Domain of complex functions

[Guest]

how to find the domain, codomain (i.e. target or range) of a complex function?

[Guest]

Just like for real valued functions of a real variable, the domain of a function of a complex variable is the set of all "x" such that f(x) is defined while the range is the set of all y such that y= f(x) for some x in the domain.

For more than that you will have to specify the function.

[GeradHum]

Similar principles are followed to find the domain and codomain of a complex function as for a real function. Here is a guide to determining the domain and codomain of a complex function:

Domain:
The domain of a complex function is made up of the complex values for which the function is defined. To determine the domain, you must take into account any constraints that may exist on the function.
a) Algebraic restrictions: Check if there are operations that are not defined for certain complex values. For example, if the function contains a square root, you must ensure that the root argument is not negative, since the square roots of negative numbers are not defined on the set of complex numbers.

b) Denominator restrictions: Make sure that the denominators of the fractions in the function are not zero. If a denominator becomes zero, the function will not be defined at that point.

In general, the domain of a complex function is a set of points in the complex plane that excludes those values that generate algebraic problems or violate the constraints of the function.

Codomain or range:
The codomain or range of a complex function is the set of complex values to which the function maps the elements of its domain. The codomain can be determined by analyzing the shape and properties of the particular function.
If the function is explicitly defined in terms of a formula, you can examine the possible results for different domain values. If the function is the result of a composition of functions, you can analyze the ranges of the component functions to determine the range of the entire function.

In some cases, the codomain can be the set of all complex numbers, that is, any complex number can be the result of the function. In other cases, the range may be restricted due to specific properties of the function.

Remember that domain and codomain are different concepts: domain refers to valid input values, while codomain refers to possible output values.

[Guest]

Domain: Every z in C where f(z) is specified.

Denominator ≠ 0

Log or √ → do not include undefined points or branch cuttings.

Codomain: Generally C (unless otherwise noted).

Range: The real set of values that f(z) takes.

Replace z=x+iy → with to find all feasible
 f(z)