If `z=x+iy=((1)/(sqrt2)-(i)/(sqrt2))^(-25), where i=sqrt(-1),` then what is the fundamental amplitude of `(z-sqrt2)/(z-sqrt2)` ?

Options

A. `pi`
B. `(pi)/(2)`
C. `(pi)/(3)`
D. `(pi)/(4)`

Explanation

To solve the problem, we start with the expression for \( z \):

\[ z = \left( \frac{1}{\sqrt{2}} - \frac{i}{\sqrt{2}} \right)^{-25} \]

### Step 1: Convert to Polar Form
First, we convert \( \frac{1}{\sqrt{2}} - \frac{i}{\sqrt{2}} \) into polar form. The modulus \( r \) and argument \( \theta \) of the complex number can be calculated as follows:

\[ r = \sqrt{\left( \frac{1}{\sqrt{2}} \right)^2 + \left( -\frac{1}{\sqrt{2}} \right)^2} = \sqrt{\frac{1}{2} + \frac{1}{2}} = \sqrt{1} = 1 \]

The argument \( \theta \) is given by:

\[ \theta = \tan^{-1}\left(\frac{-\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}}\right) = \tan^{-1}(-1) = -\frac{\pi}{4} \]

Thus, we can express the complex number in polar form:

\[ \frac{1}{\sqrt{2}} - \frac{i}{\sqrt{2}} = 1 \cdot \text{cis}\left(-\frac{\pi}{4}\right) \]

where \( \text{cis}(\theta) = \cos(\theta) + i\sin(\theta) \).

### Step 2: Raise to the Power of -25
Now we raise this expression to the power of -25:

\[ z = \left( \text{cis}\left(-\frac{\pi}{4}\right) \right)^{-25} = \text{cis}\left(25 \cdot \frac{\pi}{4}\right) \]

### Step 3: Simplify the Argument
Next, we simplify the argument:

\[ 25 \cdot \frac{\pi}{4} = \frac{25\pi}{4} \]

To find the equivalent angle within the range \( [0, 2\pi) \), we can subtract \( 2\pi \):

\[ \frac{25\pi}{4} - 6\pi = \frac{25\pi}{4} - \frac{24\pi}{4} = \frac{\pi}{4} \]

Thus, we have:

\[ z = \text{cis}\left(\frac{\pi}{4}\right) \]

### Step 4: Calculate \( z - \sqrt{2} \) and \( z + \sqrt{2} \)
Now we need to calculate \( z - \sqrt{2} \) and \( z + \sqrt{2} \):

\[ z - \sqrt{2} = \text{cis}\left(\frac{\pi}{4}\right) - \sqrt{2} \]
\[ z + \sqrt{2} = \text{cis}\left(\frac{\pi}{4}\right) + \sqrt{2} \]

### Step 5: Calculate the Fundamental Amplitude
The fundamental amplitude of \( \frac{z - \sqrt{2}}{z + \sqrt{2}} \) is given by:

\[ \text{arg}\left(\frac{z - \sqrt{2}}{z + \sqrt{2}}\right) = \text{arg}(z - \sqrt{2}) - \text{arg}(z + \sqrt{2}) \]

Using the properties of the argument function, we can find the arguments of \( z - \sqrt{2} \) and \( z + \sqrt{2} \). However, since both terms involve the same \( z \), we can simplify our calculations.

### Final Result
The fundamental amplitude can be expressed as:

\[ \text{arg}\left(\frac{z - \sqrt{2}}{z + \sqrt{2}}\right) = \text{arg}(z) = \frac{\pi}{4} \]