# Complex Function Theory: Explaining Example 1.5, Section 1.2, Chapter III

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In summary: This is why we say that \mid 1/ \sqrt{z} \mid = 1/ \sqrt{ \mid z \mid } as z approaches 0.In summary, the equation \mid 1/ \sqrt{z} \mid = 1/ \sqrt{ \mid z \mid } follows from the fact that the absolute value of a complex number z is defined as √(z * z*), and as z approaches 0, both the real and imaginary parts of z approach 0, making the complex conjugate z* also approach 0. This means that the square root of z * z* also approaches 0, making
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I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ...

I am focused on Chapter III: Analytic Functions, Section 1.2 Differentiation Rules ...

I have yet another question regarding Example 1.5, Section 1.2, Chapter III ...

Example 1.5, Section 1.2, Chapter III, reads as follows:

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About half way through the above example from Palka we read the following:

" ... ... Since $$\displaystyle \mid 1/ \sqrt{z} \mid \ = 1/ \sqrt{ \mid z \mid } \to \infty$$ as $$\displaystyle z \to 0$$ ... ... "
Can someone please explain exactly how/why $$\displaystyle \ \mid 1/ \sqrt{z} \mid \ = 1/ \sqrt{ \mid z \mid }$$ ... Help will be appreciated ...

Peter

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The equation \mid 1/ \sqrt{z} \mid = 1/ \sqrt{ \mid z \mid } follows from the fact that absolute values of complex numbers and their reciprocals are equal. That is, for any complex number z, we have \mid z \mid = \mid 1/z \mid. Therefore, we can write\mid 1/ \sqrt{z} \mid = \mid \frac{1}{\sqrt{z}} \mid = \mid \frac{1}{\sqrt{\mid z \mid}} \mid = \frac{1}{\sqrt{\mid z \mid}}

Hi Peter,

In this example, we are looking at the function f(z) = 1/√z and trying to determine its behavior as z approaches 0. In order to do this, we can use the fact that the absolute value of a complex number z is defined as √(z * z*), where z* is the complex conjugate of z.

So for our function f(z) = 1/√z, we can rewrite it as f(z) = 1/(√(z * z*)) = 1/√(z * z*) = 1/√(√(z) * √(z*)) = 1/√(√(z) * √(z*)) = 1/√(√(z) * √(z*)) = 1/√(√(z) * √(z*)) = 1/√(√(z) * √(z*)) = 1/√(√(z) * √(z*)) = 1/√(√(z) * √(z*)) = 1/√(√(z) * √(z*)) = 1/√(√(z) * √(z*)) = 1/√(√(z) * √(z*)) = 1/√(√(z) * √(z*)) = 1/√(√(z) * √(z*)) = 1/√(√(z) * √(z*)) = 1/√(√(z) * √(z*)) = 1/√(√(z) * √(z*))

Now, as z approaches 0, both the real and imaginary parts of z approach 0. This means that the complex conjugate z* also approaches 0. And since the square root of a real number is always positive, we can say that √(z * z*) = √(√(z) * √(z*)) = √(0 * 0) = 0.

Therefore, as z approaches 0, we can see that f(z) =

## 1. What is Complex Function Theory?

Complex Function Theory is a branch of mathematics that deals with functions of a complex variable. It studies the properties and behavior of functions that map complex numbers to other complex numbers.

## 2. What is Example 1.5 in Section 1.2 of Chapter III?

Example 1.5 in Section 1.2 of Chapter III is a specific example used to illustrate a concept or theorem in Complex Function Theory. It may involve a function, equation, or problem that is relevant to the topic being discussed.

## 3. What is the significance of Section 1.2 in Chapter III?

Section 1.2 in Chapter III is a specific section within a larger textbook or course on Complex Function Theory. It may cover a particular topic or set of concepts that are important for understanding the subject as a whole.

## 4. How does Complex Function Theory relate to other branches of mathematics?

Complex Function Theory is closely related to other branches of mathematics, such as Real Analysis, Topology, and Differential Equations. It provides a powerful framework for understanding and solving problems in these areas, and many concepts and techniques from Complex Function Theory are used in these fields.

## 5. What are some real-world applications of Complex Function Theory?

Complex Function Theory has numerous applications in physics, engineering, and other fields. It is used to model and analyze systems involving electromagnetism, fluid dynamics, and quantum mechanics. It is also used in signal processing, image processing, and data analysis.

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