Thevtaylor expansion of f(z) =z-1/z+1 at z=0 is

To find the **Taylor expansion** of the function \( f(z) = \frac{z - 1}{z + 1} \) at \( z = 0 \), we can proceed as follows:

### Step 1: Express the function as a series
We start by rewriting the function in a form that can be expanded as a Taylor series around \( z = 0 \).

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

We now attempt to expand \( f(z) \) into a power series around \( z = 0 \). First, divide the numerator by the denominator using long division or by factoring:

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

This is the expansion of \( \frac{1}{1+z} \) as a geometric series for \( |z| < 1 \), which yields:

\[
f(z) = (z - 1) \cdot \left( 1 - z + z^2 - z^3 + \dots \right)
\]

### Step 2: Multiply the series
Now distribute the \( z - 1 \) term:

\[
f(z) = (z - 1) - (z - 1)z + (z - 1)z^2 - (z - 1)z^3 + \dots
\]

Simplifying:

\[
f(z) = z - 1 - z^2 + z - z^3 + z^2 - z^4 + \dots
\]

Collecting like terms:

\[
f(z) = (2z - 1) - (z^2 - z^3 + z^4 - \dots)
\]

### Final Taylor expansion:
Thus, the **Taylor series expansion** of \( f(z) \) around \( z = 0 \) is:

\[
f(z) = 2z - 1 - z^2 + z^3 - z^4 + \dots
\]

This series converges for \( |z| < 1 \).

Would you like more details or any further clarifications?

Here are some related questions for practice:

1. What is the radius of convergence for the Taylor series of \( f(z) \)?
2. How would the Taylor expansion change if we expanded around \( z = 1 \)?
3. How can we apply the Taylor expansion to approximate values of \( f(z) \) for small \( z \)?
4. What is the general form of the geometric series, and how is it applied here?
5. How does the remainder term in a Taylor series expansion influence the accuracy of approximations?

**Tip:** When finding Taylor expansions, it's important to identify functions that can be expanded using known series, such as geometric or binomial series.

Discover the step-by-step solution for finding the Taylor expansion of the function f(z) = (z - 1) / (z + 1) at z = 0. Learn how to expand the function using geometric series and simplify to get a power series. Ideal for students studying complex analysis.

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