Complex Analysis Harmonic functions

In summary, the conversation discusses how to prove that a function w(x,y) = u(x,y) + cv(x,y) is also harmonic by using the Laplace Equation and basic properties of partial derivatives. The idea of using differentiation as a linear operator is also mentioned.
  • #1
Alvis
9
1
Suppose u(x,y) and v(x,y) are harmonic on G and c is an element of R. Prove u(x,y) + cv(x,y) is also harmonic.

I tried using the Laplace Equation of Uxx+Uyy=0

I have:
du/dx=Ux
d^2u/dx^2=Uxx

du/dy=Uy
d^2u/dy^2=Uyy

dv/dx=cVx
d^2v/dx^2=cVxx

dv/dy=cVy
d^2v/dy^2=cVyy
I'm not really sure how to prove these are harmonic...am I missing a relationship?
 
Physics news on Phys.org
  • #2
Define w(x,y) = u(x,y) + cv(x,y) and calculate wxx + wyy. Basic properties of the partial derivative should give you the answer.
 
  • #3
Wow, that's a really good idea. I was trying to do the harmonic conjugate but was getting nowhere. Thank you!
 
  • #4
do you understand what it means to say that differentiation is a linear operator?
 
  • Like
Likes FactChecker

Related to Complex Analysis Harmonic functions

1. What is complex analysis and how does it relate to harmonic functions?

Complex analysis is a branch of mathematics that deals with functions of a complex variable. Harmonic functions are those that satisfy the Laplace's equation, which is a fundamental equation in complex analysis. Therefore, complex analysis is essential in understanding and analyzing harmonic functions.

2. What are the key properties of harmonic functions?

Harmonic functions have the following key properties: they are twice continuously differentiable, satisfy the Laplace's equation, and have the mean value property. They also have a real and an imaginary part, and their real part is a harmonic function.

3. How do harmonic functions relate to physical phenomena?

Harmonic functions have a wide range of applications in physics, as they describe the behavior of waves, vibrations, and other physical phenomena. For example, the electric potential in electrostatics and the temperature distribution in heat conduction follow harmonic functions.

4. Can harmonic functions be expressed in terms of other functions?

Yes, harmonic functions can be expressed in terms of other functions, such as polynomials, trigonometric functions, and logarithmic functions. This is because they satisfy the Laplace's equation, which has various analytical solutions.

5. How are harmonic functions used in engineering and technology?

Harmonic functions have numerous applications in engineering and technology, such as in signal processing, image processing, and control systems. They are also used in designing and analyzing electrical circuits and in solving problems related to fluid flow, elasticity, and electromagnetism.

Similar threads

I Why is this function not ##L^1(\mathbb{R} \times \mathbb{R})##?
    • Topology and Analysis
Replies
6
Views
913
Proving that a "composition" is harmonic
    • Calculus and Beyond Homework Help
Replies
2
Views
1K
I Proof of Harmonic Function Infinitely Differentiable
    • Topology and Analysis
Replies
4
Views
1K
I How does one solve Uxx+Uyy+Uzz=C when C is non-zero?
    • Differential Equations
Replies
12
Views
3K
How should I use the Jacobi equation to determine the nature of this?
    • Calculus and Beyond Homework Help
Replies
5
Views
665
A Laurent series for algebraic functions
    • Topology and Analysis
Replies
9
Views
2K
Harmonic Functions: Laplace's Equations & Analytic Functions
    • Topology and Analysis
Replies
4
Views
1K
Show that the real part of a certain complex function is harmonic
    • Calculus and Beyond Homework Help
Replies
7
Views
1K
MHB Solving a complex value equation
    • Topology and Analysis
Replies
4
Views
1K
Solve the given second order differential equation
    • Calculus and Beyond Homework Help
Replies
14
Views
469

Hot Threads

Recent Insights

Back
Top