An integral on a circle is a mathematical concept that involves calculating the area under a curve on a circular path. This is commonly used in complex analysis and can be represented using complex numbers.
To show that a function is integral on a circle, you need to use the Cauchy integral formula, which states that the value of an integral around a closed curve is equal to the value of the function at any point inside the curve multiplied by the circumference of the circle. In this case, the function is already in the form of $\frac{1}{z}$ which is a known integral on a circle, so you can use the substitution $z = \frac{1}{w}$ to show that the function $\frac{1}{1-|z|^2}$ is integral on a circle.
Showing that a function is integral on a circle is important because it allows us to use techniques from complex analysis to solve problems in other areas of mathematics. It also helps to understand the behavior of the function and its singularities.
Sure. Let's say we have the function $f(z) = \frac{1}{z}$ and we want to find the value of the integral on a circle with radius 2 centered at the origin. We can use the formula $\int_C f(z) dz = 2\pi i f(0)$ to calculate the value of the integral, which in this case would be $2\pi i \cdot \frac{1}{0} = \infty$.
Yes, there are limitations to using the integral on a circle formula. It can only be used for functions that are analytic on the interior of the circle and have no singularities within the circle. It also assumes that the curve is closed and does not intersect itself.