Principal value

 ( Complex Analysis ) 

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 $\backslash sqrt\{4\}.$

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Motivation Consider the complex logarithm function . It is defined as the complex number such that

$e^w\; =\; z.$

Now, for example, say we wish to find . This means we want to solve

$e^w\; =\; i$

for $w$. The value $i\backslash pi/2$ is a solution.

However, there are other solutions, which is evidenced by considering the position of in the complex plane and in particular its argument $\backslash arg\; i$. We can rotate counterclockwise $\backslash pi/2$ radians from 1 to reach initially, but if we rotate further another $2\backslash pi$ we reach again. So, we can conclude that $i(\backslash pi/2\; +\; 2\backslash pi)$ is also a solution for . It becomes clear that we can add any multiple of $2\backslash pi$ to our initial solution to obtain all values for .

But this has a consequence that may be surprising in comparison of real valued functions: does not have one definite value. For , we have

$\backslash log\{z\}\; =\; \backslash ln$ + i\left(\mathrm{arg}\ z \right)

= \ln + i\left(\mathrm{Arg}\ z+2\pi k\right) for an integer , where is the (principal) argument of defined to lie in the interval $(-\backslash pi,\backslash \; \backslash pi]$. Each value of determines what is known as a branch (or sheet), a single-valued component of the multiple-valued log function. When the focus is on a single branch, sometimes a branch cut is used; in this case removing the non-positive real numbers from the domain of the function and eliminating $\backslash pi$ as a possible value for . With this branch cut, the single-branch function is continuous and analytic everywhere in its domain.

The branch corresponding to is known as the principal branch, and along this branch, the values the function takes are known as the principal values.

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General case In general, if is multiple-valued, the principal branch of is denoted

$\backslash mathrm\{pv\}\backslash ,f(z)$

such that for in the domain of , is single-valued.

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Principal values of standard functions Complex valued elementary functions can be multiple-valued over some domains. The principal value of some of these functions can be obtained by decomposing the function into simpler ones whereby the principal value of the simple functions are straightforward to obtain.

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Logarithm function We have examined the logarithm function above, i.e.,

$\backslash log\{z\}\; =\; \backslash ln$ + i\left(\mathrm{arg}\ z\right).

Now, is intrinsically multivalued. One often defines the argument of some complex number to be between $-\backslash pi$ (exclusive) and $\backslash pi$ (inclusive), so we take this to be the principal value of the argument, and we write the argument function on this branch (with the leading capital A). Using instead of , we obtain the principal value of the logarithm, and we write

(2025). 9780763757724, Jones & Bartlett Learning. . ISBN 9780763757724

$\backslash mathrm\{pv\}\backslash log\{z\}\; =\; \backslash mathrm\{Log\}\backslash ,z\; =\; \backslash ln$ + i\left(\mathrm{Arg}\,z\right).

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Square root For a complex number $z\; =\; r\; e^\{i\; \backslash phi\}\backslash ,$ the principal value of the square root is:

$\backslash mathrm\{pv\}\backslash sqrt\{z\}\; =\; \backslash exp\backslash left(\backslash frac\{\backslash mathrm\{pv\}\backslash log\; z\}\{2\}\backslash right)\; =\; \backslash sqrt\{r\}\backslash ,\; e^\{i\; \backslash phi\; /\; 2\}$

with argument Sometimes a branch cut is introduced so that negative real numbers are not in the domain of the square root function and eliminating the possibility that $\backslash phi\; =\; \backslash pi.$

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Inverse trigonometric and inverse hyperbolic functions Inverse trigonometric functions (, , , etc.) and inverse hyperbolic functions (, , , etc.) can be defined in terms of logarithms and their principal values can be defined in terms of the principal values of the logarithm.

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Complex argument The principal value of complex number argument measured in can be defined as:

• values in the range $[0,\; 2\backslash pi)$
• values in the range $(-\backslash pi,\; \backslash pi]$

For example, many computing systems include an function. The value of will be in the interval $(-\backslash pi,\; \backslash pi].$ In comparison, is typically in $(\backslash tfrac\{-\backslash pi\}\{2\},\; \backslash tfrac\{\backslash pi\}\{2\}].$

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See also

• Principal branch
• Branch point

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