8. Under the transformation w =
1
z
, find the image of the region (i) x > c, where c > 0, and (ii) y > c,
where c < 0.

Given the transformation \( w = \frac{1}{z} \), where \( z = x + iy \) is a complex number with real part \( x \) and imaginary part \( y \), we need to find the images of the regions under this transformation.

### Step 1: Analyze the Transformation

For \( w = \frac{1}{z} \), let's express \( w \) in terms of \( u + iv \), where \( u \) and \( v \) are the real and imaginary parts of \( w \), respectively:

\[
w = \frac{1}{z} = \frac{1}{x + iy}
\]

Multiply the numerator and the denominator by the conjugate of the denominator:

\[
w = \frac{x - iy}{(x + iy)(x - iy)} = \frac{x - iy}{x^2 + y^2}
\]

So, the real and imaginary parts of \( w \) are:

\[
u = \frac{x}{x^2 + y^2}, \quad v = \frac{-y}{x^2 + y^2}
\]

### Step 2: Find the Image of the Region \( x > c \) where \( c > 0 \)

The region \( x > c \) represents the half-plane to the right of the line \( x = c \). We need to find how this region transforms under \( w = \frac{1}{z} \).

1. Consider \( u = \frac{x}{x^2 + y^2} \). Since \( x > c \) and \( x^2 + y^2 > 0 \), it follows that \( u > \frac{c}{x^2 + y^2} \).

2. As \( x \) increases, \( u \) increases, and as \( x \) approaches \( c \), \( u \) approaches \( \frac{c}{c^2 + y^2} \). 

3. Therefore, the image of \( x > c \) under this transformation will be the region \( u > \frac{c}{x^2 + y^2} \).

### Step 3: Find the Image of the Region \( y > c \) where \( c < 0 \)

The region \( y > c \) represents the half-plane above the line \( y = c \). Since \( c < 0 \), this region is below the real axis. We again analyze the transformation:

1. For \( v = \frac{-y}{x^2 + y^2} \), since \( y > c \) and \( c < 0 \), \( -y < -c \) implies \( v > \frac{-c}{x^2 + y^2} \).

2. As \( y \) approaches \( c \), \( v \) approaches \( \frac{-c}{x^2 + c^2} \).

3. Therefore, the image of \( y > c \) under this transformation will be the region \( v > \frac{-c}{x^2 + y^2} \).

### Summary of Results:

1. The image of the region \( x > c \) where \( c > 0 \) under the transformation \( w = \frac{1}{z} \) is the region \( u > \frac{c}{x^2 + y^2} \).
  
2. The image of the region \( y > c \) where \( c < 0 \) under the transformation \( w = \frac{1}{z} \) is the region \( v > \frac{-c}{x^2 + y^2} \).

Would you like any more details or explanations on this problem?

Here are five related questions you might find interesting:

1. How would the transformation \( w = \frac{1}{z} \) map the region \( x < c \) where \( c < 0 \)?
2. What is the image of the entire right half-plane under the transformation \( w = \frac{1}{z} \)?
3. How does the transformation \( w = \frac{1}{z} \) affect the unit circle \( |z| = 1 \)?
4. What happens to the imaginary axis under the transformation \( w = \frac{1}{z} \)?
5. How does the transformation \( w = \frac{1}{z} \) map the first quadrant of the complex plane?

**Tip:** When dealing with complex transformations, it can be helpful to break down the transformation into its real and imaginary components to better understand how regions are mapped in the complex plane.

Explore how the transformation w = 1/z maps regions x > c and y > c under complex analysis. Understand the mathematical steps involved and the implications for complex variables. Suitable for advanced undergraduate students studying complex analysis.

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