tests whether is an analytic function for x∈Reals.
tests whether is an analytic function for x∈dom.
tests whether are analytic functions for x1,x2,…∈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∈ℛ there is an and a sequence such that for all , .
- A function is analytic in an open set if for all there is an and a sequence such that for all , .
- 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 $Assumptions assumptions on parameters GenerateConditions True whether to generate conditions on parameters PerformanceGoal $PerformanceGoal whether to prioritize speed or quality
- Possible settings for GenerateConditions include:
Automatic nongeneric conditions only True all conditions False no conditions None return unevaluated if conditions are needed
- Possible settings for PerformanceGoal are "Speed" and "Quality".
Examplesopen allclose all
Basic Examples (4)
FunctionAnalytic cannot find the answer for arbitrary values of the parameter :
With the assumption that , FunctionAnalytic succeeds:
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:
Use PerformanceGoal to avoid potentially expensive computations:
The default setting uses all available techniques to try to produce a result:
Classes of Analytic Functions (6)
Sin, Cos and Exp are analytic:
These functions are analytic in the complex plane as well:
Visualize these functions over :
Functions that are analytic in the plane are called entire functions and can be considered infinite-degree polynomials:
No discontinuous function is analytic:
Visualize some of the preceding 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 :
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:
But as the zeros of Cosh and Sinh lie on the imaginary axis, more exclusions are needed for analyticity over :
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:
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:
However, its first derivative is not continuous:
While goes smoothly to zero, its derivative oscillates wildly at the origin:
Properties & Relations (7)
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 is excluded 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:
Sum of residues of an analytic function is zero:
Use ResidueSum to verify this property:
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:
Wolfram Research (2020), FunctionAnalytic, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionAnalytic.html.
Wolfram Language. 2020. "FunctionAnalytic." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FunctionAnalytic.html.
Wolfram Language. (2020). FunctionAnalytic. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FunctionAnalytic.html