An analytic function f (z) of complex variable z = x + iy may be written as f (z) = u (x, y) + iv (x, y). Then, u (x, y) and v (x, y) must satisfy

An analytic function is a fundamental concept in complex analysis. A complex function $f(z)$ of a complex variable $z=x+iy$ is considered analytic at a point if it is differentiable not only at that point but also in some neighborhood around that point. When an analytic function $f(z)$ is expressed in terms of its real and imaginary parts, $f(z)=u(x,y)+iv(x,y)$, where $u(x,y)$ is the real part and $v(x,y)$ is the imaginary part, these two real-valued functions must satisfy specific conditions known as the Cauchy-Riemann equations.

Cauchy-Riemann Equations for Analytic Functions

For a complex function $f(z)=u(x,y)+iv(x,y)$ to be analytic at a point $z=x+iy$, the first-order partial derivatives of $u$ and $v$ with respect to $x$ and $y$ must exist and satisfy the following two conditions. These conditions are known as the Cauchy-Riemann equations and are essential for defining the analyticity of a function.

- The first Cauchy-Riemann equation states that the partial derivative of the real part $u$ with respect to $x$ must be equal to the partial derivative of the imaginary part $v$ with respect to $y$. This can be written as: $$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$$
- The second Cauchy-Riemann equation states that the partial derivative of the real part $u$ with respect to $y$ must be equal to the negative of the partial derivative of the imaginary part $v$ with respect to $x$. This can be written as: $$\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$$

These two equations are the necessary conditions for a function $f(z)$ to be analytic. If, in addition, these partial derivatives are continuous in a region, then these conditions become sufficient for the analyticity of $f(z)$ in that region.

Partial Derivatives and Analyticity Conditions

The Cauchy-Riemann equations provide a crucial link between the real and imaginary parts of an analytic function. They ensure that the derivative of $f(z)$ at any point is unique, regardless of the direction from which $z$ approaches that point. This consistency in the derivative is a hallmark of analytic functions.

Identifying the Correct Cauchy-Riemann Conditions

We need to find the option that correctly represents the Cauchy-Riemann equations. Let's compare the standard equations with the given choices:

- Standard Cauchy-Riemann Equations: $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$ $\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$

Now, let's examine the provided options:By comparing the standard Cauchy-Riemann equations with the options, it is clear that Option 2 accurately represents the necessary conditions that $u(x,y)$ and $v(x,y)$ must satisfy for the function $f(z)$ to be analytic.