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

Click For Summary
SUMMARY

The discussion centers on understanding the expression for the change in a complex function, specifically \(\delta w(z, z^*)\), in the context of complex analysis. The participant clarifies that \(\delta w(z, z^*)\) represents the change in the function \(w\) when the complex variable \(z\) and its conjugate \(z^*\) are perturbed by small amounts. The mean value theorem is suggested as a method to prove the equation involving \(\delta w\), which is expressed as \(w(x+\delta x, y+\delta y) - w(x, y)\). The participant confirms that the derivatives of the real and imaginary components of \(w\) are equal to 1 and 0, respectively, leading to the conclusion that \(\delta w = \delta x + i\delta y\).

PREREQUISITES
  • Understanding of complex numbers and their properties
  • Familiarity with the mean value theorem in calculus
  • Knowledge of partial derivatives and their application in complex functions
  • Basic concepts of complex analysis, particularly with respect to functions of complex variables
NEXT STEPS
  • Study the mean value theorem and its applications in complex analysis
  • Explore the concept of differentiability in the context of complex functions
  • Learn about the Cauchy-Riemann equations and their significance in complex analysis
  • Investigate the implications of perturbations in complex variables on function behavior
USEFUL FOR

Students and professionals in mathematics, particularly those focusing on complex analysis, as well as anyone interested in the behavior of complex functions under perturbations.

Bat1
Messages
4
Reaction score
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
 
Physics news on Phys.org
? There is NO [math]\delta* w(z, Z)[/math]. If you meant [math]\delta w(z, z*)[/math] then it is the change in w when z, and its conjugate z*, change by a slight amount.
 
I know it is NOT A MULTIPLICATION, I know what "delta" means... I don't know how to prove that equation (17)
 
Write $\delta w= w(x+\delta x, y+\delta y, z+ \delta z)- w(x, y, z)$. Use the mean value theorem.
 
\[ if: w=w(x,y), and: w=u+iv, then: u==x, v==y, then, w=x+iy (??) \]
\[ \delta w=w(x+\delta x, y+\delta y) - w(x, y)=\delta x +i\delta y \]

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

Is this the right solution??
 

Thread 'Confusion about the Moyal-Weyl twist'

I am studying the mathematical formalism behind non-commutative geometry approach to quantum gravity. I was reading about Hopf algebras and their Drinfeld twist with a specific example of the Moyal-Weyl twist defined as F=exp(-iλ/2θ^(μν)∂_μ⊗∂_ν) where λ is a constant parametar and θ antisymmetric constant tensor. {∂_μ} is the basis of the tangent vector space over the underlying spacetime Now, from my understanding the enveloping algebra which appears in the definition of the Hopf algebra...
View full post »

Similar threads

MHB Is z + z¯ and z × z¯ Real for Any Complex Number z?
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad Fundamental theorem of arithmetic from Jordan-Hölder theorem
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Did I figure-out z-translation of 2D-coordinates correctly?
  • · Replies 4 ·
Replies
4
Views
2K
High School Complex Number Solutions for |z+1| = |z+i| and |z| = 5
  • · Replies 5 ·
Replies
5
Views
2K
Graduate Anti-dual numbers and what are their properties?
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Apparent counterexample to Cauchy-Goursat theorem (Complex Analysis)
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Quintic Polynomial Tschirnhaus Transform: Possible complex Bring Radicals
  • · Replies 0 ·
Replies
0
Views
2K
High School Understanding Bases of a Vector Space
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Fourier Transformation on a Ring
  • · Replies 11 ·
Replies
11
Views
3K
Complex numbers problem |z| - iz = 1-2i
  • · Replies 20 ·
Replies
20
Views
2K