in a complex domain.
Proc. Roy. Soc. London Ser. A 460, pp. 2569–2586.
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7: Bibliography K
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Connection formulae for the first Painlevé transcendent in the complex domain.
Lett. Math. Phys. 27 (4), pp. 243–252.
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Calculation of modified Bessel functions in a complex domain.
Zh. Vychisl. Mat. i Mat. Fiz. 24 (5), pp. 650–664.
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8: 3.10 Continued Fractions
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►However, other continued fractions with the same limit may converge in a much larger domain of the complex plane than the fraction given by (3.10.4) and (3.10.5).
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9: 3.3 Interpolation
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►If is analytic in a simply-connected domain
(§1.13(i)), then for ,
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►If is analytic in a simply-connected domain
, then for ,
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10: Bibliography H
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Development of a Gaussian hypergeometric function code in complex domains.
Internat. J. Modern Phys. C 4 (4), pp. 805–840.
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