around infinity
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7 matching pages ♦
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7 matching pages
1: 1.9 Calculus of a Complex Variable
2: 25.5 Integral Representations
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►where the integration contour is a loop around the negative real axis; it starts at , encircles the origin once in the positive direction without enclosing any of the points , , …, and returns to .
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3: 2.10 Sums and Sequences
4: 5.11 Asymptotic Expansions
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5.11.3
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5: 25.11 Hurwitz Zeta Function
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►where the integration contour (see Figure 5.9.1) is a loop around the negative real axis as described for (25.5.20).
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25.11.39
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►As with fixed, ,
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►As in the sector , with and fixed, we have the asymptotic expansion
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►Similarly, as in the sector .
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6: 1.10 Functions of a Complex Variable
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►A function whose only singularities, other than the point at infinity, are poles is called a meromorphic function.
If the poles are infinite in number, then the point at infinity is called an essential singularity: it is the limit point of the poles.
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►where and are respectively the numbers of zeros and poles, counting multiplicity, of within , and is the change in any continuous branch of as passes once around
in the positive sense.
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►Branches of can be defined, for example, in the cut plane obtained from by removing the real axis from to and from to ; see Figure 1.10.1.
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►The product , with for all , converges iff converges; and it converges absolutely iff converges.
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7: 18.39 Applications in the Physical Sciences
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►allows anharmonic, or amplitude dependent, frequencies of oscillation about , and also escape of the particle to with dissociation energy .
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►where the orthogonality measure is now ,
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►Orthogonality, with measure for , for fixed
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►normalized with measure , .
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►For applications of Legendre polynomials in fluid dynamics to study the flow around the outside of a puff of hot gas rising through the air, see Paterson (1983).
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