# Calculate imaginary part if real part is following

• Chromosom
In summary, the conversation is about calculating the imaginary part of a function given its real part and using the Cauchy-Riemann differential equations to relate the two parts. The question at hand is to determine the real part when the imaginary part is given. The solution involves carefully considering the Cauchy-Riemann equations and determining which part of the function is given in order to find the appropriate solution.
Chromosom

## Homework Statement

$$f:\ v(x,y)=4xy+2x$$

The task is to calculate the imaginary part.

## The Attempt at a Solution

I have no idea what to do because in my opinion u(x,y) can be anything. For example: $$f(x,y)=4xy+2x+(3x-4y)\text i$$. But I must be wrong. I would appreciate if you just tell me what to do or where is my mistake, because I want to solve it alone :)

Read some information on the Cauchy Riemann equations

1 person
Well, usually one writes
$$f(z)=f(x+\mathrm{i} y)=u(x,y)+\mathrm{i} v(x,y).$$
If you now assume that $f$ is a analytic function, you have the Cauchy-Riemann differential equations, relating the real and imaginary parts
$$\partial_x u=\partial_y v, \quad \partial_y u=-\partial_x v.$$
So if you have given $u$ (real part) you can determine the imaginary part (up to a constant) and vice versa.

Read the question carefully again, because it seems as if the imaginary part is given and you look for the real part and not the other way around.

1 person

## 1. What is the formula for calculating the imaginary part?

The formula for calculating the imaginary part is Im(z) = b, where z is a complex number of the form a + bi and b is the imaginary part.

## 2. Can the imaginary part be calculated if only the real part is given?

Yes, the imaginary part can be calculated if only the real part is given using the formula Im(z) = b.

## 3. How do you determine the real and imaginary parts of a complex number?

For a complex number in the form a + bi, the real part is a and the imaginary part is b.

## 4. Is the imaginary part always a number?

Yes, the imaginary part is always a number. It can be positive, negative, or zero.

## 5. How does calculating the imaginary part relate to practical applications in science?

Calculating the imaginary part is essential in many scientific applications, such as in electrical engineering and physics, where complex numbers are used to represent and analyze alternating current circuits and quantum mechanics, respectively.

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