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►►►Figure 19.3.2:
and the Cauchyprincipalvalue of for .
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►Figure 19.3.5:
as a function of and for , .
Cauchyprincipalvalues are shown when .
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►Figure 19.3.6:
as a function of and for , .
Cauchyprincipalvalues are shown when .
…If (), then the function reduces to with Cauchyprincipalvalue
, which tends to as .
…If (), then by (19.7.4) it reduces to , , with Cauchyprincipalvalue
, , by (19.6.5).
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►The integral for is well defined if , and the Cauchyprincipalvalue (§1.4(v)) of is taken if vanishes at an interior point of the integration path.
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►If , then the integral in (19.2.11) is a Cauchyprincipalvalue.
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►where the Cauchyprincipalvalue is taken if .
Formulas involving that are customarily different for circular cases, ordinary hyperbolic cases, and (hyperbolic) Cauchyprincipalvalues, are united in a single formula by using .
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►The Cauchyprincipalvalue is hyperbolic:
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►If , then the Cauchyprincipalvalue satisfies
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►Circular and hyperbolic cases, including Cauchyprincipalvalues, are unified by using .
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►For the Cauchyprincipalvalue of when , see §19.7(iii).
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