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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:

View attachment 9335

View attachment 9336

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

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:

View attachment 9335

View attachment 9336

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