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Complex Derivative ... Remark in Apostol, Section 16.1 ... ...

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,918
I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ...

I am focused on Chapter 16: Cauchy's Theorem and the Residue Calculus ...

I need help in order to fully understand a remark of Apostol in Section 16.1 ...

The particular remark reads as follows:




Apostol -  Remark in Section 16.1 ... .png


Could someone please demonstrate (in some detail) how it is the case that the complex function \(\displaystyle f\) has a derivative at \(\displaystyle 0\) but at no other point of \(\displaystyle \mathbb{C}\) ... ...


Help will be much appreciated ...

Peter
 

castor28

Well-known member
MHB Math Scholar
Oct 18, 2017
247
Hi Peter ,

With $z=x+iy$, you have $f(z)=u + iv$, with $u=x^2+y^2$ and $v=0$.

What do the Cauchy-Riemann equations tell you ?
 

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,918
Hi Peter ,

With $z=x+iy$, you have $f(z)=u + iv$, with $u=x^2+y^2$ and $v=0$.

What do the Cauchy-Riemann equations tell you ?


Oh! Indeed ... Cauchy-Riemann equations are only satisfied at (0,0) ... therefore the only possible point where the derivative of f can exist is (0,0) ... and, given that the functions u and v ere continuous and have continuous first-order partial derivatives then f has a derivative at (0,0) ...

Thanks fir the help ... it is much appreciated ...

Peter