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 value of
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
NIntegrate
together with the option "Method" -> 
"PrincipalValue"
.
