Complex and Real Differentiability .... Remmert, Section 2, Ch. 1 .... ....

[Math Amateur]

I am reading Reinhold Remmert's book "Theory of Complex Functions" ...I am focused on Chapter 1: Complex-Differential Calculus ... and in particular on Section 2: Complex and Real Differentiability ... ... ...I need help in order to fully understand the relationship between complex and real differentiability ... ...Remmert's section on complex and real differentiability reads as follows:View attachment 8529
View attachment 8530
View attachment 8531In the above text from Remmert, we read the following ... ... just below 1. Characterization of complex-differentiable functions ... ...

" ... ... If $$f : D \to C$$ is complex-differentiable at $$c$$ then ...$\displaystyle \lim_{ h \to 0 } \frac{ f(c + h ) - f(c) - f\, ' (c) h }{ h} = 0$From this and (1) it follows immediately that complex-differentiable mappings are real differentiable and have $$\mathbb{C}$$-linear differentials ... ...

... ... ... "
Can someone please explain (formally and rigorously) how/why

(i) it follows from the limit immediately above and (1) that complex-differentiable mappings are real differentiable ... ...

(ii) it follows from the limit immediately above and (1) that complex-differentiable mappings have $$\mathbb{C}$$-linear differentials ... ... (***NOTE: I suspect the answer to (i) is that the form of the two limits is essentially the same ... although I'm concerned about the presence of norms in one and not the other ... and also that we can identify $$\mathbb{C}$$ with $$\mathbb{R}^2$$ as a vector space ... is that correct?)Help will be appreciated ...

Peter

=======================================================================================
MHB readers of the above post may benefit from access to Remmert's section defining R-linear and C-linear mappings ... so I am providing access to that text ... as follows:
View attachment 8532
View attachment 8533Hope that helps ...

Peter

 

[vacuum]

Dear Peter,

Thank you for reaching out for help in understanding the relationship between complex and real differentiability in Remmert's book. I understand the importance of fully understanding mathematical concepts in order to apply them effectively in our work.

To answer your first question, it follows from the limit given in the text that if a function is complex-differentiable at a point c, then it must also be real differentiable at that point. This is because the limit is essentially the same as the definition of real differentiability, with the only difference being the use of complex numbers instead of real numbers. This means that if a function is complex-differentiable, it must also have a well-defined tangent at that point and therefore be real differentiable.

For your second question, the limit given in the text also implies that complex-differentiable mappings have $\mathbb{C}$-linear differentials. This is because the limit is essentially the definition of the derivative in the complex plane, and the derivative of a complex function is $\mathbb{C}$-linear. This means that the differential of a complex-differentiable function is also $\mathbb{C}$-linear.

I hope this helps to clarify the relationship between complex and real differentiability for you. It is important to note that while we can identify $\mathbb{C}$ with $\mathbb{R}^2$ as a vector space, it is not necessary to do so in order to understand the concept of complex-differentiability.

If you need further assistance, I would recommend consulting Remmert's section on R-linear and C-linear mappings, as you suggested. I wish you the best of luck in your studies.