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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.
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►
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Figure 4.37.1:
-plane.
Branchcuts for the inverse hyperbolic functions.
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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.
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►
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Figure 4.23.1:
-plane.
Branchcuts for the inverse trigonometric functions.
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►The branch obtained by introducing a cut from to on the real -axis, that is, the branch in the sector , is the principal
branch (or principal value) of .
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►The difference between the principal branches on the two sides of the branchcut (§4.2(i)) is given by
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►For example, by converting the Maclaurin expansion of (4.24.3), we obtain a continued fraction with the same region of convergence (, ), whereas the continued fraction (4.25.4) converges for all except on the branchcuts from to and to .
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W. Kahan (1987)BranchCuts for Complex Elementary Functions or Much Ado About Nothing’s Sign Bit.
In The State of the Art in Numerical Analysis (Birmingham, 1986), A. Iserles and M. J. D. Powell (Eds.),
Inst. Math. Appl. Conf. Ser. New Ser., Vol. 9, pp. 165–211.