# FunctionDomain

FunctionDomain[f,x]

finds the largest domain of definition of the real function f of the variable x.

FunctionDomain[f,x,dom]

considers f to be a function with arguments and values in the domain dom.

FunctionDomain[funs,vars,dom]

finds the largest domain of definition of the mapping funs of the variables vars.

FunctionDomain[{funs,cons},vars,dom]

finds the domain of funs with the values of vars restricted by constraints cons.

# Details and Options • funs should be a list of functions of variables vars.
• Possible values for dom are Reals and Complexes. The default is Reals.
• If dom is Reals then all variables, parameters, constants, and function values are restricted to be real.
• cons can contain equations, inequalities, or logical combinations of these.
• The following options can be given:
•  GeneratedParameters C how to name parameters that are generated Method Automatic what method should be used WorkingPrecision Automatic precision to be used in computations

# Examples

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## Basic Examples(2)

Find the largest domain of definition of a real function:

The largest domain of definition of a complex function:

## Scope(4)

Real univariate functions:

Domain restricted by constraints:

Complex univariate functions:

Real multivariate functions:

Complex multivariate functions:

## Options(2)

### GeneratedParameters(1)

FunctionDomain may introduce new parameters to represent the domain:

Use GeneratedParameters to control how the parameters are generated:

### Method(1)

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:

## Applications(13)

### Basic Applications(6)

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:

### Solving Equations and Optimization(2)

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:

### Calculus(5)

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:

## Possible Issues(2)

All subexpressions of need to be real-valued for a point to belong to the real domain of :

Negative reals are not in the real domain of because is not real valued: is real valued for all real :

The real domain information for mathematical functions is accurate up to lower-dimensional sets:

There is no full-dimensional subset of the space on which HankelH1 is real valued:

Here is a 1-dimensional subset of the space on which HankelH1 is real valued:

FunctionDomain is unable to detect that is real valued: