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

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The discussion centers on John B. Conway's "Functions of a Complex Variable I," specifically Chapter IV, Section 1, which addresses rectifiable paths in the complex plane, denoted as $$\mathbb{C}$$. The key point is the clarification that the function $$\phi$$ does not need to be non-decreasing for the composition $$\gamma \circ \phi$$ to maintain the same trace as $$\gamma$$; rather, non-decreasing is a sufficient condition, not a necessity. This distinction is crucial for understanding the properties of rectifiable paths and their integration in complex analysis.

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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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