b) i) Show that the series 



1
(1 )
n
n z z converges for z  1 and find its sum. (3)
 ii) Obtain the analytic function whose real part u(x, y ) e cos y.
x  (2)

### (b) i) Show that the series \( \sum_{n=1}^{\infty} (-1)^{n-1} z^n \) converges for \( |z| < 1 \) and find its sum.

We are given the series:

\[
S = \sum_{n=1}^{\infty} (-1)^{n-1} z^n = z - z^2 + z^3 - z^4 + z^5 - \dots
\]

This is an alternating geometric series with the first term \( a = z \) and common ratio \( r = -z \). The general form for the sum of an infinite geometric series \( \sum_{n=0}^{\infty} ar^n \) is:

\[
S = \frac{a}{1 - r}
\]

For our series, \( a = z \) and \( r = -z \), so the sum is:

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

Thus, the series converges for \( |z| < 1 \) and its sum is:

\[
S = \frac{z}{1 + z}
\]

### (b) ii) Obtain the analytic function whose real part is \( u(x, y) = e^x \cos y \).

To find the analytic function \( f(z) = u(x, y) + iv(x, y) \), where \( u(x, y) \) is the real part and \( v(x, y) \) is the imaginary part, we apply the Cauchy-Riemann equations. The Cauchy-Riemann equations state that for \( f(z) \) to be analytic, the following conditions must hold:

\[
\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad \text{and} \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}
\]

Given that \( u(x, y) = e^x \cos y \), we calculate the partial derivatives:

\[
\frac{\partial u}{\partial x} = e^x \cos y \quad \text{and} \quad \frac{\partial u}{\partial y} = -e^x \sin y
\]

Using the Cauchy-Riemann equations, we get:

1. \( \frac{\partial v}{\partial y} = e^x \cos y \)
2. \( \frac{\partial v}{\partial x} = e^x \sin y \)

Now, we integrate each of these with respect to the appropriate variable:

- Integrating \( \frac{\partial v}{\partial y} = e^x \cos y \) with respect to \( y \), we get:

\[
v(x, y) = e^x \sin y + C(x)
\]

- Differentiating this with respect to \( x \), we have:

\[
\frac{\partial v}{\partial x} = e^x \sin y + C'(x)
\]

- But from the second Cauchy-Riemann equation, we know that \( \frac{\partial v}{\partial x} = e^x \sin y \), so \( C'(x) = 0 \), which implies \( C(x) \) is a constant.

Thus, \( v(x, y) = e^x \sin y \).

Therefore, the analytic function is:

\[
f(z) = u(x, y) + iv(x, y) = e^x \cos y + i e^x \sin y = e^x (\cos y + i \sin y)
\]

Using Euler's formula, \( \cos y + i \sin y = e^{iy} \), we can express the function as:

\[
f(z) = e^x e^{iy} = e^{x + iy}
\]

Since \( z = x + iy \), the analytic function is:

\[
f(z) = e^z
\]

---

Would you like more details on these steps, or do you have any specific questions?

Here are five related questions to explore further:
1. What is the radius of convergence of a power series?
2. How are the Cauchy-Riemann equations derived from the definition of differentiability in complex analysis?
3. Can you generalize the method for finding an analytic function given any real part \( u(x, y) \)?
4. What is the significance of the Euler's formula in complex analysis?
5. How do alternating series differ in their convergence behavior from non-alternating series?

**Tip:** To check if a complex function is analytic, always verify the Cauchy-Riemann equations in the region of interest.

b) i) Show that the series ∑ (-1)^(n-1) z^n converges for |z| < 1 and find its sum. ii) Obtain the analytic function whose real part is u(x, y) = e^x cos y.

Explore the convergence of the alternating series ∑ (-1)^(n-1) z^n for |z| < 1 and learn how to find the analytic function whose real part is u(x, y) = e^x cos y using Cauchy-Riemann equations. This university-level problem covers complex analysis concepts like series convergence and analytic functions.

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