Principal values

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See also Cauchy principal value for its use in describing improper integrals

In considering complex multiple-valued functions in complex analysis, the principal values of a function are the values along one chosen branch of that function, so it is single-valued.

Motivation

Consider the complex logarithm function log z. It is defined as the complex number w such that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^w = z\,\!}

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^w = i\,\!}

for w. Clearly iπ/2 is a solution. But is it the only solution?

Of course, there are other solutions, which is evidenced by considering the position of i in the Argand plane and thus its argument. We can rotate anticlockwise π/2 radians from 1 to reach i initially, but if we rotate further another 2π we reach i again>. So, we can conclude that i(π/2 + 2π) is also a solution for log i. It becomes clear that we can add any multiple of 2πi to our initial solution to obtain all values for log i.

But this has a consequence that may be surprising in comparison of real valued functions - log i does not have one definite value! For log z, we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \log{z} = \ln{|z|} + i\left(\mathrm{arg}\ z+2\pi k\right)}

for some integer k. Each value of k determines what is known a branch (or sheet), where a multiple-valued function is single-valued.

For simplicity, the branch corresponding to k=0 is known as the principal branch, and along this branch, the values the function takes are known as the principal values.

General case

In general, if f(z) is multiple-valued, the principal branch of f is denoted

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such that for z in the domain of f, f(z) is single-valued.

Principal values of standard functions

Complex valued elementary functions can be multiple valued over some domains. Determining 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.

We have examined the logarithm function above, ie.,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \log{z} = \ln{|z|} + i\left(\mathrm{arg}\ z\right).}

Now, arg z is intrinsically multivalued. One often defines the argument of some complex number to be between -π and π, so we take this to be the principal value of the argument, and we write the argument function on this branch Arg z (with the leading capital). Using Arg z instead of arg z, it should be clear that we obtain the principal value of the logarithm, and we write

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{pv}\ \log{z} = \mathrm{Log}\ z = \ln{|z|} + i\left(\mathrm{Arg}\ z\right).}

So far we have only considered the logarithm function. What about exponents? Consider zα, with α in C. One usually defines zα to be eα log z. Yet eα log z is multiple-valued since we are using log as opposed to Log. Using Log we obtain the principal value of zα, ie.,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{pv}\ z^\alpha = e^{\alpha \mathrm{Log}\ z}.}