91Ó°ÊÓ

In complex analysis, an **analytic function** is a beautiful concept that extends the idea of differentiability from real-valued functions to complex-valued functions. For a complex function to be analytic at a point, it must not only be differentiable at that point but also in a neighborhood surrounding it.
An analytic function can be infinitely differentiated and often expressed as a power series. This capability allows it to be approximated very accurately and studied thoroughly.

• A function is called holomorphic if it is analytic at every point in its domain.
• Analytic functions display properties such as conformality, meaning they preserve angles between curves.

To check if a function like the one given, \( f(z) = 4z - 6\bar{z} + 3 \), is analytic, we use the Cauchy-Riemann equations. If these equations don't hold, the function isn't analytic at any point, as demonstrated in the sample problem.

The **Cauchy-Riemann equations** are pivotal in determining whether a complex function is differentiable and therefore analytic. They provide a set of conditions that must be satisfied.
For a function \( f(z) = u(x, y) + iv(x, y) \), where \( z = x + iy \), these equations are:

• \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \)
• \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \)

To be analytic at a point, both equations need to be satisfied simultaneously at that point and in a surrounding region. If any one fails, the function is non-analytic there.
This condition is why we derived the partial derivatives in the exercise. The exercise calculated them, showing that the equation \( -2 = 10 \) does not satisfy the Cauchy-Riemann condition. Therefore, demonstrating why the function is not analytic.

In complex analysis, **complex differentiation** extends the notion of differentiating functions of a real variable into the realm of complex numbers. Unlike real differentiation, complex differentiation depends heavily on the function being analytic.
To perform the differentiation, high emphasis is placed on:

• Ensuring that the function is locally linear, which essentially means, at small scales, it behaves like its linear approximation.
• Verifying compliance with the Cauchy-Riemann equations, as previously discussed.

The existence of a complex derivative at a point implies differentiability in a neighborhood around that point, making this definition stricter than in real analysis.
This was illustrated by the given function, \( f(z) = 4z - 6\bar{z} + 3 \), which failed the complex differentiation test due to not satisfying the Cauchy-Riemann equations.