Understand Proposition 1.3 in Conway's Functions of Complex Variables I

[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 aspects of Proposition 1.3 ...Proposition 1.3 and its proof read as follows:View attachment 7436
https://www.physicsforums.com/attachments/7437
https://www.physicsforums.com/attachments/7438In the above text from Palka, we read the following:

" ... ... Also, we may choose \(\displaystyle \delta_2 \gt 0\) such that if \(\displaystyle P = \{ a = t_0 \lt t_1 \lt \ ... \ \lt t_m = b \}\) and \(\displaystyle \lvert \lvert P \rvert \rvert = \text{ max } \{ (t_k - t_{ k - 1 } ) : \ 1 \le k \le m \} \lt \delta_2\) then \(\displaystyle \left\lvert \int_a^b \lvert \gamma' (t) \rvert \ dt - \sum_{ k = 1 }^m \lvert \gamma' ( \tau_k ) \rvert ( t_k - t_{ k - 1 } ) \right\rvert \lt \epsilon\)

where \(\displaystyle \tau_k\) is any point in \(\displaystyle [ t_{ k - 1 } , t_k ]\). ... ... "Can someone please explain how/why the above quoted statement is true ... ...?

Is it simply a restatement of the condition for the Riemann integral to exist ... ?
Help will be much appreciated ... ...

Peter

 

[Opalg]

" ... ... Also, we may choose \(\displaystyle \delta_2 \gt 0\) such that if \(\displaystyle P = \{ a = t_0 \lt t_1 \lt \ ... \ \lt t_m = b \}\) and \(\displaystyle \lvert \lvert P \rvert \rvert = \text{ max } \{ (t_k - t_{ k - 1 } ) : \ 1 \le k \le m \} \lt \delta_2\) then

\(\displaystyle \left\lvert \int_a^b \lvert \gamma' (t) \rvert \ dt - \sum_{ k = 1 }^m \lvert \gamma' ( \tau_k ) \rvert ( t_k - t_{ k - 1 } ) \right\rvert \lt \epsilon\)

where \(\displaystyle \tau_k\) is any point in \(\displaystyle [ t_{ k - 1 } , t_k ]\). ... ... "Can someone please explain how/why the above quoted statement is true ... ...?

Is it simply a restatement of the condition for the Riemann integral to exist ... ?

Yes!

 

[math771]

Hi Peter,

I have also read this section and can understand why Proposition 1.3 might be a bit confusing. To fully understand this proposition, we need to first understand the definition of the Riemann-Stieltjes integral.

The Riemann-Stieltjes integral is a generalization of the Riemann integral where the integrand is multiplied by a function of a variable. This function is called the integrator and is denoted by \alpha. In this case, we are dealing with the complex integral, so \alpha is a complex-valued function.

Now, Proposition 1.3 states that if we choose a partition P of the interval [a,b] such that the maximum length of the subintervals is less than a certain value \delta_2, then the difference between the Riemann-Stieltjes integral and the corresponding Riemann sum will be less than a given value \epsilon.

To understand why this statement is true, we need to look at the definition of the Riemann-Stieltjes integral and the Riemann sum. The Riemann-Stieltjes integral is defined as the limit of the Riemann sums as the maximum length of the subintervals approaches 0. So, if we choose a partition P such that the maximum length of the subintervals is less than \delta_2, then we are essentially making the subintervals smaller and smaller, approaching 0. This means that the Riemann sum will approach the Riemann-Stieltjes integral, and the difference between them will be less than \epsilon.

In simpler terms, the statement is saying that if we choose a small enough partition, the Riemann sum will be a good approximation of the Riemann-Stieltjes integral.

I hope this helps clarify Proposition 1.3 for you. Let me know if you have any other questions or need further clarification.