Show f(z) = √|xy| is not analytic at origin although the C-R equations are satisfied at that point.

Complex Analysis Theorem from Analytic function
Statement/Theorem /Prove that :-

Show that the function f(z) = sqrt(|xy|) or √|xy| is not analytic at the origin although the Cauchy-Riemann equations are satisfied at that point.

Solution.
Let f(z) = u(x, y) + iv(x, y) =√|xy|
Here

u(x,y)= √|xy| and v(x,y)=0

At the origin zₒ=(0,0),
(∂u/∂x) = lim ₓ→ₒ {u(x, 0) - u(0, 0)}/x
⇒ (∂u/∂x)= (0 - 0)/x
⇒(∂u/∂x) = 0

and

(∂v/∂y) = lim ᵧ→ₒ {v(0, y) - v(0, 0)}/y
⇒(∂v/∂y)= (0 - 0)/y
⇒(∂v/∂y) = 0

Here ,
∵(∂u/∂x) = 0 and (∂v/∂y) = 0
∴(∂u/∂x) = (∂v/∂y)
Hence Cauchy-Riemann equations are satisfied at the origin zₒ=(0,0).

Now by property of differentiability...

f'(0)=lim z→o {f(z)-f(0)}/z
⇒f'(0)=lim z→o {√|xy|-0}/(x+iy)
⇒f'(0)=lim z→o √|xy|/(x+iy)
Now let a line y=mx, z→o we get

⇒f'(0)=lim z→o √|mx²|/(x+imx)
Simplify
⇒f'(0)=lim z→o √|m|/(1+im)
Here f'(0) is dependent on slop m thus the limit is not unique

∴ differentiability is not exist so that f(z) is not Analytic function at the origin zₒ=(0,0).

Hence Proved...!!!
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