intersection indices
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1: 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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2: 21.7 Riemann Surfaces
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►On this surface, we choose
cycles (that is, closed oriented curves, each with at most a finite number of singular points) , , , such that their intersection indices satisfy
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3: 4.37 Inverse Hyperbolic Functions
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►In (4.37.3) the integration path may not intersect
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►The principal values (or principal branches) of the inverse , , and are obtained by introducing cuts in the -plane as indicated in Figure 4.37.1(i)-(iii), and requiring the integration paths in (4.37.1)–(4.37.3) not to cross these cuts.
…Each is two-valued on the corresponding cut(s), and each is real on the part of the real axis that remains after deleting the intersections with the corresponding cuts.
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►Except where indicated otherwise, it is assumed throughout the DLMF that the inverse hyperbolic functions assume their principal values.
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►This section also indicates conformal mappings, and surface plots for complex arguments.
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4: 4.23 Inverse Trigonometric Functions
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►In (4.23.3) the integration path may not intersect
.
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►The principal values (or principal branches) of the inverse sine, cosine, and tangent are obtained by introducing cuts in the -plane as indicated in Figures 4.23.1(i) and 4.23.1(ii), and requiring the integration paths in (4.23.1)–(4.23.3) not to cross these cuts.
…Each is two-valued on the corresponding cuts, and each is real on the part of the real axis that remains after deleting the intersections with the corresponding cuts.
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►Except where indicated otherwise, it is assumed throughout the DLMF that the inverse trigonometric functions assume their principal values.
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5: Mathematical Introduction
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►However, in many cases the coloring of the surface is chosen instead to indicate the quadrant of the plane to which the phase of the function belongs, thereby achieving a 4D effect.
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complex plane (excluding infinity). | |
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intersection. | |
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6: 3.5 Quadrature
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►with saddle point at , and when the vertical path intersects the real axis at the saddle point.
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►Table 3.5.20 gives the results of applying the composite trapezoidal rule (3.5.2) with step size ;
indicates the number of function values in the rule that are larger than (we exploit the fact that the integrand is even).
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7: 4.2 Definitions
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►where the integration path does not intersect the origin.
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►where the path does not intersect
; see Figure 4.2.1.
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►However, in the absence of any indication to the
contrary it is assumed that the definition is the closed one.
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►Unless indicated otherwise, it is assumed throughout the DLMF that a power assumes its principal value.
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