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

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In John B. Conway's "Functions of a Complex Variable I," the discussion centers on rectifiable paths in complex integration, specifically regarding the Riemann-Stieljes integral. The query focuses on the necessity of the function $$\phi$$ being non-decreasing for the path $$\gamma \circ \phi$$ to maintain the same trace as $$\gamma$$. It is clarified that Conway states non-decreasing as a sufficient condition rather than a necessity for this property. Understanding this distinction is crucial for grasping the implications of rectifiable paths in complex analysis. The conversation emphasizes the importance of interpreting mathematical conditions accurately in the context of complex integration.
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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 $$\mathbb{C}$$ 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 $$\phi$$ need to be non-decreasing in order for $$\gamma \circle \phi$$ to be a path with the same trace as $$\gamma$$ ... ... ?Help will be much appreciated ... ...

Peter
 
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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.
 

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