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

Mylo O'Moore

Mylo O'Moore

Answered question

2021-02-08

State Cauchy-Riemann equations. Show that f(z) x*+ iy

Answer & Explanation

mhalmantus

mhalmantus

Skilled2021-02-09Added 105 answers

Step 1
Cauchy-Riemann Equations:
A necessary condition that the function f=u+iv is differentiable at a point z0=x0+iy0 is that the partial derivatives ux,uy,vx,vy exists and ux=vy,uy=vx at the point (x0,y0)
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,
ux=1
uy=0
vx=0
vy=1
So, ux=vyanduy=vx 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 z0.
So, the given function is not analytic.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?