Complex Analysis - Finding the equation of a circle

In summary: So the equation for the circle is correct and the center is at (-\frac{3}{2}, 0) with a radius of \frac{3}{2}. In summary, the expression z/(z+3) being purely imaginary means that the real part of the numerator is zero, which leads to the equation of a circle with center at (-\frac{3}{2}, 0) and radius \frac{3}{2}.
  • #1
NewtonianAlch
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Homework Statement


If [itex]\frac{z}{z + 3}[/itex] is purely imaginary, show that z lies on a certain circle and find the equation of that circle.

The Attempt at a Solution



So,

[itex]\frac{z}{z + 3}[/itex] = [itex]\frac{x + iy}{x + iy + 3}[/itex]

Multiplying by the complex conjugate (and simplifying), we get,

[itex]\frac{x^{2} + y^{2} + 3x + 3iy}{x^{2} + y^{2} + 6x + 9}[/itex]

Since we're only interested in the imaginary part here, I take,

[itex]\frac{3iy}{x^{2} + 6x + 9 + y^{2}}[/itex]

I am not too sure what to do from here, also...does "z lies on a certain circle" mean on the boundary line or anywhere in that enclosed zone including the boundary?
 
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  • #2
You're not "only interested in the imaginary part". You are told z/(z+3) is pure imaginary; this means the real part is zero. That's probably where you want to start.

A circle is the set of points equidistant from its center; the circle does not include its interior. So "on a circle" means on the curvy line.
 
  • #3
So if the real part is going to be zero, it means the numerator can only be zero. From that we can get an equation for a circle, which I believe is (x + [itex]\frac{3}{2}[/itex])[itex]^{2}[/itex] + [itex]y^{2}[/itex] = [itex]\frac{9}{4}[/itex] - but is this the circle I'm looking for?

If so, why am I getting an equation for a circle from the real part, when the expression in question is supposedly purely imaginary?
 
  • #4
It's _because_ the expression is pure imaginary that the real part is zero, which gives you the equation you are after.
 
  • #5
Ah I see, that makes it more clear now.
 

Related to Complex Analysis - Finding the equation of a circle

1. What is complex analysis?

Complex analysis is a branch of mathematics that deals with complex numbers and their functions. It is a powerful tool for solving problems in various fields such as physics, engineering, and economics.

2. How do you find the equation of a circle using complex analysis?

The equation of a circle in complex analysis can be found by using the formula z = x+iy, where z represents a complex number, x and y represent the coordinates of a point on the circle, and i is the imaginary unit. The equation can be written as (z-a)(z-ā) = r², where a is the center of the circle and r is the radius.

3. What are some applications of finding the equation of a circle using complex analysis?

Finding the equation of a circle using complex analysis has various applications, such as in physics, where it is used to describe the motion of a particle in a circular path. It is also used in engineering to design circular structures, and in economics to model circular flow of income and expenditure.

4. Can the equation of a circle be written in polar form using complex analysis?

Yes, the equation of a circle can be written in polar form using complex analysis. It is given by z = re^(iθ), where r is the radius and θ is the angle between the positive real axis and the line joining the center of the circle to a point on the circle.

5. What are some key concepts in complex analysis that are important for finding the equation of a circle?

Some key concepts in complex analysis that are important for finding the equation of a circle include complex numbers, polar form, and the concept of modulus and argument. It is also important to understand the properties of circles, such as their center, radius, and equation in Cartesian and polar forms.

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