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 Chow to name parameters that are generated
    Method Automaticwhat method should be used
    WorkingPrecisionAutomaticprecision 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 TemplateBox[{{x, -, 1}}, AlternatingFactorial]:

The function has pole singularities at negative integers:

TemplateBox[{3, 2, x}, EllipticTheta] 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 TemplateBox[{1, 2, x}, EllipticTheta]:

TemplateBox[{1, 2, x}, EllipticTheta] 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 TemplateBox[{6, {-, 1}, x}, LegendreP3]:

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 TemplateBox[{x}, Gamma] 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 TemplateBox[{x, y}, BesselK]:

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

Over its real domain, TemplateBox[{x, y}, BesselK] is real analytic:

Check complex analyticity of TemplateBox[{x, 4, 4, 1}, Hypergeometric2F1]:

A function has to be defined to be complex analytic:

Over its domain, TemplateBox[{x, 4, 4, 1}, Hypergeometric2F1] 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:

Wolfram Research (2014), FunctionDomain, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionDomain.html.

Text

Wolfram Research (2014), FunctionDomain, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionDomain.html.

CMS

Wolfram Language. 2014. "FunctionDomain." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FunctionDomain.html.

APA

Wolfram Language. (2014). FunctionDomain. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FunctionDomain.html

BibTeX

@misc{reference.wolfram_2025_functiondomain, author="Wolfram Research", title="{FunctionDomain}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/FunctionDomain.html}", note=[Accessed: 17-July-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_functiondomain, organization={Wolfram Research}, title={FunctionDomain}, year={2014}, url={https://reference.wolfram.com/language/ref/FunctionDomain.html}, note=[Accessed: 17-July-2025 ]}