# determine if pole is inside unit circle

i would like to know how to determine if pole of given function is inside unit circle contour? for example let us take this function

$f(z)=(i-1)/(z+i)$ and we have contour $z=\gamma(t)=2e^{it}$ where $0\leq t\leq\pi$.it just a example,in this case pole is equal to $z=-i$,but suppose that pole is equal to $z=a+b*i$,actual if we have circle with some radius and given point,we can calculate distance between center and this point and see if this distance is less then radius and after this we can say is it this point inside,on boundary or outside of circle,but what abut unit circle?please help me

• You're asking how to find out whether a point is inside the unit circle? That's the same as finding out whether it is inside any other circle. – Henning Makholm Aug 24 '13 at 10:25
• but in any other circle we have coordinates and we can calculate distants – dato datuashvili Aug 24 '13 at 10:31
• What coordinates are you missing? By definition the unit circle has its center at $0$ and radius $1$. – Henning Makholm Aug 24 '13 at 10:33
• Compute the distance from the center of the circle and see if it is smaller than the radius or not. – Henning Makholm Aug 24 '13 at 10:38
• Let $z = a + ib$ where $a$ and $b$ are real how do you calculate $|z|$? and what's the relationship between $|z|$ and a circle? – Warren Hill Aug 24 '13 at 10:40

## 1 Answer

Let $z = a + ib$ where $a$ and $b$ are real how do you calculate $|z|$? and what's the relationship between $|z|$ and a circle?

When we plot a complex point we plot with real part $a$ along the $x$ axis and the imaginary part $b$ on the $y$ axis. We therefore have the coordinates of a point.

We know that $|z| = \sqrt{a^2+b^2}$ so if $|z| \lt 1$ it's inside the unit circle.

• if $z=5$,then $a=5$ and $b=0$ right – dato datuashvili Aug 24 '13 at 10:56
• yes that is correct – Warren Hill Aug 24 '13 at 10:57
• i wanted to make sure about this – dato datuashvili Aug 24 '13 at 10:59