Principal Value

The principal value of an analytic multivalued function is the single value chosen by convention to be returned for a given argument. Complex multivalued functions have multiple branches in the complex plane, with those corresponding to the principal values known as the principal branch. For example, the principal branch of the natural logarithm, sometimes denoted , is the one for which , and hence is equal to the value  for all (Knopp 1996, p. 111). All values of  then consist of

with , ..., with the principal branch corresponding to . Since  has only a single branch point, all branches can be plotted to give the Riemann surface.

The term "principal value" also occurs in the theory of integration (e.g., Vladimirov 1971, p. 75), where it means something completely different and is more properly known as the Cauchy principal value. The Cauchy principal valueof an integral is implemented in the Wolfram Language using the command Integrate together with the option PrincipalValue -> True. Similarly, Cauchy principal values can be computed numerically using NIntegratetogether with the option "Method" -> "PrincipalValue".

REFERENCES:

Knopp, K. Theory of Functions Parts I and II, Two Volumes Bound as One. New York: Dover, Part I, p. 111, 1996.

Vladimirov, V. S. Equations of Mathematical Physics. New York: Dekker, 1971.