Coefficient Of Magnification For W = Z^2 At Z = 1 + I

For the conformal transformation w = z^2, what is the coefficient of magnification at z = 1 + i?

Introduction to Conformal Transformations

In the realm of complex analysis, conformal transformations play a pivotal role in mapping one complex plane to another while preserving angles locally. These transformations are essential in various fields, including fluid dynamics, electromagnetism, and even cartography. Understanding the properties of these transformations, such as magnification and rotation, is crucial for solving complex problems in these areas.

Our main focus in this article revolves around a specific conformal transformation: w = z^2. We aim to explore the behavior of this transformation, particularly concerning the coefficient of magnification at a specific point in the complex plane, namely z = 1 + i. Before diving into the specifics, let's briefly discuss the general concept of magnification in conformal transformations.

When a conformal transformation maps a region from the z-plane to the w-plane, the size of infinitesimal shapes can change. The magnification factor quantifies this change in size. It represents the ratio of the size of the transformed shape in the w-plane to the size of the original shape in the z-plane. This factor varies from point to point and is a crucial aspect of understanding how the transformation distorts the original geometry. We need to understand how the w = z^2 transformation affects the magnification around the point z = 1 + i to determine the coefficient of magnification, and select the correct answer from the options provided: (a)

${ \sqrt{2} }$

(b)

${ 2\sqrt{2} }$

(c) 2 (d) None of these. In the subsequent sections, we will delve into the mathematical details of calculating the magnification coefficient for this specific transformation and point. This exploration will not only provide a solution to the given problem but also offer a deeper understanding of conformal transformations and their applications.

Understanding Magnification in Conformal Transformations

The concept of magnification in conformal transformations is intrinsically linked to the derivative of the transformation function. Let's consider a general conformal transformation represented by the complex function w = f(z), where z is a complex variable in the z-plane, and w is its image in the w-plane. The derivative of this function, f'(z), provides crucial information about how the transformation behaves locally.

The magnitude of the derivative, |f'(z)|, is the magnification factor at the point z. This value represents the factor by which infinitesimal lengths are scaled under the transformation. In simpler terms, if |f'(z)| = 2 at a particular point, it means that a small line segment in the z-plane near that point will be stretched by a factor of 2 in the w-plane. The argument of the derivative, arg(f'(z)), represents the angle of rotation at the point z. This signifies how much a small shape is rotated under the transformation.

The magnification factor is crucial in understanding the local behavior of a conformal transformation. It allows us to visualize how the transformation distorts shapes as they are mapped from one plane to another. A magnification factor greater than 1 indicates stretching, while a factor less than 1 indicates shrinking. A factor equal to 1 implies that the size of the shape remains unchanged, at least infinitesimally close to the point under consideration. To apply this concept to our problem, we need to calculate the derivative of our specific transformation, w = z^2, and then evaluate its magnitude at the point z = 1 + i. This will give us the coefficient of magnification at that point. This process involves complex differentiation and modulus calculation, which we will explore in detail in the following sections.

Calculating the Derivative and Magnification Factor for w = z^2

To determine the magnification factor for the conformal transformation w = z^2 at the point z = 1 + i, the first step is to find the derivative of the transformation function with respect to z. This derivative, denoted as dw/dz or f'(z), will provide us with the necessary information to calculate the magnification.

The derivative of w = z^2 with respect to z is straightforward and can be found using basic calculus rules. Applying the power rule, we get:

${ \frac{dw}{dz} = f'(z) = 2z }$

Now that we have the derivative, we can evaluate it at the specific point of interest, z = 1 + i. Substituting z = 1 + i into the derivative, we get:

${ f'(1 + i) = 2(1 + i) = 2 + 2i }$

The magnification factor at a point is given by the magnitude of the derivative at that point. Therefore, to find the magnification factor at z = 1 + i, we need to calculate the magnitude of f'(1 + i) = 2 + 2i. The magnitude of a complex number a + bi is given by

${ \sqrt{a^2 + b^2} }$

Applying this formula to our case:

${ |f'(1 + i)| = |2 + 2i| = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} }$

Thus, the magnification factor for the transformation w = z^2 at the point z = 1 + i is 2√2. This result indicates that in the neighborhood of the point z = 1 + i, the transformation stretches the complex plane by a factor of 2√2. In the next section, we will interpret this result in the context of the given problem and identify the correct answer choice.

Determining the Coefficient of Magnification at z = 1 + i

In the previous section, we meticulously calculated the magnification factor for the conformal transformation w = z^2 at the point z = 1 + i. Our calculations revealed that the magnitude of the derivative at this point, which represents the magnification factor, is 2√2.

Now, we need to interpret this result in the context of the original problem, which asks for the coefficient of magnification at z = 1 + i. The coefficient of magnification, as we have established, is precisely the magnification factor, which we found to be 2√2. Considering the options provided:

(a)

${ \sqrt{2} }$

(b)

${ 2\sqrt{2} }$

(c) 2

(d) None of these

It is clear that our calculated magnification factor, 2√2, corresponds directly to option (b). Therefore, the coefficient of magnification for the conformal transformation w = z^2 at z = 1 + i is 2√2. This implies that small shapes around the point z = 1 + i in the z-plane will be magnified by a factor of 2√2 when mapped to the w-plane under this transformation. The correct answer is (b) 2√2.

In conclusion, by systematically applying the principles of complex analysis and conformal mappings, we have successfully determined the magnification coefficient for the given transformation at the specified point. This exercise highlights the importance of understanding derivatives and their magnitudes in the context of conformal transformations.

Conclusion and Key Takeaways

In this detailed exploration, we have successfully determined the coefficient of magnification for the conformal transformation w = z^2 at the point z = 1 + i. Through a step-by-step approach, we first established the foundational concepts of conformal transformations and magnification factors. We then calculated the derivative of the transformation function, evaluated it at the given point, and found the magnitude of the derivative, which represents the magnification factor. Our calculations led us to the result of 2√2, which corresponds to option (b) in the provided choices.

This exercise underscores several key takeaways in the realm of complex analysis and conformal mappings:

1. Conformal transformations are essential tools for mapping complex planes while preserving angles locally, making them invaluable in various scientific and engineering applications.
2. The magnification factor, given by the magnitude of the derivative of the transformation function, quantifies the local scaling of shapes under the transformation.
3. Calculating the derivative of the transformation function is a crucial step in determining the magnification factor at a specific point.
4. The magnitude of a complex number, representing the magnification factor, provides a direct measure of how much the transformation stretches or shrinks shapes in the vicinity of that point.
5. Understanding these concepts allows us to analyze and predict the behavior of conformal transformations, which is crucial for solving problems in fields like fluid dynamics, electromagnetism, and cartography.

By mastering these principles, one can gain a deeper appreciation for the power and elegance of complex analysis in solving real-world problems. The example we explored serves as a valuable illustration of how these concepts can be applied to determine specific properties of conformal transformations, such as the coefficient of magnification. Understanding these concepts is crucial for anyone delving into advanced mathematics, physics, or engineering, where conformal mappings play a significant role. This knowledge provides not just the ability to solve specific problems but also a broader perspective on how complex systems behave under transformations, paving the way for innovative solutions and a deeper understanding of the world around us.