Overview of Complex Function

A complex function is a function that converts complex numbers to complex numbers. In other words, it is a function that has a subset of the complex numbers as a domain and the complex numbers as a codomain. Complex functions are commonly assumed to have a domain that covers a nonempty open subset of the complex plane.

Complex functions are examined in the framework of Complex analysis.  Complex analysis is an area of mathematics concerned with complex numbers, functions, and calculus. In a nutshell, complex analysis is an extension of real-number calculus to the complex domain. We will apply the calculus concepts of continuity, derivatives, and integrals to the case of complex functions of a complex variable. We’ll come across analytic functions along the way, which will be the focus of this introduction. In reality, complex analysis is, to a considerable degree, the study of analytic functions.

Complex function

A complex function is one that relates complex numbers to complex numbers. In other terms, it is a function with a domain of complex numbers and a codomain of complex numbers. In general, complex functions are assumed to have a domain that contains a nonempty open subset of the complex plane.

For any complex function, the domain values z and their pictures f(z) in the range can be divided into real and imaginary parts.

A complex variable’s complex function z can be divided into two functions, as in 

f(z)=u(z)+iv(z) or f(x,y)=u(x,y)+iv(x,y) (x,y).

Because such functions are clearly reliant on two independent variables and have two separable functions, visualizing the function would necessitate a four-dimensional space, which is difficult to conceive.

Some complex-valued function qualities (such as continuity) are just the corresponding properties of vector-valued functions with two real variables.

Other complex analysis notions, such as differentiability, are straightforward generalizations of related real-function concepts but may have completely different features.

Every differentiable complex function, in particular, is analytic, and two differentiable functions that are equal in a neighborhood of a point are equal on the intersection of their domains if the domains are connected.

The latter condition is the foundation of the analytic continuation principle, which permits extending every real analytic function in a unique fashion to obtain a complex analytic function whose domain is the entire complex plane with a finite number of curve arcs removed.

Determining complex function

Assume S is a collection of complex numbers. A function f defined on S is a rule that allocates a complex number w to each z in S. The number w is known as the value of f at z and is indicated by the symbol f(z); that is, w=f (z). The domain of definition of f is denoted by the set S.

If only one value of w corresponds to each value of z, we call w a single-valued function of z or f(z). If more than one value of w corresponds to each value of z, w is referred to as a multiple-valued or many-valued function of z.

A multiple-valued function can be thought of as a collection of single-valued functions, each of which is referred to as a function branch. In general, we regard one particular member to be the multiple-valued function’s principal branch, and the value of the function corresponding to this branch to be the principal value.

Examples:

The function w=z2 is a z function with a single value. If, on the other hand, w=z1/2, then there are two values of w for each value of z. As a result, the function

Solution:

w=z1/2  is a multiple-valued (two-valued) function of z.

Assume w=u+iv is the value of a function f at z=x+iy, thus

u+iv=f(x+iy)

Because each of the real numbers u and v is dependent on the real variables x and y, f(z) can be represented in terms of a pair of real-valued functions of x and y:

      f(z)=u(x,y)+iv(x,y)

 If the polar coordinates r and   are used instead of x and y, then

u+iv=f(reiθ)

where w=u+iv and z= reiθ In this instance, we write

f(z)=u(r,θ)+iv(r,θ)

Conclusion

In this article we conclude that Complex functions are functions with complex variables that include both real and imaginary numbers. Complex functions are investigated in the context of Complex analysis. Complex analysis is the field of mathematics that studies holomorphic functions, that is, functions that are defined in some region of the complex plane, have complex values, and may be differentiated as complex functions.