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)
By default, FunctionAnalytic may generate conditions on symbolic parameters:
Use PerformanceGoal to avoid potentially expensive computations:
Classes of Analytic Functions (6)
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 following integral is nonzero, so Log cannot be analytic:
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:
Thus, a series solution can be found using AsymptoticDSolveValue:
Properties & Relations (7)
Use D to compute derivatives:
Use Series to compute initial terms of Taylor series:
Use Solve to find the roots of in the unit disk:
Use FunctionContinuous to check whether a function is continuous:
Use FunctionMeromorphic to check whether a function is meromorphic:
Use ResidueSum to verify this property:
Possible Issues (3)
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