# Differentiating Complex Square Root Function: Bruce P. Palka, Ex. 1.5, Ch. III

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In summary, Example 1.5, Section 1.2, Chapter III of Bruce P. Palka's book "An Introduction to Complex Function Theory" discusses writing \sqrt{z} in terms of re^{i\theta} for complex numbers z, with r and \theta real numbers. The purpose is to find the two square roots of z, which can be expressed as 2 e^{i\pi/4}= \sqrt{2}+ i\sqrt{2} and 2 e^{3i\pi/4}= \sqrt{2}- i\sqrt{2}. This is illustrated in the example by taking z= 4i and finding r= 4 and \theta= \
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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 need help with an aspect of Example 1.5, Section 1.2, Chapter III ...

Example 1.5, Section 1.2, Chapter III, reads as follows:
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At the start of the above example we read the following:

" ... ... Write $$\displaystyle \theta (z) = \text{Arg } z$$. Then $$\displaystyle \sqrt{z} = \sqrt{ \mid z \mid } e^{ i \theta(z)/2 }$$. ... "My question is as follows:

How exactly is $$\displaystyle \sqrt{z} = \sqrt{ \mid z \mid } e^{ i \theta(z)/2 }$$In particular ... surely it should be $$\displaystyle \sqrt{z} = \mid \sqrt{ z } \mid e^{ i \theta(z)/2 }$$ ... ...

Peter

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In particular ... surely it should be $$\displaystyle \sqrt{z} = \mid \sqrt{ z } \mid e^{ i \theta(z)/2 }$$ ... ...

Peter
It wouldn't make much sense to write $$\displaystyle \sqrt{z}$$ in terms of $$\displaystyle |\sqrt{z}|$$!

For z a complex number, $$\displaystyle |z|$$ is a positive real number and so is $$\displaystyle \sqrt{|z|}$$. The purpose here is to write $$\displaystyle \sqrt{z}$$ in the form $$\displaystyle re^{i\theta}$$ with r and $$\displaystyle \theta$$ real numbers.

For example, taking z= 4i, r= 4 and $$\displaystyle \theta= \pi/2$$ so that |z|= 4 and $$\displaystyle \sqrt{|z|}= 2$$. The two square roots of 4i are $$\displaystyle 2 e^{i\pi/4}= \sqrt{2}+ i\sqrt{2}$$ and
$$\displaystyle 2 e^{i(\pi+ 2pi)/4}= 2e^{3i\pi/4}= \sqrt{2}- i\sqrt{2}$$.

HallsofIvy said:
It wouldn't make much sense to write $$\displaystyle \sqrt{z}$$ in terms of $$\displaystyle |\sqrt{z}|$$!

For z a complex number, $$\displaystyle |z|$$ is a positive real number and so is $$\displaystyle \sqrt{|z|}$$. The purpose here is to write $$\displaystyle \sqrt{z}$$ in the form $$\displaystyle re^{i\theta}$$ with r and $$\displaystyle \theta$$ real numbers.

For example, taking z= 4i, r= 4 and $$\displaystyle \theta= \pi/2$$ so that |z|= 4 and $$\displaystyle \sqrt{|z|}= 2$$. The two square roots of 4i are $$\displaystyle 2 e^{i\pi/4}= \sqrt{2}+ i\sqrt{2}$$ and
$$\displaystyle 2 e^{i(\pi+ 2pi)/4}= 2e^{3i\pi/4}= \sqrt{2}- i\sqrt{2}$$.

THanks so much for the help ...

.. it is much appreciated...

Peter

## 1. What is a complex square root function?

A complex square root function is a mathematical function that takes a complex number as its input and returns a complex number as its output. It is similar to a regular square root function, but it can handle complex numbers, which are numbers that have a real and imaginary component.

## 2. How do you differentiate a complex square root function?

To differentiate a complex square root function, you can use the chain rule. First, rewrite the function as a power of 1/2, then use the power rule to differentiate the function. Finally, multiply the result by the derivative of the inner function.

## 3. Why is it important to differentiate complex square root functions?

Differentiating complex square root functions is important in many areas of science and engineering, such as signal processing, control systems, and electromagnetism. It allows us to find the rate of change of a complex variable, which can help us understand and analyze complex systems.

## 4. Can you provide an example of differentiating a complex square root function?

Sure, let's take the function f(z) = √(z+3). First, rewrite it as f(z) = (z+3)^1/2. Then, using the power rule, the derivative is f'(z) = 1/2(z+3)^-1/2. Finally, multiply by the derivative of the inner function, which is 1. So the final derivative is f'(z) = 1/2(z+3)^-1/2.

## 5. What are some common applications of complex square root functions?

Complex square root functions have many applications in science and engineering. They are used in signal processing to analyze and manipulate complex signals, in control systems to model and control complex systems, and in electromagnetism to analyze and design complex circuits. They are also used in physics and quantum mechanics to describe the behavior of complex systems at the quantum level.

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