MHB Complex Valued Functions BV: John B. Conway Prop 1.3 Explained

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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 another aspect of the proof of Proposition 1.3 ...Proposition 1.3 and its proof read as follows:View attachment 7439
View attachment 7440
View attachment 7441
In the above text from Conway we read the following:

" ... ... Hence

$$
\int_a^b \lvert \gamma' (t) \rvert \ dt \le \epsilon + \sum_{ k = 1 }^m \lvert \gamma' ( \tau_k ) \rvert ( t_k - t_{ k - 1 } )$$$$ = \epsilon + \sum_{ k = 1 }^m \left\lvert \int_{ t_{ k-1 }}^{ t_k } \gamma' ( \tau_k ) dt \right\rvert $$ ... ..."
Can someone please explain exactly how/why $$\sum_{ k = 1 }^m \lvert \gamma' ( \tau_k ) \rvert ( t_k - t_{ k - 1 } ) = \sum_{ k = 1 }^m \left\lvert \int_{ t_{ k-1 }}^{ t_k } \gamma' ( \tau_k ) dt \right\rvert$$as is implied by the above quote from Conway ... ...?*** NOTE *** Seems as if Conway is treating $$\gamma' ( \tau_k )$$ as a constant ... but why ...?Help will be much appreciated ... ...

Peter
 
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It's because, for all $k$, $$\lvert \gamma'(\tau_k)\rvert(t_k - t_{k-1}) = \lvert \gamma'(\tau_k)(t_k-t_{k-1})\rvert = \left\lvert \int_{t_{k-1}}^{t_k} \gamma'(\tau_k)\, dt\right\rvert$$
 
Euge said:
It's because, for all $k$, $$\lvert \gamma'(\tau_k)\rvert(t_k - t_{k-1}) = \lvert \gamma'(\tau_k)(t_k-t_{k-1})\rvert = \left\lvert \int_{t_{k-1}}^{t_k} \gamma'(\tau_k)\, dt\right\rvert$$
Sorry to be slow Euge ...

... but I do not see why $$\lvert \gamma'(\tau_k)(t_k-t_{k-1})\rvert = \left\lvert \int_{t_{k-1}}^{t_k} \gamma'(\tau_k)\, dt\right\rvert$$Can you help further ...

hmm ... but maybe I guess that although $$\gamma' $$ is not a constant ... $$\gamma'( \tau_k)$$ is a constant, say $$\gamma'( \tau_k) = K$$ and so

$$\int_{t_{k-1}}^{t_k} \gamma'(\tau_k) dt = \int_{t_{k-1}}^{t_k}K dt = K ( {t_{k-1}} - {t_k} )
$$Is that correct?

Peter
 
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Since $\gamma'$ is a complex-valued function, $\gamma'(\tau_k)$ is a complex number, not a complex function as it appeared you were thinking. Your integral calculation is correct.
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
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