# Proving Entirety of conj(f(conj(z))) for an Entire Function f

• tylerc1991
In summary, if a function f(z) = u(x,y) + iv(x,y) is entire, then the function conj(f(conj(z))) is also entire. This is because the Cauchy-Riemann equations hold for entire functions, and the conjugate of a function f(z) is denoted by conj(f(z)). Therefore, by showing that u1(x,y) = u(x,-y) and v1(x,y) = (-v(x,-y)) satisfy the Cauchy-Riemann equations, we can conclude that conj(f(conj(z))) is entire. This is due to the continuity of the partial derivatives of f(z).
tylerc1991

## Homework Statement

Show that if a function f(z) = u(x,y) +iv(x,y) is entire, then the function conj(f(conj(z))) is entire.

## Homework Equations

(i) The Cauchy-Riemann (CR) equations hold for functions that are entire: u_x = v_y and u_y = -v_x

(ii) conj(_) is the conjugate (i.e. there is a conjugate bar over f and over z)

## The Attempt at a Solution

Since f is entire, CR is satisfied:
so u_x = v_y and u_y = -v_x

this implies:
u_x = -(-v_y) and -u_y = -(-v_x)

this implies:
CR is satisfied for a function g(z) = u(x,-y) - iv(x,-y)

but g(z) = conj(f(conj(z)))

this implies:
CR is satisfied for conj(f(conj(z)))

Since f is entire, the partial derivatives are continuous

this implies:
conj(f(conj(z))) is entire.

You've got the right idea. The presentation is a little confusing. Might be best if you say conj(f(conj(z))=u1(x,y)+i*v1(x,y) where u1(x,y)=u(x,-y) and v1(x,y)=(-v(x,-y)) and show u1 and v1 satisfy CR.

OK I can reword it. But there are no holes in the proof? Thank you very much for your help!

It's all there. I just have to read between the lines a bit to figure out u_x=(-(-v_y)) is important. Do rewrite with different symbols for the real and imaginary parts of conj(f(conj(z)).

## What is a complex analysis proof?

A complex analysis proof is a mathematical argument that demonstrates the validity of a statement or theorem in complex analysis. It involves using the properties and operations of complex numbers to prove the given statement.

## What are the key concepts in complex analysis that are used in proofs?

The key concepts in complex analysis that are used in proofs include the properties of complex numbers such as addition, subtraction, multiplication, and division, as well as concepts like continuity, differentiability, and analyticity.

## How do complex analysis proofs differ from real analysis proofs?

Complex analysis proofs differ from real analysis proofs in that they involve the use of complex numbers and their properties, while real analysis proofs only consider real numbers. Additionally, complex analysis deals with concepts like holomorphic functions and contour integration, which are not present in real analysis.

## What are some common techniques used in complex analysis proofs?

Some common techniques used in complex analysis proofs include direct proofs, proof by contradiction, proof by induction, and the use of Cauchy-Riemann equations. Other techniques may involve using symmetry, analytic continuation, and the Cauchy integral formula.

## What are some applications of complex analysis proofs in science and technology?

Complex analysis proofs have various applications in science and technology, including in fields such as physics, engineering, and computer science. They are used to solve problems involving electric circuits, fluid dynamics, and signal processing. They are also used in the development of algorithms and software for complex systems.

Replies
6
Views
1K
Replies
8
Views
6K
Replies
14
Views
2K
Replies
17
Views
2K
Replies
5
Views
2K
Replies
8
Views
708
Replies
7
Views
1K
Replies
27
Views
2K
Replies
1
Views
4K
Replies
7
Views
1K