Principal value

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:$e^w\; =\; z,!$Now, for example, say we wish to find log i. This means we want to solve:$e^w\; =\; i,!$ 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: $log\{z\}\; =\; ln\; +\; ileft(mathrm\{Arg\}\; z\; ight).$

So far we have only considered the logarithm function. What about exponents? Consider "z"^α, with α in C. One usually defines "z"^α to be "e"^{α log "z"}. Yet "e"^{α log "z"} is multiple-valued since we are using log as opposed to Log. Using Log we obtain the principal value of "z"^α, ie.,: $mathrm\{pv\}\; z^alpha\; =\; e^\{alpha\; mathrm\{Log\}\; z\}.$

ee also

*Principal value of square roots of negative and complex numbers
*Principal branch
*Branch point