- #1

Math Amateur

Gold Member

MHB

- 3,998

- 48

I am reading Reinhold Remmert's book "Theory of Complex Functions" ...

I am focused on Chapter 0: Complex Numbers and Continuous Functions ... and in particular on Section 1.3: Scalar Product and Absolute Value ... ...

I need help in order to fully understand Remmert's derivation of the Law of Cosines for \(\displaystyle \mathbb{C}\)

The relevant part of Remmert's section on Scalar Product and Absolute Value reads as follows:View attachment 8548In the above text from Remmert we read the following:

" ... ... From the Cauchy-Schwarz Inequality it follows that

\(\displaystyle -1 \le \frac{ \langle w, z \rangle }{ \mid w \mid \mid z \mid } \le 1 \) for all \(\displaystyle w,z \in \mathbb{C}^{ \times } \)According to (non-trivial) results of calculus, for each \(\displaystyle w,z \in \mathbb{C}^{ \times }\) therefore a unique real number \(\displaystyle \phi\) with \(\displaystyle 0 \le \phi \le \pi\), exists satisfying \(\displaystyle \text{ cos } \phi = \frac{ \langle w, z \rangle }{ \mid w \mid \mid z \mid }\) ... ... "

Can someone please demonstrate formally and rigorously exactly how ...

\(\displaystyle -1 \le \frac{ \langle w, z \rangle }{ \mid w \mid \mid z \mid } \le 1 \)

implies

\(\displaystyle \text{ cos } \phi = \frac{ \langle w, z \rangle }{ \mid w \mid \mid z \mid }\) ... ...

Help will be appreciated ...

Peter========================================================================================It may help MHB readers of the above post to have access to the start of Remmert's Section 1.3 as it will help with the context and notation of the post ... so I am providing access to the same ... as follows:View attachment 8549Hope that helps ...

Peter

I am focused on Chapter 0: Complex Numbers and Continuous Functions ... and in particular on Section 1.3: Scalar Product and Absolute Value ... ...

I need help in order to fully understand Remmert's derivation of the Law of Cosines for \(\displaystyle \mathbb{C}\)

The relevant part of Remmert's section on Scalar Product and Absolute Value reads as follows:View attachment 8548In the above text from Remmert we read the following:

" ... ... From the Cauchy-Schwarz Inequality it follows that

\(\displaystyle -1 \le \frac{ \langle w, z \rangle }{ \mid w \mid \mid z \mid } \le 1 \) for all \(\displaystyle w,z \in \mathbb{C}^{ \times } \)According to (non-trivial) results of calculus, for each \(\displaystyle w,z \in \mathbb{C}^{ \times }\) therefore a unique real number \(\displaystyle \phi\) with \(\displaystyle 0 \le \phi \le \pi\), exists satisfying \(\displaystyle \text{ cos } \phi = \frac{ \langle w, z \rangle }{ \mid w \mid \mid z \mid }\) ... ... "

Can someone please demonstrate formally and rigorously exactly how ...

\(\displaystyle -1 \le \frac{ \langle w, z \rangle }{ \mid w \mid \mid z \mid } \le 1 \)

implies

\(\displaystyle \text{ cos } \phi = \frac{ \langle w, z \rangle }{ \mid w \mid \mid z \mid }\) ... ...

Help will be appreciated ...

Peter========================================================================================It may help MHB readers of the above post to have access to the start of Remmert's Section 1.3 as it will help with the context and notation of the post ... so I am providing access to the same ... as follows:View attachment 8549Hope that helps ...

Peter