Understanding Analytic Functions in Complex Analysis

In the domain of complex analysis, the concept of an analytic function is of fundamental importance. An analytic function is a function that is locally given by a convergent power series. This means that for every point in its domain, the function is differentiable and can be represented as a power series in a neighborhood of that point. The key to understanding when a function is analytic lies in checking whether it satisfies the Cauchy-Riemann equations, a necessary and sufficient condition for a complex function to be analytic.

Polynomial Function: Z^2

The function Z^2 (where Z is a complex variable) is a polynomial function. Polynomial functions have a special property: they are analytic everywhere in the complex plane. This means that no matter where you are in the complex plane, Z^2 will be differentiable and can be represented as a power series. Therefore, Z^2 is an analytic function.

The Square of Modulus: Z^2 Z * Overline{Z}

The function Z^2 Z * Overline{Z} (the square of the modulus of Z) is another case to consider. This function does not conform to the properties of analytic functions because it depends on both Z and its complex conjugate Overline{Z}. The Cauchy-Riemann equations, which must be satisfied by an analytic function, cannot be met here because the derivative of Z * Overline{Z} with respect to Overline{Z} is not equal to the derivative with respect to Z. Therefore, Z^2 Z * Overline{Z} is not an analytic function.

The Square Root Function: √Z

The function √Z (the square root of Z) is also subject to analysis. While the square root is not a polynomial, it is analytic in its domain. However, to avoid branches, the square root function is typically defined with a specific domain, such as the complex plane minus the negative real axis. Within this domain, √Z meets the criteria for being an analytic function, meaning it is differentiable at every point in its domain. Thus, √Z is analytic in its specified domain.

The Complex Conjugate: Z^ Conjugate

The function Z^ Conjugate (denoted as Overline{Z}) is not analytic because it depends on both Z and its complex conjugate Overline{Z}. Similar to the previous case, the Cauchy-Riemann equations cannot be satisfied for Z^ Conjugate. Therefore, it is not an analytic function.

Conclusion

Among the functions discussed, Z^2 and √Z are the analytic functions, with √Z being analytic in a specific domain. The polynomials Z^2 and the function Z^ Conjugate are not analytic functions due to their dependence on the complex conjugate rather than just the variable itself.

The definition of an analytic function in complex analysis requires that a function is differentiable at all points z x iy in a domain D. For a function to be differentiable, the limit of the difference quotient must exist, meaning that the following limit must be the same from all directions in the complex plane:

(f(z h) - f(z)) / h as h s it → 0.

For example, consider the function Z^2. The difference quotient is:

(Z h)^2 - Z^2) / h 2Zh as h → 0.

Similarly, for the square root function:

(√(Z h) - √Z) / h 1 / [2√Z] as h → 0, provided Z is not 0.

The function Z^2 can be shown to have no derivative unless Z 0, as:

(Z h)^2 - Z^2) / h Z h as h → 0, from the real direction. (Z - ih)^2 - Z^2) / h -Z - h as h → 0, from the imaginary direction.

This lack of a consistent limit from different directions indicates that Z^2 is not analytic.

In summary, the distinction between analytic and non-analytic functions in complex analysis is crucial for understanding the behavior and properties of functions in the complex plane.