Definition:Argument of Complex Number/Principal Argument

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Let $R$ be the principal range of the complex numbers $\C$.

The unique value of $\theta$ in $R$ is known as the principal value of the argument, or just principal argument, of $z$.

This is denoted $\Arg z$.

Note the capital $A$.

The standard practice is for $R$ to be $\hointl {-\pi} \pi$.

This ensures that the principal argument is continuous on the real axis for positive numbers.

Thus, if $z$ is represented in the complex plane, the principal argument $\Arg z$ is intuitively defined as the angle which $z$ yields with the real ($y = 0$) axis.

Also known as

Some sources refer to the principal argument as just the principal value.

Some sources use the term principal phase.

Linguistic Note

The word principal is (except in the context of economics) an adjective which means main.

Do not confuse with the word principle, which is a noun.


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