Let be real and nonnegative, with at most one of them 0.
Change-of-parameter relations can be used to shift the parameter of
from either circular region to the other, or from either hyperbolic
region to the other (§19.20(iii)). The latter case allows evaluation
of Cauchy principal values (see (19.20.14)).
19.21.12 |
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where
19.21.13 |
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and may be permuted. Also,
19.21.14 |
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For each value of , permutation of produces three values
of , one of which lies in the same region as and two lie in the other
region of the same type. In (19.21.12), if is the largest
(smallest) of , and , then and lie in the same region if it
is circular (hyperbolic); otherwise and lie in different regions, both
circular or both hyperbolic. If , then and
; hence
19.21.15 |
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