# Differentiation of the Complex Square Root Function ... ... Palka, Example 1.5, Chapter III

#### Peter

##### Well-known member
MHB Site Helper
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:  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

#### HallsofIvy

##### Well-known member
MHB Math Helper
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
$$\displaystyle 2 e^{i(\pi+ 2pi)/4}= 2e^{3i\pi/4}= \sqrt{2}- i\sqrt{2}$$.

#### Peter

##### Well-known member
MHB Site Helper
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