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.

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.