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11: 4.37 Inverse Hyperbolic Functions
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►These functions are analytic in the cut plane depicted in Figure 4.37.1(iv), (v), (vi), respectively.
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§4.37(iv) Logarithmic Forms
… ►On the cuts … ►On the part of the cuts from to … ►On the cuts …12: 30.6 Functions of Complex Argument
13: 4.2 Definitions
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►Most texts extend the definition of the principal value to include the branch cut
…where is the excess of the number of times the path in (4.2.1) crosses the negative real axis in the positive sense over the number of times in the negative sense.
►In the DLMF we allow a further extension by regarding the cut as representing two sets of points, one set corresponding to the “upper side” and denoted by , the other set corresponding to the “lower side” and denoted by .
…Consequently is two-valued on the cut, and discontinuous across the cut.
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►This is an analytic function of on , and is two-valued and discontinuous on the cut shown in Figure 4.2.1, unless .
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14: 4.39 Continued Fractions
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►where is in the open cut plane of Figure 4.37.1(i).
…where is in the open cut plane of Figure 4.37.1(iii).
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15: 30.18 Software
16: 10.72 Mathematical Applications
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►These expansions are uniform with respect to , including the turning point and its neighborhood, and the region of validity often includes cut neighborhoods (§1.10(vi)) of other singularities of the differential equation, especially irregular singularities.
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►These asymptotic expansions are uniform with respect to , including cut neighborhoods of , and again the region of uniformity often includes cut neighborhoods of other singularities of the differential equation.
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►These approximations are uniform with respect to both and , including , the cut neighborhood of , and .
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17: 14.27 Zeros
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(either side of the cut) has exactly one zero in the interval if either of the following sets of conditions holds:
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18: 4.25 Continued Fractions
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►valid when lies in the open cut plane shown in Figure 4.23.1(i).
…valid when lies in the open cut plane shown in Figure 4.23.1(ii).
…valid when lies in the open cut plane shown in Figure 4.23.1(iv).
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19: 4.3 Graphics
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►Figure 4.3.2 illustrates the conformal mapping of the strip onto the whole -plane cut along the negative real axis, where and (principal value).
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20: Philip J. Davis
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►Moreover, a cutting plane feature allows users to track curves of intersection produced as a moving plane cuts through the function surface.
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