FunctionAnalytic

FunctionAnalytic[f,x]

tests whether is an analytic function for xReals.

FunctionAnalytic[f,x,dom]

tests whether is an analytic function for xdom.

FunctionAnalytic[{f1,f2,},{x1,x2,},dom]

tests whether are analytic functions for x1,x2,dom.

FunctionAnalytic[{funs,cons},xvars,dom]

tests whether are analytic functions for xvars in an open set containing the solutions of the constraints cons over the domain dom.

Details and Options

  • Complex analytic functions are also known as holomorphic functions.
  • A function is analytic in an open set if for all y there is an and a sequence such that for all TemplateBox[{{x, -, y}}, Abs]<r(y), we have .
  • A function is analytic in an open set if for all there is an and a sequence such that for all TemplateBox[{{{, {{{x, _, 1}, -, {y, _, 1}}, ,, ..., ,, {{x, _, n}, -, {y, _, n}}}, }}}, Norm]<r(y_1,...,y_n), we have f(x_1,...,x_n)=sum_(k in TemplateBox[{}, NonNegativeIntegers]^n)a_k(y_1,...,y_n)(x_1-y_1)^(k_1) ... (x_n-y_n)^(k_n) .
  • If funs contains parameters other than xvars, the result is typically a ConditionalExpression.
  • 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 inequalities or logical combinations of these.
  • The following options can be given:
  • Assumptions$Assumptionsassumptions on parameters
    GenerateConditionsTruewhether to generate conditions on parameters
    PerformanceGoal$PerformanceGoalwhether to prioritize speed or quality
  • Possible settings for GenerateConditions include:
  • Automaticnon-generic conditions only
    Trueall conditions
    Falseno conditions
    Nonereturn unevaluated if conditions are needed
  • Possible settings for PerformanceGoal are "Speed" and "Quality".

Examples

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Basic Examples  (4)

Test analyticity of real functions:

Test analyticity of complex functions:

Test analyticity over restricted domains:

Test analyticity of multivariate functions:

Scope  (6)

Real univariate functions:

Complex univariate functions:

Functions with restricted domains:

Real multivariate functions:

Complex multivariate functions:

Functions with symbolic parameters:

Options  (4)

Assumptions  (1)

FunctionAnalytic cannot find the answer for arbitrary values of the parameter :

With the assumption that , FunctionAnalytic succeeds:

GenerateConditions  (2)

By default, FunctionAnalytic may generate conditions on symbolic parameters:

With GenerateConditions->None, FunctionAnalytic fails instead of giving a conditional result:

This returns a conditionally valid result without stating the condition:

By default, all conditions are reported:

With GenerateConditions->Automatic, conditions that are generically true are not reported:

PerformanceGoal  (1)

Use PerformanceGoal to avoid potentially expensive computations:

The default setting uses all available techniques to try to produce a result:

Applications  (11)

Classes of Analytic Functions  (6)

Polynomials are analytic:

Sin, Cos and Exp are analytic:

Visualize these functions:

These functions are analytic in the complex plane as well:

Visualize these functions over :

Functions that are analytic in plane are called entire functions and can be considered infinite-degree polynomials:

No discontinuous function is analytic:

Visualize some of the above functions:

Some continuous functions are not analytic, like the real absolute value function RealAbs:

The problem with RealAbs is the "kink" at the origin:

The complex absolute value function Abs is analytic nowhere in the complex plane:

It suffers from a different problem than RealAbs, namely it is nowhere differentiable:

The reciprocal of an analytic function is analytic wherever :

Thus, rational functions may or may not be continuous over the reals:

However, as every nonconstant polynomial has a root in the plane, rational functions are never analytic on TemplateBox[{}, Complexes]:

Instead, rational functions are the prototype of the larger class of meromorphic functions in the complex plane:

Visualizing the function in the complex plane shows the blowup at :

As Cot and Csc are rational functions of Sin and Cos, they are analytic when sine is nonzero:

