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find the upper bound of the function mod(z-2)/mod(z^2+2iz-3);modz

Vaidehi , 7 Years ago
Grade 12th pass
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Askiitians Tutor Team

Last Activity: 17 Days ago

To find the upper bound of the function |z2||z2+2iz3| for |z|, we can utilize the properties of complex numbers and the triangle inequality. Let's break this down step by step.

Understanding the Function

The function we are analyzing is a ratio of two expressions involving complex numbers. The numerator is the modulus of z2, and the denominator is the modulus of z2+2iz3. To find the upper bound, we need to analyze how these moduli behave as |z| varies.

Step 1: Analyzing the Numerator

The numerator, |z2|, can be interpreted as the distance from the point z in the complex plane to the point 2. Using the triangle inequality, we can express this as:

  • |z2||z|+|2|=|z|+2

Step 2: Analyzing the Denominator

Next, we need to consider the denominator, |z2+2iz3|. We can rewrite this expression by factoring or simplifying it, but a direct approach using the triangle inequality is often more straightforward. We can express it as:

  • \(|z^2 + 2iz - 3| \geq ||z^2| - |2iz| - |3||\

Calculating each term, we find:

  • |z2|=|z|2
  • |2iz|=2|z|
  • |3|=3

Thus, we can write:

  • \(|z^2 + 2iz - 3| \geq ||z|^2 - 2|z| - 3|\

Step 3: Finding the Lower Bound of the Denominator

To ensure we have a valid lower bound, we can analyze the expression |z|22|z|3. This is a quadratic expression in terms of |z|. The roots of the equation |z|22|z|3=0 can be found using the quadratic formula:

  • \(|z| = \frac{2 \pm \sqrt{(2)^2 - 4(1)(-3)}}{2(1)} = \frac{2 \pm \sqrt{16}}{2} = \frac{2 \pm 4}{2}\

This gives us roots at |z|=3 and |z|=1 (which we discard since |z| cannot be negative). Thus, for |z|3, the expression |z|22|z|3 is positive, ensuring that:

  • |z2+2iz3||z|22|z|3

Step 4: Establishing the Upper Bound

Now we can combine our findings. The upper bound of the function can be expressed as:

  • |z|+2|z|22|z|3

As |z| increases, the numerator grows linearly while the denominator grows quadratically. Therefore, we can conclude that the upper bound of the function approaches zero as |z| becomes very large.

Final Thoughts

In summary, the upper bound of the function |z2||z2+2iz3| can be analyzed effectively using the triangle inequality and properties of quadratic functions. As |z| increases, the function approaches zero, indicating that the upper bound diminishes with larger values of |z|. This analysis provides a clear understanding of how the function behaves in the complex plane.

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