Difference between Riemann-Stieltjes and Riemann Integral

In summary, the main difference between Riemann-Stieltjes integral and Riemann integral is that in Riemann-Stieltjes, the partitions are defined by the integrator function, while in Riemann, the intervals are of equal length. The integrator function g(x) in Riemann-Stieltjes can also be used to calculate the density of the function f(x).
  • #1
dpa
147
0
Hi all,

Homework Statement



Is the difference between riemann stieltjes integral and riemann integral that in riemann integral, the intervals are of equal length and in riemann stieltjes, the partitions are defined by the integrator function?

If not so what exactly is it that integrator function defines?

Homework Equations


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The Attempt at a Solution


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Thank You
 
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  • #2
dpa said:
Hi all,

Homework Statement



Is the difference between riemann stieltjes integral and riemann integral that in riemann integral, the intervals are of equal length and in riemann stieltjes, the partitions are defined by the integrator function?

There is nothing in the definition of the Riemann integral which requires equal intervals. A Riemann sum for [itex]f[/itex] on [itex][a,b][/itex] is
[tex]
\sum_{i=1}^{n} f(\xi_i)(x_i - x_{i-1})
[/tex]
where [itex]x_{i-1} \leq \xi_i \leq x_i[/itex], [itex]x_0 = a[/itex] and [itex]x_n = b[/itex]. The corresponding expression for the Riemann-Stieltjes integral with integrator [itex]g[/itex] is
[tex]
\sum_{i=1}^{n} f(\xi_i)(g(x_i) - g(x_{i-1}))
[/tex]
Thus the Riemann integral is the special case of the Riemann-Stieltjes integral where [itex]g(x) = x[/itex].
 
  • #3
So, what exactly is it when people refer to "density" or similar notions when they discuss about Riemann Stieltjes integral. Is it how how fast alpha(x_i) grows? in the interval?
 

1. What is the main difference between Riemann-Stieltjes and Riemann Integral?

Riemann-Stieltjes integral is a generalization of Riemann integral that allows for a more flexible integration method by using a different type of integration function (the Stieltjes integral function). This function can be used to integrate functions that are not necessarily continuous, and it can also handle integrands that have points of discontinuity.

2. What is the role of the Stieltjes integral function in the Riemann-Stieltjes integral?

The Stieltjes integral function is used to define the Riemann-Stieltjes integral. It is a function that takes in two variables - the integrand and the Stieltjes function - and returns a value that represents the area under the curve of the integrand with respect to the Stieltjes function.

3. Can you give an example of a function that can be integrated using Riemann-Stieltjes integral but not Riemann integral?

Yes, a function with a point of discontinuity can be integrated using Riemann-Stieltjes integral but not Riemann integral. For example, the function f(x) = 1 at x=1 and 0 for all other values, can be integrated using the Stieltjes integral function with the Stieltjes function being the Heaviside step function. However, this function cannot be integrated using Riemann integral as it is not continuous.

4. What are the advantages of using Riemann-Stieltjes integral over Riemann integral?

Riemann-Stieltjes integral allows for a more flexible integration method as it can handle integrands with points of discontinuity and non-continuous functions. It also allows for integration over a wider range of functions, making it a more powerful tool in mathematical analysis and applications.

5. Are there any limitations or drawbacks of using Riemann-Stieltjes integral?

The main limitation of Riemann-Stieltjes integral is the increased complexity compared to Riemann integral. The Stieltjes function must be carefully chosen to ensure the integral function exists, and integration must be performed using special techniques such as Darboux sums. Additionally, Riemann-Stieltjes integral may not always converge, even for functions that are continuous and have no points of discontinuity.

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