Understanding the Problem: Fundamental Amplitude of a Complex Expression

The problem asks us to find the fundamental amplitude (or principal argument) of a complex number expression involving \(z\), where \(z\) itself is defined as a power of another complex number. The fundamental amplitude is the unique value of the argument of a complex number that lies within the interval \((-\pi, \pi]\).

Step 1: Simplify the Complex Number \(z\)

The given complex number is \({\rm{z}} = {\left( {\frac{1}{{\sqrt 2 }} - \frac{{\rm{i}}}{{\sqrt 2 }}} \right)^{ - 25}}\). Let's first simplify the base of the power, which is \(w = \frac{1}{{\sqrt 2 }} - \frac{{\rm{i}}}{{\sqrt 2 }}\).

We can convert \(w\) into polar form \(r(\cos \theta + i \sin \theta)\).

• The magnitude \(r\) is given by \(r = \sqrt{{\left(\frac{1}{\sqrt{2}}\right)^2} + {\left(-\frac{1}{\sqrt{2}}\right)^2}} = \sqrt{\frac{1}{2} + \frac{1}{2}} = \sqrt{1} = 1\).
• To find the argument \(\theta\), we look at the real and imaginary parts: \(\cos \theta = \frac{1}{\sqrt{2}}\) and \(\sin \theta = -\frac{1}{\sqrt{2}}\). This corresponds to an angle in the fourth quadrant. The principal argument is \(\theta = -\frac{\pi}{4}\).

So, \(w\) in polar form is \(1 \cdot \left(\cos\left(-\frac{\pi}{4}\right) + i \sin\left(-\frac{\pi}{4}\right)\right)\). Using Euler's formula, this is also \(e^{-i\pi/4}\).

Now, we can calculate \(z\):

\(z = w^{-25} = \left(e^{-i\pi/4}\right)^{-25} = e^{i(-25)(-\pi/4)} = e^{i(25\pi/4)}\)

The argument of \(z\) is \(\frac{25\pi}{4}\). To find the fundamental amplitude, we need to find the equivalent angle in \((-\pi, \pi]\) by subtracting multiples of \(2\pi\).

\(\frac{25\pi}{4} = \frac{24\pi + \pi}{4} = 6\pi + \frac{\pi}{4}\)

Since \(6\pi\) is a multiple of \(2\pi\), the fundamental amplitude of \(z\) is \(\frac{\pi}{4}\).

Therefore, \(z = \cos\left(\frac{\pi}{4}\right) + i \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} + i \frac{1}{\sqrt{2}}\).

Step 2: Evaluate the Expression \(\frac{{\rm{z}} - \sqrt 2 }}{{{\rm{z}} - {\rm{i}}\sqrt 2 }}\)

Now we substitute the value of \(z\) we found into the given expression:

\(\frac{{\rm{z}} - \sqrt 2 }}{{{\rm{z}} - {\rm{i}}\sqrt 2 }} = \frac{\left(\frac{1}{\sqrt{2}} + i \frac{1}{\sqrt{2}}\right) - \sqrt{2}}{\left(\frac{1}{\sqrt{2}} + i \frac{1}{\sqrt{2}}\right) - i\sqrt{2}}\)

Simplify the numerator:

Numerator \( = \frac{1}{\sqrt{2}} - \sqrt{2} + i \frac{1}{\sqrt{2}} = \frac{1 - \sqrt{2} \cdot \sqrt{2}}{\sqrt{2}} + i \frac{1}{\sqrt{2}} = \frac{1 - 2}{\sqrt{2}} + i \frac{1}{\sqrt{2}} = -\frac{1}{\sqrt{2}} + i \frac{1}{\sqrt{2}}\)

Simplify the denominator:

Denominator \( = \frac{1}{\sqrt{2}} + i \left(\frac{1}{\sqrt{2}} - \sqrt{2}\right) = \frac{1}{\sqrt{2}} + i \left(\frac{1 - \sqrt{2} \cdot \sqrt{2}}{\sqrt{2}}\right) = \frac{1}{\sqrt{2}} + i \left(\frac{1 - 2}{\sqrt{2}}\right) = \frac{1}{\sqrt{2}} - i \frac{1}{\sqrt{2}}\)

