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.

Logarithm function

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

Now, arg z is intrinsically multivalued. One often defines the argument of some complex number to be between -π (exclusive) and π (inclusive), 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 A). Using Arg z instead of arg z, we obtain the principal value of the logarithm, and we write

Exponential function

So far we have only considered the logarithm function. What about exponents?

Consider with . 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, i.e.,

Square root

For a complex number the principal value of the square root is:

with argument

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