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

The correct answer is:

$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$

Step-by-Step Explanation:

1. Definition of analyticity:
A complex-valued function $f\left(z\right)=u(x,y)+iv(x,y)$ defined on a domain in the complex plane is analytic (or holomorphic) if it is complex-differentiable at every point in that domain.

2. Differentiability condition:
The derivative of $f\left(z\right)$ with respect to $z=x+iy$ is defined as:
$f\text{'}\left(z\right)=\underset{\Delta z\to 0}{lim}\frac{f(z+\Delta z)-f\left(z\right)}{\Delta z}$
For the derivative to exist, the limit must be the same regardless of the path along which $\Delta z=\Delta x+i\Delta y$ approaches 0.

3. Approaching along the real axis ($\Delta y=0$):
If we approach the limit horizontally so that $\Delta z=\Delta x$, the derivative is:
$f\text{'}\left(z\right)=\frac{\partial f}{\partial x}=\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}$

4. Approaching along the imaginary axis ($\Delta x=0$):
If we approach the limit vertically so that $\Delta z=i\Delta y$, the derivative is:
$f\text{'}\left(z\right)=\frac{\partial f}{\partial \left(iy\right)}=\frac{1}{i}\frac{\partial f}{\partial y}=-i\left(\frac{\partial u}{\partial y}+i\frac{\partial v}{\partial y}\right)=\frac{\partial v}{\partial y}-i\frac{\partial u}{\partial y}$

5. Equating the real and imaginary parts:
For $f\text{'}\left(z\right)$ to be uniquely defined, the expressions obtained from both paths must be equal:
$\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}=\frac{\partial v}{\partial y}-i\frac{\partial u}{\partial y}$
Equating the real parts gives:
$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$
Equating the imaginary parts gives:
$\frac{\partial v}{\partial x}=-\frac{\partial u}{\partial y}\Rightarrow \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$
These are the fundamental partial differential equations known as the Cauchy-Riemann equations, which are represented in the second option (image_1.png).