World Library  
Flag as Inappropriate
Email this Article

Principal branch

Article Id: WHEBN0000972441
Reproduction Date:

Title: Principal branch  
Author: World Heritage Encyclopedia
Language: English
Subject: Euler's continued fraction formula, Mercator series, Lagrange inversion theorem, Incomplete gamma function, Polylogarithm
Publisher: World Heritage Encyclopedia

Principal branch

In mathematics, a principal branch is a function which selects one branch ("slice") of a multi-valued function. Most often, this applies to functions defined on the complex plane.


  • Examples 1
    • Trigonometric inverses 1.1
    • Exponentiation to fractional powers 1.2
    • Complex logarithms 1.3
  • See also 2
  • External links 3


Principal branch of arg(z)

Trigonometric inverses

Principal branches are used in the definition of many inverse trigonometric functions, such as the selection either to define that cos−1:ℝ↦(-π,π] or that cos−1:ℝ↦[0,2π).

Exponentiation to fractional powers

A more familiar principal branch function, limited to real numbers, is that of a positive real number raised to the power of 1/2.

For example, take the relation y = x1/2, where x is any positive real number.

This relation can be satisfied by any value of y equal to a square root of x (either positive or negative). By convention, x is used to denote the positive square root of x.

In this instance, the positive square root function is taken as the principal branch of the multi-valued relation x1/2.

Complex logarithms

One way to view a principal branch is to look specifically at the exponential function, and the logarithm, as it is defined in complex analysis.

The exponential function is single-valued, where ez is defined as:

e^z = e^a \cos b + i e^a \sin b

where z = a + i b.

However, the periodic nature of the trigonometric functions involved makes it clear that the logarithm is not so uniquely determined. One way to see this is to look at the following:

\operatorname{Re} (\log z) = \log \sqrt{a^2 + b^2}


\operatorname{Im} (\log z) = \operatorname{atan2}(b, a) + 2 \pi k

where k is any integer and atan2 is arctangent with the appropriate sign correction.

Any number log z defined by such criteria has the property that elog z = z.

In this manner log function is a multi-valued function (often referred to as a "multifunction" in the context of complex analysis). A branch cut, usually along the negative real axis, can limit the imaginary part so it lies between −π and π. These are the chosen principal values.

This is the principal branch of the log function. Often it is defined using a capital letter, Log z.

See also

External links

This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Hawaii eBook Library are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.