Visualize the functions along with sine:

Since sine's only zeros are on the real line, this means the Cot and Csc are analytic except on the multiples of :

Similarly, Tan and Sec are continuous when cosine is nonzero:

This same principle applies to the hyperbolic trigonometric functions Coth and Csch:

As well as Tanh and Sech:

But as the zeros of Cosh and Sinh lie on the imaginary axis, more exclusions are needed for analyticity over TemplateBox[{}, Complexes]:

Plots of the hyperbolic functions are just rotations by along with a phase shift by the same amount:

The compositions of analytic functions are analytic:

Multivariate polynomials are analytic over the reals and complexes:

So are multivariate polynomials in entire functions:

Rational multivariate functions may or may not be analytic over the reals:

They are always nonanalytic over the complexes:

By composing with analytic univariate functions, many more analytic functions can be generated:

Visualize the analytic functions:

Calculus  (5)

Analytic functions can be represented by power series:

In this case, the sum converges for all values of :

Consider the following function:

It is analytic on the open disk of radius 5:

Therefore it can be expressed as a power series about any point in the disk, say :

However, this sum does not converge for all values of :

Substituting a value of outside this range produces a divergent sum:

Visualize the function along with the domains of analyticity and convergence:

The integral of an analytic function around a close contour is zero:

The following integral is nonzero, so Log cannot be analytic:

Visualize the functions and the contour:

If and are analytic in a region in the complex plane, only has simple zeros and is nonzero. A sum over the zeros of can be computed as . Consider :

This function is analytic in a disk of radius 4:

Let . It is analytic and nonzero in the disk of radius 4:

The function has two simple roots within the disk, at and :

Thus, the sum is easy to compute:

The integral gives the same answer:

Differential equations with analytic coefficients have solutions that are analytic at most points, which makes series solutions a viable method of approach. Consider the following differential equation:

There is no closed-form solution to this equation:

However, all coefficients are analytic:

Thus, a series solution can be found using AsymptoticDSolveValue:

The function is continuous:

However, its first derivative is not continuous:

Therefore is not analytic:

While goes smoothly to zero, its derivative oscillates wildly at the origin:

Visualize and its first derivative:

Properties & Relations  (6)

An analytic function is differentiable arbitrarily many times:

Use D to compute derivatives:

An analytic function can be expressed as a Taylor series at each point of its domain:

Use Series to compute initial terms of Taylor series:

The resulting polynomial approximates near :

Zeros of an analytic function cannot have an accumulation point in the domain:

Zeros of have an accumulation point at :

is continuous, but not analytic:

is analytic if we exclude from the domain:

An analytic function can have only finitely many zeros in a closed and bounded region:

Use Solve to find the roots of in the unit disk:

Use FunctionContinuous to check whether a function is continuous:

Continuous functions may not be analytic:

Analytic functions are continuous:

Use FunctionMeromorphic to check whether a function is meromorphic:

Meromorphic functions may not be complex analytic:

A quotient of complex analytic functions is meromorphic:

Possible Issues  (3)

A function needs to be defined everywhere to be analytic:

A function needs to be real-valued to be analytic over the real domain:

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 :

Wolfram Research (2020), FunctionAnalytic, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionAnalytic.html.

Text

Wolfram Research (2020), FunctionAnalytic, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionAnalytic.html.

BibTeX

@misc{reference.wolfram_2021_functionanalytic, author="Wolfram Research", title="{FunctionAnalytic}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/FunctionAnalytic.html}", note=[Accessed: 28-July-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_functionanalytic, organization={Wolfram Research}, title={FunctionAnalytic}, year={2020}, url={https://reference.wolfram.com/language/ref/FunctionAnalytic.html}, note=[Accessed: 28-July-2021 ]}

CMS

Wolfram Language. 2020. "FunctionAnalytic." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FunctionAnalytic.html.

APA

Wolfram Language. (2020). FunctionAnalytic. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FunctionAnalytic.html