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►The principalvalue, or principal branch, is defined by
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►We regard this as the closed definition of the principalvalue.
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►The principalvalue is
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►Another example of a principalvalue is provided by
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►Analytic continuation of the principalvalue of yields a multi-valued function with branch points at and .
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►Unless indicated otherwise, in the rest of this chapter and elsewhere in the DLMF the functions , , , , and assume their principalvalues, that is, the branches that are real on the positive real axis and two-valued on the negative real axis.
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►►►Figure 4.29.2: Principalvalues of and .
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Magnify
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►►►Figure 4.29.4: Principalvalues of and .
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►►►Figure 4.29.6: Principalvalues of and .
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►►►Figure 4.15.2:
and .
Principalvalues are shown with thickened lines.
Magnify
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►►►Figure 4.15.4:
and .
Only principalvalues are shown.
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Magnify
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►►►Figure 4.15.6:
and .
Only principalvalues are shown.
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Magnify
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►►
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►Compare the principalvalue of the logarithm (§4.2(i)).
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►The principalvalues of the inverse hyperbolic cosecant, hyperbolic secant, and hyperbolic tangent are given by
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►Graphs of the principalvalues for real arguments are given in §4.29.
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►Throughout this subsection all quantities assume their principalvalues.
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