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21: 6.12 Asymptotic Expansions
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22: 9.5 Integral Representations
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9.5.3
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23: 3.7 Ordinary Differential Equations
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►Assume that we wish to integrate (3.7.1) along a finite path
from to in a domain .
The path is partitioned at points labeled successively , with , .
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►Now suppose the path
is such that the rate of growth of along is intermediate to that of two other solutions.
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24: 15.19 Methods of Computation
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►As noted in §3.7(ii), the integration path should be chosen so that the wanted solution grows in magnitude at least as fast as all other solutions.
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25: 11.5 Integral Representations
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►In (11.5.8) and (11.5.9) the path of integration separates the poles of the integrand at from those at .
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26: 31.11 Expansions in Series of Hypergeometric Functions
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§31.11(v) Doubly-Infinite Series
►Schmidt (1979) gives expansions of path-multiplicative solutions (§31.6) in terms of doubly-infinite series of hypergeometric functions. …27: 8.21 Generalized Sine and Cosine Integrals
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►(obtained from (5.2.1) by rotation of the integration path) is also needed.
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►In these representations the integration paths do not cross the negative real axis, and in the case of (8.21.4) and (8.21.5) the paths also exclude the origin.
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28: 3.5 Quadrature
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►For these cases the integration path may need to be deformed; see §3.5(ix).
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§3.5(ix) Other Contour Integrals
… ►For example, steepest descent paths can be used; see §2.4(iv). … ►with saddle point at , and when the vertical path intersects the real axis at the saddle point. … ►A special case is the rule for Hilbert transforms (§1.14(v)): …29: 1.10 Functions of a Complex Variable
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►The function on is said to be analytically continued along the path
, , if there is a chain , .
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►Here and elsewhere in this subsection the path
is described in the positive sense.
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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 .
►If the path circles a branch point at , times in the positive sense, and returns to without encircling any other branch point, then its value is denoted conventionally as .
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►Thus if is continued along a path that circles
times in the positive sense and returns to without circling , then .
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