Harmonic function and its conjugate function. Hello friends, today I’ll talk about the harmonic function and its conjugate function in complex analysis.

**Harmonic function and its conjugate function**

Let’s say that is a function of two real variables and . And it will be a harmonic function if it satisfies the Laplace equation

Now if is a harmonic function, then there will be a function where

Now here . In many books, it’s also written as .

And the function is the conjugate of the harmonic function .

Then as per the Cauchy-Riemann equations,

Now I’ll give some examples of a harmonic function and its conjugate.

**Solved examples of a harmonic function and its conjugate function**

Disclaimer: None of these examples are mine. I have chosen these from some book or books. I have also given the due reference at the end of the post.

So here is the first example.

**Example 1**

According to Stroud and Booth (2011)*, “Demonstrate that the following is harmonic and obtain the conjugate function.

”

**Solution**

Now here the given function is:

(1)

And I have to prove that is a harmonic function. Also, I have to get the conjugate function of . So I’ll start with the value of .

**Step 1**

First of all, I’ll differentiate equation (1) partially with respect to to get

(2)

Read also: **First-order partial derivative of functions with two variables**

Next, I’ll differentiate equation (2) partially with respect to to get

(3)

Read also: **Second-order partial derivative of functions with two variables**

Now I’ll get the value of . So I’ll differentiate equation (1) partially with respect to to get

(4)

Then I’ll differentiate equation (4) partially with respect to to get

(5)

Now the function will be a harmonic function if it satisfies the Laplace equation

Therefore I’ll add equations (3) and (5) to get

Hence I can conclude that the function is harmonic. Now I’ll get its conjugate function.

**Step 2**

Let’s say is the conjugate function of . Then there exists a function for which

Now I’ll use Cauchy-Riemann equations to get the value of . As per this equation,

Also from equation (2), I can see that

So I can say that

Next, I’ll integrate it with respect to to get

(6)

Also, as per Cauchy-Riemann equation,

And from equation (4), I can see that

So I can say that

Next, I’ll integrate it with respect to to get

(7)

Now I’ll compare equations (6) and (7) to get the value of .

Thus I can say that

So the conjugate function is

Hence I can conclude that this is the answer to the given example.

Now I’ll give you another example.

**Example 2**

According to Stroud and Booth (2011)*, “Find the conjugate function of the following.

”

**Solution**

Now here the given function is

(8)

And I have to find out its conjugate function. So I’ll start with the partial differentiation.

**Step 1**

Now I’ll differentiate equation (8) partially with respect to to get

(9)

Next, I’ll differentiate equation (8) partially with respect to to get

(10)

Suppose is the conjugate function of . Then as per Cauchy-Riemann equation,

Therefore I can use equation (9) to say that

Then I’ll integrate to get

(11)

Also, I can use equation (10) to say that

Now I’ll integrate to get

(12)

Now I’ll compare equations (11) and (12) to get the value of .

Thus I can say that

Hence I can conclude that this is the answer to the given example.

Dear friends, this is the end of today’s post. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

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