How to Determine Complex Differentiability and Holomorphicity of a Function?

[cummings12332]

Homework Statement

f(z)=z(bar(z))^2+2(bar(z))z^2 ,then calculate the total differential of f viewed as a map from R^2->R^2 . determine the points at which f is complex differentiable , is f holomorhpic anywhere?

2. The attempt at a solution
i did the first part and for secund part i use the Jocobian ,for if it is differentiable then if follow the cachy rieman equations which is 9x^2+3y^2=x^2+3y^2, and 6xy=-2xy the solution for the equation systems is x=o y is real , so it can be differentiable at (o,y) y is real number ,does it right? if it is ,then How should i found holomorphic anywhere?

 

[Dick]

Homework Statement

f(z)=z(bar(z))^2+2(bar(z))z^2 ,then calculate the total differential of f viewed as a map from R^2->R^2 . determine the points at which f is complex differentiable , is f holomorhpic anywhere?

2. The attempt at a solution
i did the first part and for secund part i use the Jocobian ,for if it is differentiable then if follow the cachy rieman equations which is 9x^2+3y^2=x^2+3y^2, and 6xy=-2xy the solution for the equation systems is x=o y is real , so it can be differentiable at (o,y) y is real number ,does it right? if it is ,then How should i found holomorphic anywhere?

That looks ok for differentiable. What's the definition of 'holomorphic'?

 

[cummings12332]

That looks ok for differentiable. What's the definition of 'holomorphic'?

f:k->C is holomorphic if f is complex differentiable at all point of the region K
For the quesiton, for x=o so z=iy then the point can be differentiable is the line of imaginary axis ,for it is a line we cannnot define holomorphic here, then that's nowhere for f to be holomophic

is my argument right??

 

[Dick]

f:k->C is holomorphic if f is complex differentiable at all point of the region K
For the quesiton, for x=o so z=iy then the point can be differentiable is the line of imaginary axis ,for it is a line we cannnot define holomorphic here, then that's nowhere for f to be holomophic

is my argument right??

It's not very well worded. Sure, it's differentiable on a line. You'll want to explain why a 'line' isn't a 'region'. What property does a region have that a line doesn't?

 

[cummings12332]

It's not very well worded. Sure, it's differentiable on a line. You'll want to explain why a 'line' isn't a 'region'. What property does a region have that a line doesn't?

if i choose a point inside the region,then choose the very close small region around that point, in the point inside the small region should be diff. what i mean is the region should be open ,but the line we cannot find here, i didnt explain it in my post sorry

 

[Dick]

if i choose a point inside the region,then choose the very close small region around that point, in the point inside the small region should be diff. what i mean is the region should be open ,but the line we cannot find here, i didnt explain it in my post sorry

Right. There's no point on the line that has a neighborhood of the point that's contained in the line. The line doesn't contain any open sets. It has empty interior.