Proving Analytic Function Derivatives with CR Equations

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In summary, the conversation was about proving that the derivative of an analytic function is also analytic using only the Cauchy-Riemann equations and the condition of continuous second partials. The attempt at a solution involved expressing the derivative in terms of the partial derivatives and using the CR equations for the original function. Ultimately, it was concluded that the derivative of an analytic function is indeed analytic.
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g1990
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Homework Statement


Let g(z) be an analytic function. I have to show that g'(z) is also analytic, using only the CR eqns. I am given that the 2nd partials are continuous


Homework Equations


let f(x,y)=u(x,y)+iv(x,y)
CR: du/dx=dv/dy and du/dy=-dv/dx
continuous 2nd partials: d/dx(du/dy)=d/dy(du/dx) and d/dx(dv/dy)=d/dy(dv/dx)


The Attempt at a Solution


I am confused as to how to express the derivative df/dz. would f'=du/dx+du/dy+idv/dx+idv/dy?
 
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  • #2
I don't think you want df/dz. If a function satisfies the CR equations and has continuous partials, what can you say about it?
 
  • #3
I have to solve this problem using only calculus. I can't use any theorems like using Taylor series. Maybe I can say that it is harmonic?
 
  • #4
nvm- got it! f'(z)=du/dx+idv/dx. Taking these and the real and imaginary parts, we can get the CR eqns for f' using the CR eqns for f and the fact that the mixed partials are equal.
 

What is an analytic function?

An analytic function is a mathematical function that is defined and differentiable at every point in its domain. It can be written as a power series, and its derivative exists at every point in its domain.

What are CR equations?

CR equations, also known as Cauchy-Riemann equations, are a set of necessary and sufficient conditions for a function to be analytic. They express the relationship between the real and imaginary parts of a complex function, and are used to prove the differentiability of a function at a point.

How do you prove analytic function derivatives using CR equations?

To prove that a function is analytic at a point, you must show that it satisfies the CR equations at that point. This involves taking the partial derivatives of the function with respect to the real and imaginary variables, and setting them equal to each other. If the equations are satisfied, then the function is analytic at that point.

What is the significance of proving analytic function derivatives?

Proving analytic function derivatives is important because it allows us to determine if a function is analytic, which has many applications in mathematics and physics. It also allows us to calculate the higher order derivatives of a function and study its behavior near a point.

Can CR equations be used to prove the differentiability of any function?

No, CR equations can only be used to prove the differentiability of analytic functions. They do not apply to functions that are not defined and differentiable at every point in their domain.

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