# Show that f is entire when u is harmonic and v=

• zardiac
It means that the Laplacian of u, $\nabla^2u$, is equal to 0. Use that.In summary, the conversation discusses the task of showing that f=u+iv is an entire analytic function, given that u is harmonic everywhere in R^2. The conversation mentions using the Cauchy Riemann equations, but the person gets stuck and is unsure of how to use the fact that u is harmonic. The expert suggests using the definition of harmonic, which states that the Laplacian of u, $\nabla^2u$, is equal to 0, to solve the problem.
zardiac

## Homework Statement

Assume that u is harmonic everywhere in R^2, and let
$$v(x,y)=\int_0^y u_x'(x,t)dt - \int_0^x u_y'(s,0)ds$$
show that f=u+iv is entire analytic.

## Homework Equations

Maybe Cauchy Riemann: $$u_x'=v_y'$$ and $$u_y'=-v_x'$$

## The Attempt at a Solution

I have only tried to see what happens if I use the Cauchy Riemann equations, but I get stuck right away. I am not sure how to use the fact that u is harmonic either.

Any hints would be very appreciated.

ux' is very strange notation. I think you mean just the derivative of u with respect to x- but that is just ux. There is no need for the ' here.

I have only tried to see what happens if I use the Cauchy Riemann equations, but I get stuck right away.
Good. Show what you did and where you got stuck.

I am not sure how to use the fact that u is harmonic either.
What does "harmonic" mean?

## 1. What does it mean for a function to be entire?

Being entire means that a function is analytic, or differentiable, at every point in its domain. This includes the points at infinity. Essentially, an entire function is a smooth, continuous function that has no singularities or breaks in its behavior.

## 2. How is a function's analyticity related to its harmonicity?

Analyticity and harmonicity are closely related properties of a function. A function is harmonic if it satisfies Laplace's equation, which essentially means that its second derivatives exist and are continuous. A function is analytic if it is differentiable at every point in its domain. Therefore, a function must be analytic in order to be harmonic.

## 3. Can a function be entire if its real and imaginary parts satisfy certain conditions?

Yes, a function can be entire if its real and imaginary parts satisfy certain conditions. In this case, we are given that the real part of the function is harmonic, which means that it satisfies Laplace's equation. If the imaginary part is also harmonic, then the function will be entire.

## 4. How can we prove that a function is entire?

One way to prove that a function is entire is to show that it is analytic at every point in its domain. This can be done by using the Cauchy-Riemann equations, which relate the partial derivatives of the real and imaginary parts of a complex function. If these equations are satisfied, then the function is analytic and therefore entire.

## 5. Are there any other conditions that must be met for a function to be entire?

Yes, in addition to being analytic, a function must also be bounded in order to be entire. This means that the function cannot grow too quickly or become arbitrarily large. If a function is both analytic and bounded, then it is entire.

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