Smooth Paths in Complex Analysis .... Palka Example 1.3, Section 1.2 in Chapter 4 .... ....

[Math Amateur]

I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ...

I am focused on Chapter 4: Complex Integration, Section 1.2 Smooth and Piecewise Smooth Paths ...

I need help with some aspects of Example 1.3, Section 1.2, Chapter 4 ...

Example 1.3, Section 1.2, Chapter 4 reads as follows:View attachment 7419In the above text from Palka, we read the following:

"... ... Since \(\displaystyle \dot{ \gamma } ( 0 )\) fails to exist, \(\displaystyle \gamma\) is not smooth ... ... "I wish to rigorously demonstrate that \(\displaystyle \dot{ \gamma } ( 0 )\) fails to exist ... but do not know how to proceed ...

Can someone please show me how to rigorously prove that \(\displaystyle \dot{ \gamma } ( 0 )\) fails to existHelp will be much appreciated ...

Peter

 

[Opalg]

Form the difference quotient $\dfrac{\gamma(t) - \gamma(0)}t$. If $t<0$, this is $1-i$. If $t>0$, it is $1+i$. The derivative $\dot{\gamma}(0)$, if it exists, is the limit as $t\to0$ of the difference quotient. If the limit exists, it would have to be both $1-i$ and $1+i$. So the conclusion is that it does not exist.

 

[Math Amateur]

Form the difference quotient $\dfrac{\gamma(t) - \gamma(0)}t$. If $t<0$, this is $1-i$. If $t>0$, this is $1+i$. The derivative $\dot{\gamma}(0)$, if it exists, is the limit as $t\to0$ of the difference quotient. If the limit exists, it would have to be both $1-i$ and $1+i$. So the conclusion is that it does not exist.

Thanks Opalg ...

... straightforward when you see how ...

Peter