Proof of Complex Numbers: Delta*w(z, z) Explained

[Bat1]

Hi,
I have this problem and its solution but i know what right size is, but i don't understand what left size (delta*w(z, z)) is equal to

 

[HOI]

? There is NO \(\displaystyle \delta* w(z, Z)\). If you meant \(\displaystyle \delta w(z, z*)\) then it is the change in w when z, and its conjugate z*, change by a slight amount.

 

[Bat1]

I know it is NOT A MULTIPLICATION, I know what "delta" means... I don't know how to prove that equation (17)

 

[HOI]

Write $\delta w= w(x+\delta x, y+\delta y, z+ \delta z)- w(x, y, z)$. Use the mean value theorem.

 

[Bat1]

\[ if: w=w(x,y), and: w=u+iv, then: u==x, v==y, then, w=x+iy (??) \]
\[ \delta w=w(x+\delta x, y+\delta y) - w(x, y)=\delta x +i\delta y \]

\[ \delta z(\partial w/\partial z)+\delta z^*(\partial w/\partial z^*) =\delta (x+iy)* 0.5*(\partial w/\partial x+i*\partial w/\partial y)+\delta (x-iy)* 0.5*(\partial w/\partial x-i*\partial w/\partial y)=\delta x*(\partial w/\partial x) +\delta y*(\partial w/\partial y)=\delta x*(\partial u/\partial x+i\partial v/y) +\delta y*(\partial u/\partial y+\partial v/y)= \]\[ if: u==x, v==y, then: \]
\[ \partial u/\partial x=1,\partial v/\partial x=0,\partial u/\partial y=0,\partial '/\partial x=1, \]\[ = \delta x +i\delta y \ \]

Is this the right solution??