Question

Verify Picard's theorem for the function cos(1/z) at z₀ = 0

Answer 1

Picard's theorem states that if a function is analytic in a punctured neighborhood of a point and has an essential singularity at that point, then the function takes on every complex number, with possibly one exception, as values in any neighborhood of the point. In this case, to verify Picard's theorem for the function cos(1/z) at z₀ = 0, we show that the function has an essential singularity at z = 0, and therefore takes on every complex value, with possibly one exception, in any neighborhood of z = 0.

Step-by-step explanation:

Picard's theorem states that if a function is analytic in a punctured neighborhood of a point z₀ and has an essential singularity at z₀, then the function takes on every complex number, with possibly one exception, as values in any neighborhood of the point. In other words, the function is dense in the complex plane near the singularity.

To verify Picard's theorem for the function cos(1/z) at z₀ = 0:

1. First, note that 1/z has an essential singularity at z = 0.
2. Then, consider the function cos(1/z). Since cos(z) is entire, it is analytic everywhere except at isolated points.
3. However, at z = 0, the function cos(1/z) has an essential singularity, which means that it takes on every complex number, with possibly one exception, as values in any neighborhood of z = 0.
4. Therefore, Picard's theorem is verified for the function cos(1/z) at z₀ = 0.