Which one of the following functions is analytic in the region |z| ≤ 1?

Understanding Analytic Functions and the Region |z| ≤ 1

An analytic function in complex analysis is a function that is complex differentiable at every point within a given region. The question asks us to identify the function that is analytic in the region defined by |z| ≤ 1. This region represents the closed unit disk, including the boundary circle |z| = 1 and its interior |z| < 1.

For rational functions, analyticity is maintained as long as the function does not have any singularities (poles) within the specified region. Singularities of a rational function occur at the points where the denominator is equal to zero.

Analyzing Options for Singularities within |z| ≤ 1

We need to check the locations of the poles for each function:

**Option 1: $\frac{z^2-1}{z}$ **

- Denominator: $z$
- Pole: $z=0$
- Modulus of the pole: $|0|=0$
- Is the pole within the region |z| ≤ 1? Yes, since $0\le 1$.
- Conclusion: This function is not analytic in the region.

**Option 2: $\frac{z^2-1}{z+2}$ **

- Denominator: $z+2$
- Pole: $z=-2$
- Modulus of the pole: $|-2|=2$
- Is the pole within the region |z| ≤ 1? No, since $2>1$. The pole is outside the unit disk.
- The numerator's zeros ($z=\pm 1$) lie on the boundary $|z|=1$. At these points, the denominator is non-zero ($1+2=3$ at $z=1$, and $-1+2=1$ at $z=-1$).
- Conclusion: This function is analytic in the region |z| ≤ 1.

**Option 3: $\frac{z^2-1}{z-0.5}$ **

- Denominator: $z-0.5$
- Pole: $z=0.5$
- Modulus of the pole: $|0.5|=0.5$
- Is the pole within the region |z| ≤ 1? Yes, since $0.5\le 1$.
- Conclusion: This function is not analytic in the region.

**Option 4: $\frac{z^2-1}{z+j0.5}$ **

- Denominator: $z+j0.5$
- Pole: $z=-j0.5$
- Modulus of the pole: $|-j0.5|=\sqrt{0^2+(-0.5{)}^2}=0.5$
- Is the pole within the region |z| ≤ 1? Yes, since $0.5\le 1$.
- Conclusion: This function is not analytic in the region.

Determining the Correct Analytic Function

The function $\frac{z^2-1}{z+2}$ is the only one among the options whose pole ($z=-2$) is located outside the specified region |z| ≤ 1. Therefore, it remains differentiable throughout the closed unit disk and is analytic in this region.