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

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

" ... ... Hence

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

Peter

 

[Euge]

It's because, for all $k$, $$\lvert \gamma'(\tau_k)\rvert(t_k - t_{k-1}) = \lvert \gamma'(\tau_k)(t_k-t_{k-1})\rvert = \left\lvert \int_{t_{k-1}}^{t_k} \gamma'(\tau_k)\, dt\right\rvert$$

 

[Math Amateur]

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 \(\displaystyle \lvert \gamma'(\tau_k)(t_k-t_{k-1})\rvert = \left\lvert \int_{t_{k-1}}^{t_k} \gamma'(\tau_k)\, dt\right\rvert\)Can you help further ...

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

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

Peter

 

[Euge]

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.