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1: 4.37 Inverse Hyperbolic Functions
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►In (4.37.1) the integration path may not pass through either of the points , and the function assumes its principal value when is real.
In (4.37.2) the integration path may not pass through either of the points , and the function assumes its principal value when .
…In (4.37.3) the integration path may not intersect .
… and have branch points at ; the other four functions have branch points at .
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►Table 4.30.1 can also be used to find interrelations between inverse hyperbolic functions.
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2: 33.6 Power-Series Expansions in
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33.6.5
►where and (§5.2(i)).
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►Corresponding expansions for can be obtained by combining (33.6.5) with (33.4.3) or (33.4.4).
3: 4.21 Identities
4: 19.34 Mutual Inductance of Coaxial Circles
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►The method of §19.29(ii) uses (19.29.18), (19.29.16), and (19.29.15) to produce
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19.34.3
►where and
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19.34.4
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19.34.5
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5: 4.38 Inverse Hyperbolic Functions: Further Properties
6: 32.10 Special Function Solutions
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►For example, if , with , then the Riccati equation is
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►Solutions for other values of are derived from by application of the Bäcklund transformations (32.7.1) and (32.7.2).
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►with , and , , independently.
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►with and .
In the case when in (32.10.15), the Riccati equation is
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7: 19.22 Quadratic Transformations
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►However, if and are complex conjugates and and are real, then the right-hand sides of all transformations in §§19.22(i) and 19.22(iii)—except (19.22.3) and (19.22.22)—are free of complex numbers and .
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►These relations need to be used with caution because is negative when .
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19.22.5
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8: 10.38 Derivatives with Respect to Order
9: 8.12 Uniform Asymptotic Expansions for Large Parameter
10: 10.59 Integrals
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10.59.1
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