Recifiable Paths in C .... Conway, Section 1, Ch. 4 ....

[Math Amateur]

I am reading John B. Conway's book, "Functions of a Complex Variable I" (Second Edition) ...

I am currently focussed on Chapter IV: Complex Integration ... Section 1: Riemann-Stieljes Integral ... ...

I need help in fully understanding some notes by Conway on rectifiable paths in \(\displaystyle \mathbb{C}\) on page 63 ... ...

The notes on rectifiable paths on page 63 read as follows:https://www.physicsforums.com/attachments/7446
My question regarding the above text from Conway is as follows:Why exactly does \(\displaystyle \phi\) need to be non-decreasing in order for \(\displaystyle \gamma \circle \phi\) to be a path with the same trace as \(\displaystyle \gamma\) ... ... ?Help will be much appreciated ... ...

Peter

 

[Euge]

He's not saying that $\varphi$ must be nondecreasing, but that if $\varphi$ is nondecreasing, then $\gamma\circ \varphi$ has the same trace as $\gamma$. It's a sufficient condition he's giving, not a necessary condition.

 

[math771]

Hi Peter,

I am also currently reading Conway's book and have come across the same notes on rectifiable paths. From my understanding, the reason why \phi needs to be non-decreasing is because the composition \gamma \circle \phi will essentially be tracing the same path as \gamma, just at a different pace.

If \phi is non-decreasing, it means that the values of \phi are increasing along the path \gamma. This ensures that as \gamma moves along its path, the values of \gamma \circle \phi also increase, creating a consistent and continuous path. If \phi was not non-decreasing, it could potentially lead to overlapping or disjointed paths, making it difficult to define a trace for \gamma \circle \phi.

I hope this helps clarify the concept for you. Let me know if you have any other questions or if I can provide further explanation. Happy reading!