107 777
Assignments Done
100%
Successfully Done
In July 2024

# Answer to Question #236275 in Chemical Engineering for Lok

Question #236275

13)Show that the function f (z)=(z̅ )²/z,z≠0 ;0, z=0 satiesfies Cauchy-Rieman equations at z=0.Does f'(0) exist? Explain the problem with step by step process?

1
2021-09-24T02:14:28-0400

Consider

"f(z)= |z|^2\\\\\ndet \\space \\Delta w = f(z + \\Delta z)- f(z)\\\\\n\\frac{ \\Delta w}{ \\Delta z}=\\frac{ |z+\\Delta z|^2-|z|^2}{ \\Delta z}\\\\\n\\frac{ \\Delta w}{ \\Delta z}=\\frac{ (z+\\Delta z)(\\bar{z}+\\bar{\\Delta z})-z \\bar{z}}{ \\Delta z}\\\\\n\\frac{ \\Delta w}{ \\Delta z}=\\bar{z}+\\bar{\\Delta z}+\\frac{ z\\bar{\\Delta z}}{ \\Delta z}\\\\"

When the horizontal line and vertical line approach "\\bar{\\Delta z}" towards the origin

"\\bar{\\Delta z}={\\Delta z}, \\bar{\\Delta z}=-{\\Delta z}"

Then

"\\frac{\\Delta w}{\\Delta z}=\\bar z+\\Delta z+z" for horizontal line

"\\frac{\\Delta w}{\\Delta z}=\\bar z-\\Delta z-z" for vertical line

as "\\Delta z \\to 0"

"\\bar z+\\Delta z=\\bar z-\\bar z" or z= 0

So f'(0) does not exist when "z \\not=0"

Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!