Evaluate numerically:

Complex number inputs:

Evaluate to high precision:

For real inputs, the result is exact:

For complex inputs, the precision of the output tracks the precision of the input:

Evaluate efficiently at high precision:

Compute the elementwise values of an array using automatic threading:

Or compute the matrix Arg function using MatrixFunction :

Arg can be used with Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around :

Values of Arg at fixed points:

Value at zero:

Values at infinity:

Exact inputs:

Evaluate for complex exponentials:

Find a value of x for which the Arg [I x]=π/2:

Visualize the result:

Plot the on the real axis:

Plot on the reals:

Plot over the complex plane:

Visualize Arg in three dimensions:

Use Arg to specify regions of the complex plane:

Arg is defined for all real and complex inputs:

Function range of Arg for real inputs:

Except on the negative reals, arg(TemplateBox[{z}, Conjugate])=-arg(z):

Arg is not a differentiable function:

The difference quotient does not have a limit in the complex plane:

There is only a limit in certain directions, for example, the real direction:

Use ComplexExpand to get differentiable expressions for real-valued variables:

Arg is not an analytic function:

It has both singularities and discontinuities:

Over the complex plane, it is singular everywhere and discontinuous on the non-positive reals:

Arg is nonincreasing:

Arg is not injective:

Arg is not surjective:

Arg is non-negative:

Arg is neither convex nor concave:

TraditionalForm formatting:

Expand assuming real variables x and y:

Simplify Abs using appropriate assumptions:

Express a non-zero complex number in term of its Arg and Abs :

is equal to :

Except for , exp(ⅈ arg(z))=TemplateBox[{z}, Sign]):