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I am reading Theodore W. Gamelin's book: "Complex Analysis" ...

I am focused on Chapter 1: The Complex Plane and Elementary Functions ...

I am currently reading Chapter 1, Section 4: The Square and Square Root Functions ... and need some help in verifying a remark by Gamelin ... ...

The relevant section from Gamelin is as follows:View attachment 9288

View attachment 9289In the above text by Gamelin we read the following ... ...

" ... ... Every value \(\displaystyle w\) in the slit plane is the image of exactly two \(\displaystyle z\) values. one in the (open) right half-plane [Re \(\displaystyle z \gt 0\)], the other in the left half-plane [Re \(\displaystyle z \lt 0\)]. ... ... "Now, I wanted to demonstrate via an example that a value of \(\displaystyle w\) was given by two values of \(\displaystyle z\) ... so I let \(\displaystyle w = 1 + i\) ... and proceeded as follows ...

\(\displaystyle w = 1 + i\)

so that

\(\displaystyle w = 2^{ \frac{1}{2} } e^{ i \frac{ \pi }{ 4} }\)So then we have ... ...

\(\displaystyle z_1 = f_1(w) = w^{ \frac{1}{2} } = 2^{ \frac{1}{4} } e^{ i \frac{ \pi }{8} }\)

... and ...

\(\displaystyle z_2 = f_2(w) = - f_1(w) = -w^{ \frac{1}{2} } = 2^{ \frac{1}{4} } e^{ - i \frac{ \pi }{8} } \)(Note that Gamelin uses \(\displaystyle f_2(w)\) for the second branch of \(\displaystyle w^{ \frac{1}{2} }\) ... and, further, notes that \(\displaystyle f_2(w) = - f_1(w)\) ... ... ... ... )My problem is that I do not believe my value or \(\displaystyle z_2\) is correct ... but I cannot see where my process for calculating \(\displaystyle z_2\) is wrong ...

Can someone please explain my mistake and show and explain the correct process for calculating \(\displaystyle z_2\) ... ...

Help will be appreciated ...

Peter

I am focused on Chapter 1: The Complex Plane and Elementary Functions ...

I am currently reading Chapter 1, Section 4: The Square and Square Root Functions ... and need some help in verifying a remark by Gamelin ... ...

The relevant section from Gamelin is as follows:View attachment 9288

View attachment 9289In the above text by Gamelin we read the following ... ...

" ... ... Every value \(\displaystyle w\) in the slit plane is the image of exactly two \(\displaystyle z\) values. one in the (open) right half-plane [Re \(\displaystyle z \gt 0\)], the other in the left half-plane [Re \(\displaystyle z \lt 0\)]. ... ... "Now, I wanted to demonstrate via an example that a value of \(\displaystyle w\) was given by two values of \(\displaystyle z\) ... so I let \(\displaystyle w = 1 + i\) ... and proceeded as follows ...

\(\displaystyle w = 1 + i\)

so that

\(\displaystyle w = 2^{ \frac{1}{2} } e^{ i \frac{ \pi }{ 4} }\)So then we have ... ...

\(\displaystyle z_1 = f_1(w) = w^{ \frac{1}{2} } = 2^{ \frac{1}{4} } e^{ i \frac{ \pi }{8} }\)

... and ...

\(\displaystyle z_2 = f_2(w) = - f_1(w) = -w^{ \frac{1}{2} } = 2^{ \frac{1}{4} } e^{ - i \frac{ \pi }{8} } \)(Note that Gamelin uses \(\displaystyle f_2(w)\) for the second branch of \(\displaystyle w^{ \frac{1}{2} }\) ... and, further, notes that \(\displaystyle f_2(w) = - f_1(w)\) ... ... ... ... )My problem is that I do not believe my value or \(\displaystyle z_2\) is correct ... but I cannot see where my process for calculating \(\displaystyle z_2\) is wrong ...

Can someone please explain my mistake and show and explain the correct process for calculating \(\displaystyle z_2\) ... ...

Help will be appreciated ...

Peter

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