Yet Another Question on the Complex Square Root Function .... Palka, Example 1.5, Chapter III

[Math Amateur]

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

 

[Jhoneine]

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}}