The expression becomes:

\(\frac{-\frac{1}{\sqrt{2}} + i \frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}} - i \frac{1}{\sqrt{2}}}\)

We can factor out \(\frac{1}{\sqrt{2}}\) from the numerator and denominator:

\(\frac{\frac{1}{\sqrt{2}}(-1 + i)}{\frac{1}{\sqrt{2}}(1 - i)} = \frac{-1 + i}{1 - i}\)

To simplify this complex fraction, multiply the numerator and denominator by the conjugate of the denominator, which is \(1 + i\):

\(\frac{-1 + i}{1 - i} \times \frac{1 + i}{1 + i} = \frac{(-1)(1) + (-1)(i) + (i)(1) + (i)(i)}{(1)^2 + (-1)^2}\)

\( = \frac{-1 - i + i + i^2}{1 + 1} = \frac{-1 - 1}{2} = \frac{-2}{2} = -1\)

So, the expression simplifies to \(-1\).

Step 3: Find the Fundamental Amplitude of the Result

We need to find the fundamental amplitude of the complex number \(-1\). In the complex plane, \(-1\) is located on the negative real axis. Its Cartesian form is \(-1 + 0i\).

For a complex number \(x+iy\), the argument \(\theta\) satisfies \(\cos \theta = \frac{x}{r}\) and \(\sin \theta = \frac{y}{r}\), where \(r = \sqrt{x^2+y^2}\).

For \(-1\), \(x = -1\) and \(y = 0\). The magnitude is \(r = \sqrt{(-1)^2 + 0^2} = \sqrt{1} = 1\).

\(\cos \theta = \frac{-1}{1} = -1\)

\(\sin \theta = \frac{0}{1} = 0\)

The angle \(\theta\) in the interval \((-\pi, \pi]\) for which \(\cos \theta = -1\) and \(\sin \theta = 0\) is \(\theta = \pi\).

The fundamental amplitude of \(\frac{{\rm{z}} - \sqrt 2 }}{{{\rm{z}} - {\rm{i}}\sqrt 2 }}\) is \(\pi\).

Summary of Steps

Here is a quick summary of the process:

1. Convert the base of the power in \(z\) to polar form.
2. Use De Moivre's Theorem (or exponential form) to calculate \(z^{-25}\).
3. Find the fundamental amplitude of \(z\).
4. Convert \(z\) back to Cartesian form.
5. Substitute the value of \(z\) into the expression \(\frac{{\rm{z}} - \sqrt 2 }}{{{\rm{z}} - {\rm{i}}\sqrt 2 }}\).
6. Simplify the complex fraction by multiplying by the conjugate of the denominator.
7. Find the fundamental amplitude of the resulting complex number.

Revision Table: Key Complex Number Concepts

Additional Information: De Moivre's Theorem and Arguments

De Moivre's Theorem is crucial when dealing with powers of complex numbers. It simplifies the process significantly compared to multiplying the Cartesian form repeatedly. When finding the argument of a complex number raised to a power, say \(z^n\), the argument is \(n\) times the argument of \(z\). However, this new argument might fall outside the fundamental interval \((-\pi, \pi]\). It is essential to add or subtract multiples of \(2\pi\) to bring the argument back into this interval to find the fundamental amplitude.

For the division of two complex numbers, say \(\frac{z_1}{z_2}\), if their arguments are \(\theta_1\) and \(\theta_2\), the argument of the quotient is \(\theta_1 - \theta_2\). In our problem, we calculated the quotient directly in Cartesian form and then found its argument. Alternatively, we could have found the magnitudes and arguments of the numerator \((z - \sqrt{2})\) and the denominator \((z - i\sqrt{2})\) and then used the property \(\text{Arg}\left(\frac{z_1}{z_2}\right) = \text{Arg}(z_1) - \text{Arg}(z_2)\) (modulo \(2\pi\)). However, simplifying the expression to \(-1\) was the most direct approach here.