# How to Solve a Harmonic Function Problem Involving Partial Derivatives?

• MHB
• jaychay
In summary, the conversation discusses how a harmonic conjugate function is related to the concept of holomorphic functions and complex differentiability. The next step in the process is to integrate the partial derivative of the harmonic conjugate function with respect to one of the variables, treating the other variable as a constant. This will result in a function that satisfies the necessary conditions for being a harmonic conjugate.
jaychay

I am really struggle with this question.

Hint: A harmonic conjugate function $v$ must have that $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$, so that the function given by $f(x+iy)=u(x,y)+iv(x,y)$ is holomorphic (complex differentiable).

Klaas van Aarsen said:
Hint: A harmonic conjugate function $v$ must have that $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$, so that the function given by $f(x+iy)=u(x,y)+iv(x,y)$ is holomorphic (complex differentiable).
Can you tell what is the next step that I should do please ?

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jaychay said:
Can you tell what is the next step that I should do please ?
It's a partial derivative (note the round d's). When we partially differentiate with respect to $x$, then that means that we treat $y$ as a constant.
More concisely:
$$\frac{\partial}{\partial x}(y)=0$$
So you can simplify your expression for $\frac{\partial u}{\partial x}$ a bit.

Next step is to integrate $\frac{\partial u}{\partial x}$ with respect to $y$ to find $v(x,y)$.
When we do so, we treat $x$ as a constant.

Let me give an example.
Suppose we have $u(x,y)=xy$. Then we have $\frac{\partial u}{\partial x}=y$.
And $v(x,y)=\int \frac{\partial u}{\partial x} \,dy = \int y\,dy = \frac 12y^2 + C(x)$, where $C(x)$ is some function of $x$.
We can verify by evaluating $\frac{\partial v}{\partial y} = \frac{\partial}{\partial y}\Big(\frac 12y^2 + C(x)\Big) = y$.
As we can see, we get indeed that $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$.

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## What is a harmonic function problem?

A harmonic function problem is a mathematical problem that involves finding a function that satisfies the Laplace equation, which states that the sum of the second derivatives of a function with respect to each variable is equal to zero. This type of problem is commonly encountered in physics, engineering, and other fields that involve the study of potential fields.

## What are some real-world applications of harmonic function problems?

Harmonic function problems have a wide range of applications, including in electrical engineering, fluid dynamics, and heat transfer. For example, harmonic functions can be used to model the flow of electricity in a circuit, the flow of fluids in a pipe, or the distribution of heat in a room.

## What techniques are commonly used to solve harmonic function problems?

There are several techniques that can be used to solve harmonic function problems, including separation of variables, the method of images, and the method of Green's functions. These techniques involve breaking down the problem into simpler parts and using known solutions or properties of harmonic functions to find a solution.

## What are some properties of harmonic functions?

Harmonic functions have several important properties, including the fact that they are infinitely differentiable, they have continuous second derivatives, and they satisfy the mean value property. They also have the property of superposition, meaning that the sum of two harmonic functions is also a harmonic function.

## Are there any limitations or challenges when working with harmonic function problems?

One limitation of harmonic function problems is that they can only be used to model systems that are in equilibrium, meaning that they are not changing over time. Additionally, finding an analytical solution to a harmonic function problem can be challenging and may require the use of numerical methods.

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