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Explain how did the z-i/z+i sign got interchaned

In the image you provided, it seems that the expression involves the inverse tangent function and a logarithmic term. The key part you're asking about is how the expression \( \frac{z - i}{z + i} \) got interchanged. 

To understand this, let's break down the logarithmic expression step by step.

### Original Expression:
\[
\tan^{-1}(z) - \frac{i}{2} \log\left(\frac{z - i}{z + i}\right)
\]

### Interchanged Expression:
\[
\tan^{-1}(z) - \frac{i}{2} \log\left(\frac{z + i}{z - i}\right)
\]

### Explanation:

The logarithmic function has a property:
\[
\log\left(\frac{a}{b}\right) = -\log\left(\frac{b}{a}\right)
\]

This property is crucial because it explains why the signs inside the logarithm change when the terms \( \frac{z - i}{z + i} \) and \( \frac{z + i}{z - i} \) are swapped.

### Step-by-Step Process:

1. **Initial Form:**
   \[
   -\frac{i}{2} \log\left(\frac{z - i}{z + i}\right)
   \]

2. **Apply the Logarithmic Property:**
   \[
   \log\left(\frac{z - i}{z + i}\right) = -\log\left(\frac{z + i}{z - i}\right)
   \]
   
3. **Substitute Back:**
   \[
   -\frac{i}{2} \log\left(\frac{z - i}{z + i}\right) = -\frac{i}{2} \cdot \left(-\log\left(\frac{z + i}{z - i}\right)\right)
   \]

4. **Simplify the Signs:**
   \[
   \frac{i}{2} \log\left(\frac{z + i}{z - i}\right)
   \]

So, what has happened is that the original negative sign in front of the fraction cancels out when the property of logarithms is applied, resulting in the fraction \( \frac{z + i}{z - i} \) inside the logarithm.

### Conclusion:
The interchange of \( \frac{z - i}{z + i} \) to \( \frac{z + i}{z - i} \) happens due to the logarithmic identity, which changes the sign when the fraction is inverted.

Do you want more details on any specific part of this process or have any questions?

### Follow-up Questions:
1. What is the inverse tangent function, and how is it related to complex numbers?
2. How does the logarithmic function behave with complex arguments?
3. Why is the factor of \( \frac{i}{2} \) important in the context of this expression?
4. How do such transformations apply in solving complex integration problems?
5. Can you explain how the concept of branch cuts might affect the evaluation of the logarithm here?

### Tip:
When dealing with logarithms of fractions in complex analysis, always remember that the logarithm of a quotient can be flipped, but it introduces a negative sign. This is a fundamental property that often simplifies or transforms expressions in significant ways.

Explore how the expression z-i/z+i gets interchanged in logarithmic functions with a detailed explanation. This problem involves complex numbers, logarithmic identities, and the inverse tangent function, crucial in advanced mathematics.

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