What is principal value: Definition and 3 Discussions

In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. A simple case arises in taking the square root of a positive real number. For example, 4 has two square roots: 2 and −2; of these the positive root, 2, is considered the principal root and is denoted as





4


.


{\displaystyle {\sqrt {4}}.}

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  1. H

    I Complex function, principal value notation

    When a variable in ##[\text { } ]## means its principal value, ##(-\pi,\pi]##, which is correct: ##Log(z^2)=log([z]^2)## or ##Log(z^2)=log([z^2])## (both, neither)?
  2. Euge

    POTW Estimate of a Principal Value Integral

    For ##x\in \mathbb{R}##, let $$A(x) = \frac{1}{2\pi}\, P.V. \int_{-\infty}^\infty e^{i(xy + \frac{y^3}{3})}\, dy$$ Show that the integral defining ##A(x)## exists and ##|A(x)| \le M(1 + |x|)^{-1/4}## for some numerical constant ##M##.
  3. brotherbobby

    Adding square roots of ##i## leading to different answers!

    Statement of the problem : We have to find what is ##\sqrt{i} + \sqrt{-i}## First Attempt (Euler's Formula) : I use the Euler's formula (see Relevant Equations 1) above which yields ##i = e^{i\frac{\pi}{2}}##. Likewise ##-i = e^{i\left(-\frac{\pi}{2}\right)}##. Now I evaluate where, in the...
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