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11—20 of 781 matching pages
11: 1.10 Functions of a Complex Variable
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►If we can assign a unique value to
at each point of , and is analytic on , then is a branch of .
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►(b) By specifying the value of
at a point (not a branch point), and requiring to be continuous on any path that begins at
and does not pass through any branch points or other singularities of .
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►Then the value of
at any other point is obtained by analytic continuation.
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►has a unique solution analytic at
, and
…in a neighborhood of , where is the residue of
at
.
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12: 20.4 Values at = 0
13: Bibliography D
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Integralen voor de -functie van Riemann.
Mathematica (Zutphen) B5, pp. 170–180 (Dutch).
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Quantum Fields and Strings: A Course for Mathematicians. Vol. 1, 2.
American Mathematical Society, Providence, RI.
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Zeros of Bernoulli, generalized Bernoulli and Euler polynomials.
Mem. Amer. Math. Soc. 73 (386), pp. iv+94.
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Handbuch der Laplace-Transformation. Bd. II. Anwendungen der Laplace-Transformation. 1. Abteilung.
Birkhäuser Verlag, Basel und Stuttgart (German).
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Complex zeros of cylinder functions.
Math. Comp. 20 (94), pp. 215–222.
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14: 16.9 Zeros
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►Then has at most finitely many zeros if and only if the can be re-indexed for in such a way that is a nonnegative integer.
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►Then has at most finitely many real zeros.
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15: Bibliography
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Some determinants of Bernoulli, Euler and related numbers.
Portugal. Math. 18, pp. 91–99.
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Sturm-Liouville Theory.
Birkhäuser Verlag, Basel.
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Classical Orthogonal Polynomials.
In Orthogonal Polynomials and Applications, C. Brezinski, A. Draux, A. P. Magnus, P. Maroni, and A. Ronveaux (Eds.),
Lecture Notes in Math., Vol. 1171, pp. 36–62.
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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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Bernoulli’s power-sum formulas revisited.
Math. Gaz. 90 (518), pp. 276–279.
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16: 7.3 Graphics
17: 26.6 Other Lattice Path Numbers
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Delannoy Number
► is the number of paths from to that are composed of directed line segments of the form , , or . … ► is the number of lattice paths from to that stay on or above the line , are composed of directed line segments of the form or , and for which there are exactly occurrences at which a segment of the form is followed by a segment of the form . … ►
26.6.12
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26.6.13
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18: Bibliography H
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Certain sums that contain cylindrical functions.
Bul. Akad. Štiince RSS Moldoven. 1972 (3), pp. 75–77, 94 (Russian).
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The Laplace transform for expressions that contain a probability function.
Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).
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Expansions for the probability function in series of Čebyšev polynomials and Bessel functions.
Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).
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Coulomb Wave Functions.
In Handbuch der Physik, Bd. 41/1, S. Flügge (Ed.),
pp. 408–465.
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Variable precision exponential function.
ACM Trans. Math. Software 12 (2), pp. 79–91.
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19: 15.3 Graphics
20: 19.2 Definitions
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►Let be a cubic or quartic polynomial in with simple zeros, and let be a rational function of and containing at least one odd power of .
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►The integral for is well defined if , and the Cauchy principal value (§1.4(v)) of is taken if vanishes at an interior point of the integration path.
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►with a branch point at
and principal branch .
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