Expand the complex function

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as Taylor series about⁡⁡𝑧 = 𝑖.

To expand the complex function \( \frac{1}{z} \) as a Taylor series about \( z = i \), we need to express \( \frac{1}{z} \) in a form that allows us to apply the Taylor series expansion around \( z = i \).

### Step 1: Express \( z \) in Terms of \( z - i \)

First, rewrite \( \frac{1}{z} \) in terms of \( z - i \):

\[
\frac{1}{z} = \frac{1}{(z - i) + i}
\]

Now, express this as a series:

\[
\frac{1}{z} = \frac{1}{i} \cdot \frac{1}{1 + \frac{z - i}{i}}
\]

This can be rewritten as:

\[
\frac{1}{z} = -\frac{1}{i} \cdot \frac{1}{1 - \frac{i - z}{i}}
\]

### Step 2: Expand Using Taylor Series

The Taylor series for \( \frac{1}{1 - w} \) around \( w = 0 \) is:

\[
\frac{1}{1 - w} = 1 + w + w^2 + w^3 + \dots
\]

In our case, \( w = \frac{i - z}{i} \), so we have:

\[
\frac{1}{z} = -\frac{1}{i} \cdot \left[ 1 + \frac{i - z}{i} + \left(\frac{i - z}{i}\right)^2 + \left(\frac{i - z}{i}\right)^3 + \dots \right]
\]

Simplifying further:

\[
\frac{1}{z} = -\frac{1}{i} \left[1 + \frac{i - z}{i} + \frac{(i - z)^2}{i^2} + \frac{(i - z)^3}{i^3} + \dots \right]
\]

Since \( \frac{1}{i} = i \), this becomes:

\[
\frac{1}{z} = i \left[1 + \frac{i - z}{i} + \frac{(i - z)^2}{i^2} + \frac{(i - z)^3}{i^3} + \dots \right]
\]

Expanding each term:

\[
\frac{1}{z} = i \left[1 + (i - z)/i - \frac{(i - z)^2}{i^2} + \frac{(i - z)^3}{i^3} - \dots \right]
\]

### Step 3: Final Form of the Taylor Series

Thus, the Taylor series expansion of \( \frac{1}{z} \) about \( z = i \) is:

\[
\frac{1}{z} = i \left[1 - \frac{z - i}{i} - \frac{(z - i)^2}{i^2} - \frac{(z - i)^3}{i^3} - \dots \right]
\]

Or, more explicitly:

\[
\frac{1}{z} = i - (z - i) - \frac{(z - i)^2}{i} - \frac{(z - i)^3}{i^2} - \dots
\]

This series converges when \( |z - i| < |i| = 1 \).

Would you like further details or have any questions?

### Relative Questions:
1. How do you derive the Taylor series of a function about a different point in the complex plane?
2. What are the conditions for the convergence of a Taylor series in the complex plane?
3. Can the Taylor series be used to approximate complex functions over larger domains?
4. How does the Laurent series differ from the Taylor series in complex analysis?
5. What is the significance of the radius of convergence in complex series expansions?

### Tip:
When expanding functions in a Taylor series around a complex point, it’s important to consider the radius of convergence to ensure the series is valid within a particular region.

Learn how to expand the complex function 1/z as a Taylor series around z = i. This problem involves complex analysis and the application of Taylor series in the complex plane. Suitable for advanced undergraduate students and researchers.

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