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 z.


For example, the principal branch of the natural logarithm, sometimes denoted Lnz, is the one for which ln1=0, and hence is equal to the value lnx for all x>0 (Knopp 1996, p. 111). The value of a function on its principal branch is known as its principal value. All values of lnz then consist of


with k=0,+/-1,+/-2, ..., with the principal branch corresponding to k=0. Since lnz has only a single branch point, all branches can be plotted to give the entire Riemann surface.

See also

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

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

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Principal Branch

Cite this as:

Weisstein, Eric W. "Principal Branch." From MathWorld--A Wolfram Web Resource.

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