Proving the Equation (left(frac{1 ix}{1 - ix}right)^n frac{1 ia}{1 - ia}) Has All Real Roots

In this article, we will demonstrate through a step-by-step mathematical analysis that the equation (left(frac{1 ix}{1 - ix}right)^n frac{1 ia}{1 - ia}) has all real roots. We will break down the proof into several key steps, each focusing on transforming the given equation into a manageable form and analyzing it rigorously. This will involve complex analysis and the consideration of magnitudes and arguments.

Step 1: Analyzing the Left-Hand Side (LHS)

Let's denote the left-hand side (LHS) of the equation as ( z ): z frac{1 ix}{1 - ix}.

First, let us express ( z ) in a more manageable form. We can multiply the numerator and the denominator by ( 1 ix ) to simplify:

z frac{1 ix}{1 - ix} cdot frac{1 ix}{1 ix} frac{1 2ix - x^2}{1 - x^2}

Further simplifying the numerator:

1 2ix - x^2 (1 - x^2) 2ix

We now have:

z frac{1 - x^2 2ix}{1 - x^2} frac{1 - x^2}{1 - x^2} frac{2ix}{1 - x^2} 1 frac{2ix}{1 - x^2}

This can be separated into its real and imaginary parts:

z 1 i cdot frac{2x}{1 - x^2}

Step 2: Analyzing the Right-Hand Side (RHS)

Next, let's analyze the right-hand side (RHS) of the equation: w frac{1 ia}{1 - ia}.

Similarly, we can simplify ( w ) by multiplying the numerator and the denominator by ( 1 ia ):

w frac{1 ia}{1 - ia} cdot frac{1 ia}{1 ia} frac{1 2ia - a^2}{1 - a^2}

Further simplifying the numerator:

1 2ia - a^2 (1 - a^2) 2ia

We now have:

w frac{1 - a^2 2ia}{1 - a^2} frac{1 - a^2}{1 - a^2} frac{2ia}{1 - a^2} 1 i cdot frac{2a}{1 - a^2}

This can also be separated into its real and imaginary parts:

w 1 i cdot frac{2a}{1 - a^2}

Step 3: Setting Up the Equation

Now we set the two sides equal:

left(1 i cdot frac{2x}{1 - x^2}right)^n 1 i cdot frac{2a}{1 - a^2}

Step 4: Analyzing the Magnitudes

For both sides to be equal, their magnitudes must also be equal. The magnitude of ( z ) is:

|z| sqrt{left(1 - frac{2x^2}{1 - x^2}right) left(frac{2x}{1 - x^2}right)^2} sqrt{frac{1 - x^2 4x^2}{(1 - x^2)^2}} sqrt{frac{1 3x^2}{(1 - x^2)^2}} 1

The magnitude of ( w ) is:

|w| sqrt{left(1 - frac{2a^2}{1 - a^2}right) left(frac{2a}{1 - a^2}right)^2} sqrt{frac{1 - a^2 4a^2}{(1 - a^2)^2}} sqrt{frac{1 3a^2}{(1 - a^2)^2}} 1

Step 5: Analyzing the Argument

Next, we consider the arguments of both sides. The argument of ( z ) is given by:

arg(z) arctanleft(frac{2x}{1 - x^2}right)

And for ( w ):

arg(w) arctanleft(frac{2a}{1 - a^2}right)

Step 6: Roots of the Equation

The equation can be rewritten in terms of arguments:

n arctanleft(frac{2x}{1 - x^2}right) arctanleft(frac{2a}{1 - a^2}right) 2kpi

for some integer ( k ). This implies:

arctanleft(frac{2x}{1 - x^2}right) frac{arctanleft(frac{2a}{1 - a^2}right) 2kpi}{n}

The left-hand side ( arctanleft(frac{2x}{1 - x^2}right) ) is a continuous function of ( x ), while the right-hand side is a constant for fixed ( a ) and ( k ). Therefore, the equation will have solutions where the values of ( x ) correspond to real values.

Thus, for each fixed ( a ), the equation (left(frac{1 ix}{1 - ix}right)^n frac{1 ia}{1 - ia}) has all real roots (boxed{x}).

This completes the proof that the equation has all real roots.