Question

# State Cauchy-Riemann equations. Show that f(z) x*+ iy' is not analytic anywhere but the Cauchy-Riemann equations are satisfied at the origin.

Equations
State Cauchy-Riemann equations. Show that f(z) x*+ iy' is not analytic anywhere but the Cauchy-Riemann equations are satisfied at the origin.

2021-02-09
Step 1
Cauchy-Riemann Equations:
A necessary condition that the function f=u+iv is differentiable at a point $$\displaystyle{z}_{{0}}={x}_{{0}}+{i}{y}_{{0}}$$ is that the partial derivatives $$\displaystyle{u}_{{x}},{u}_{{y}},{v}_{{x}},{v}_{{y}}$$ exists and $$\displaystyle{u}_{{x}}={v}_{{y}},{u}_{{y}}=−{v}_{{x}}$$ at the point $$\displaystyle{\left({x}_{{0}},{y}_{{0}}\right)}$$
Step 2
Given equation is f=x+iy
Step 3
Let, f(z)=u(x,y)+iv(x,y)
Comparing,
u(x,y)=x
v(x,y)=y
Step 4
Then,
$$\displaystyle{u}_{{x}}={1}$$
$$\displaystyle{u}_{{y}}={0}$$
$$\displaystyle{v}_{{x}}={0}$$
$$\displaystyle{v}_{{y}}={1}$$
So, $$\displaystyle{u}_{{x}}={v}_{{y}}{\quad\text{and}\quad}{u}_{{y}}=−{v}_{{x}}$$ at (0,0).
So, Cauchy-Riemann equations are satisfied at the origin.
Step 5
But the Cauchy-Riemann equations are satisfied only at the point z=0.
Hence, f(z)=x+iy can not have derivative at any point $$\displaystyle{z}\ne{0}$$.
So, the given function is not analytic.