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

In summary, the conversation discusses the meaning of the left size (delta*w(z, z)) in comparison to the right size (delta*w(z, z*)). The speaker clarifies that there is no multiplication involved and explains the equation (17) using the mean value theorem. They also provide an example of the solution and break down the variables involved.
  • #1
Bat1
4
0
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
IMG_20211025_180923.jpg
 
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  • #2
? 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.
 
  • #3
I know it is NOT A MULTIPLICATION, I know what "delta" means... I don't know how to prove that equation (17)
 
  • #4
Write $\delta w= w(x+\delta x, y+\delta y, z+ \delta z)- w(x, y, z)$. Use the mean value theorem.
 
  • #5
\[ if: w=w(x,y), and: w=u+iv, then: u==x, v==y, then, w=x+iy (??) \]
\[ \delta w=w(x+\delta x, y+\delta y) - w(x, y)=\delta x +i\delta y \]

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

Is this the right solution??
 

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

1. What is the purpose of using Delta*w(z, z) in proving complex numbers?

The use of Delta*w(z, z) in proving complex numbers is to show the relationship between the real and imaginary parts of a complex number. It helps to simplify calculations and understand the properties of complex numbers.

2. How does Delta*w(z, z) help in solving complex number equations?

Delta*w(z, z) allows us to break down complex number equations into simpler forms, making it easier to solve and understand. It also helps to identify patterns and properties of complex numbers.

3. Can Delta*w(z, z) be used to prove all properties of complex numbers?

No, Delta*w(z, z) is only one method of proving properties of complex numbers. There are other methods such as geometric proofs and algebraic proofs that can also be used.

4. Is Delta*w(z, z) applicable to all types of complex numbers?

Yes, Delta*w(z, z) can be applied to all types of complex numbers, including real numbers, imaginary numbers, and complex numbers with both real and imaginary parts.

5. How can understanding Delta*w(z, z) benefit in the study of complex analysis?

Understanding Delta*w(z, z) can help in the study of complex analysis by providing a deeper understanding of the properties and behavior of complex numbers. It also allows for easier manipulation and calculation of complex functions and equations.

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