When the variables are real and distinct, the various cases of
are called circular (hyperbolic) cases if
is positive (negative), because they typically occur in
conjunction with inverse circular (hyperbolic) functions. Cases encountered in
dynamical problems are usually circular; hyperbolic cases include Cauchy
principal values. If are permuted so that , then the
Cauchy principal value of is given by
19.20.14 |
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valid when
19.20.15 |
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or
19.20.16 |
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Since , is in a hyperbolic region. In the complete case
() (19.20.14) reduces to
19.20.17 |
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, . |
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