FunctionDomain may introduce new parameters to represent the domain:

Use GeneratedParameters to control how the parameters are generated:

By default, domains of real univariate functions are given in a reduced form:

Domains of other functions are not reduced:

Use Method to specify whether the domain should be given in a reduced form:

Compute the real domain of :

The imaginary part of the function is zero on the real domain:

Compute the real domain of :

The complement of the domain is the open disk of radius 3, centered at :

Outside the real domain, a function may be complex, singular or undefined:

Outside the real domain, the function is complex valued:

Negative integers are not in the domain of :

The function has pole singularities at negative integers:

is undefined outside its real domain:

Compute the complex domain of :

has pole singularities at points that do not belong to the domain:

Compute the complex domain of :

has an essential singularity at zero:

Compute the complex domain of :

is undefined outside the domain:

Solve over the reals:

The solutions must belong to the real domain of :

The plot of shows that there is one solution:

Solve automatically uses the domain information and finds the solution:

Find the global minimum of :

The minimum must belong to the real domain of :

Find the roots of in the interior of the real domain:

Select the root at which the value of is minimal:

Check that the value of at is less than the values of at the domain endpoints:

Visualize the result:

Minimize automatically uses the domain information and finds the minimum:

If the limit of a function over points from its real domain exists, it must be a real number or a real infinity:

Use Limit to verify that the limits of at along real directions are real infinities:

If the integral over a subset of the real domain exists, it is a real number or a real infinity:

Use Integrate to compute the integral of :

Verify that the integral is indeed a real number:

If the derivative of a function at a point in its real domain exists, it is real valued:

Compute the derivative:

The derivative is indeed real valued over the domain of :

Check real analyticity of :

A function has to be defined and real valued in order to be real analytic:

Over its real domain, is real analytic:

Check complex analyticity of :

A function has to be defined to be complex analytic:

Over its domain, is complex analytic: