TOPICS

Principal Branch

The principal branch of an analytic multivalued function, also called a principal sheet, is a single-valued "slice" (i.e., branch) of the function chosen that is for convenience in referring to a specific canonical value (a so-called principal value) of the function for each complex .

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). The value of a function on its principal branch is known as its principal value. 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 entire Riemann surface.

Branch, Branch Cut, Branch Point, Lambert W-Function, Multivalued Function, Principal Value

References

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

Principal Branch

Cite this as:

Weisstein, Eric W. "Principal Branch." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PrincipalBranch.html