Proving Analyticity of u_x - iu_y in Complex Analysis

In summary, to show that u_x-iu_y is analytic for all z, we can use the definition of a harmonic function, which involves second partial derivatives being equal to 0. Then, we can use the Cauchy-Riemann equations to write the form of the derivative of u_x-iu_y, and relate these two ideas. By manipulating these equations through integration and differentiation, we can show that u_x-iu_y is indeed analytic for all z.
  • #1
bballife1508
37
0
Suppose that u(x,y) is harmonic for all (x,y). Show that u_x-iu_y is analytic for all z.

(Assume that all derivatives in the question exist and are continuous)

I have no idea where to start with this? Something with the Cauchy Riemann equations is required but I'm not sure exactly how to incorporate them.
 
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  • #2
Look up what it means to be harmonic ( Hint: It involves second partial derievatives).

Then you cauchy riemann equations to write the form of your derievative of u_x - iu_y. Relate the two ideas.
 
  • #3
i'm aware of what harmonic is. the second partials of u add to 0, but how do i relate this to the C.R equations
 
  • #4
What are the cauchy riemann equations ? Can you write them out ?
 
  • #5
Write down the definition of a Harmonic Function, write down the C-R equations.
now can you manipulate them to get what is needed?
(HINT: try integration and differentiation).
 

FAQ: Proving Analyticity of u_x - iu_y in Complex Analysis

What is the definition of analyticity in complex analysis?

Analyticity in complex analysis refers to the property of a complex-valued function f(z) being differentiable at every point in its domain. This means that the function has a well-defined derivative at each point in its domain, which is a complex number.

How is analyticity related to the Cauchy-Riemann equations?

The Cauchy-Riemann equations are a set of necessary and sufficient conditions for a complex-valued function to be analytic. These equations express the relationship between the partial derivatives of a complex function and must be satisfied for a function to be analytic.

Is the function ux - iuy always analytic?

No, the function ux - iuy is only analytic if it satisfies the Cauchy-Riemann equations. If the partial derivatives of u and v (where f(z) = u(x,y) + iv(x,y)) do not satisfy these equations, then the function is not analytic.

How can I prove the analyticity of ux - iuy?

The analyticity of ux - iuy can be proven by showing that the Cauchy-Riemann equations are satisfied for the function. This can be done by calculating the partial derivatives of u and v and substituting them into the equations.

What are some applications of proving analyticity in complex analysis?

Proving analyticity is important in complex analysis as it allows for the use of powerful techniques such as Cauchy's integral formula, Cauchy's integral theorem, and the residue theorem. These techniques are used in various fields such as physics, engineering, and economics to solve complex problems involving complex functions.

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