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21—30 of 103 matching pages
21: 2.11 Remainder Terms; Stokes Phenomenon
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►When a rigorous bound or reliable estimate for the remainder term is unavailable, it is unsafe to judge the accuracy of an asymptotic expansion merely from the numerical rate of decrease of the terms at the point of truncation.
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►For large the integrand has a saddle point at
.
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22: 1.8 Fourier Series
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►Let be an absolutely integrable function of period , and continuous except at a finite number of points in any bounded interval.
…at every point at which has both a left-hand derivative (that is, (1.4.4) applies when ) and a right-hand derivative (that is, (1.4.4) applies when ).
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►If a function is periodic, with period , then the series obtained by differentiating the Fourier series for term by term converges at every point to .
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23: 6.4 Analytic Continuation
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►Analytic continuation of the principal value of yields a multi-valued function with branch points at
and .
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24: 19.14 Reduction of General Elliptic Integrals
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►A similar remark applies to the transformations given in Erdélyi et al. (1953b, §13.5) and to the choice among explicit reductions in the extensive table of Byrd and Friedman (1971), in which one limit of integration is assumed to be a branch point of the integrand at which the integral converges.
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25: 21.1 Special Notation
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positive integers. | |
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intersection index of and , two cycles lying on a closed surface. if and do not intersect. Otherwise gets an additive contribution from every intersection point. This contribution is if the basis of the tangent vectors of the and cycles (§21.7(i)) at the point of intersection is positively oriented; otherwise it is . | |
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26: 4.2 Definitions
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►This is a multivalued function of with branch point at
.
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►In all other cases, is a multivalued function with branch point at
.
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27: 18.1 Notation
28: 8.21 Generalized Sine and Cosine Integrals
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►From §§8.2(i) and 8.2(ii) it follows that each of the four functions , , , and is a multivalued function of with branch point at
.
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29: 28.33 Physical Applications
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►For points
that are at intersections of with the characteristic curves or , a periodic solution is possible.
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